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BEGIN:VEVENT
SUMMARY:Shalom Eliahou (Universite du Littoral Cote d'Opale)
DTSTART:20200710T150000Z
DTEND:20200710T160000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/1
DESCRIPTION:Title: Iterated sumsets and Hilbert functions\nby Shal
om Eliahou (Universite du Littoral Cote d'Opale) as part of New York Numbe
r Theory Seminar\n\n\nAbstract\nLet $A\,B \\subset \\Z$. Denote $A+B=\\{a+
b \\mid a \\in A\, b \\in B\\}$\, the \\emph{sumset} of $A\,B$. For $A=B$\
, denote $2A=A+A$. More generally\, for $h \\ge 2$\, denote $hA=A+(h-1)A$\
, the $h$-fold \\emph{iterated sumset} of $A$. If $A$ is finite\, how does
the sequence $|hA|$ behave as $h$ grows? This is a typical problem in add
itive combinatorics. In this talk\, we focus on the following specific que
stion: if $|hA|$ is known\, what can one say about $|(h-1)A|$ and $|(h+1)A
|$? It is known that $$|(h-1)A| \\ge |hA|^{(h-1)/h}\,$$ a consequence of P
l\\"unnecke's inequality derived from graph theory. Here we propose a new
approach\, by modeling the sequence $|hA|$ with the Hilbert function of a
suitable standard graded algebra $R(A)$. We then apply Macaulay's 1927 the
orem on the growth of Hilbert functions. This allows us to recover and str
engthen Pl\\"unnecke's estimate on $|(h-1)A|$. This is joint work with Es
hita Mazumdar.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20200903T190000Z
DTEND:20200903T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/2
DESCRIPTION:Title: Sums of finite sets of integers\, II\nby Mel Na
thanson (CUNY) as part of New York Number Theory Seminar\n\nAbstract: TBA\
n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20200910T190000Z
DTEND:20200910T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/3
DESCRIPTION:Title: Chromatic sumsets\nby Mel Nathanson (CUNY) as p
art of New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20200917T190000Z
DTEND:20200917T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/4
DESCRIPTION:Title: A curious convergent series of integers with missin
g digits\nby Mel Nathanson (CUNY) as part of New York Number Theory Se
minar\n\n\nAbstract\nBy a classical theorem of Kempner\, the sum of the re
ciprocals of integers with missing digits converges. This result is exten
ded to a much larger family of ``missing digits'' sets of positive integer
s with convergent harmonic series. Related sets with divergent harmonic s
eries are also constructed.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Helfgott (Gottigen)
DTSTART:20200924T190000Z
DTEND:20200924T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/5
DESCRIPTION:Title: Expansion in a prime divisibility graph\nby Har
ald Helfgott (Gottigen) as part of New York Number Theory Seminar\n\n\nAbs
tract\n(Joint with M. Radziwill.)\nLet $\\mathbf{N} = \\mathbb{Z} \\cap (N
\, 2N]$ and $\\mathbf{P} \\subset [1\,H]$ a set of primes\n with $H \\l
eq \\exp(\\sqrt{\\log N}/2)$. Given any subset $\\mathcal{X} \\subset \\ma
thbf{N}$\,\ndefine the linear operator\n $$\n (A_{|\\mathcal{X}} f)(n) =
\\sum_{\\substack{p \\in \\mathbf{P} : p | n \\\\ n\, n \\pm p \\in \\math
cal{X}}} f(n \\pm p) - \\sum_{\\substack{p \\in \\mathbf{P} \\\\ n\, n \\p
m p \\in \\mathcal{X}}} \\frac{f(n \\pm p)}{p}\n $$\non functions $f:\\ma
thbf{N}\\to \\mathbb{C}$. Let $\\mathcal{L} = \\sum_{p \\in \\mathbf{P}} \
\frac{1}{p}$.\n\nWe prove that\, for any $C > 0$\, there exists a subset $
\\mathcal{X} \\subset \\mathbf{N}$ of density $1 - O(e^{-C \\mathcal{L}})$
in $\\mathbf{N}$ such that\n$A_{|\\mathcal{X}}$ has a strong expander pro
perty:\nevery eigenvalue of $A_{|\\mathcal{X}}$ is $O(\\sqrt{\\mathcal{L}}
)$.\nIt follows immediately that\, for any bounded\n $f\,g:\\mathbf{N}\\
to \\mathbb{C}$\,\n \\begin{equation}\\label{eq:bamidyar}\n \\frac{1}{
N \\mathcal{L}} \\Big|\n \\sum_{\\substack{n \\in \\mathbf{N} \\\\ p \\in
\\mathbf{P} : p | n}} f(n) \\overline{g(n\\pm p)} -\n \\sum_{\\substack{
n \\in \\mathbf{N} \\\\ p \\in \\mathbf{P}}} \\frac{f(n)\\overline{g(n\\pm
p)}}{p} \\Big| =\n O\\Big(\\frac{1}{\\sqrt{\\mathcal{L}}}\\Big).\n \\en
d{equation}\n This bound is sharp up to constant factors.\n\n Specializi
ng the above bound to $f(n) = g(n) = \\lambda(n)$ with $\\lambda(n)$ the L
iouville function\, and using a result in (Matom\\"aki-Radziwi\\l\\l-Tao\,
2015)\,\n we obtain\n \\begin{equation}\\label{eq:cruciator}\n \\fra
c{1}{\\log x} \\sum_{n\\leq x} \\frac{\\lambda(n) \\lambda(n+1)}{n} =\n
O\\left(\\frac{1}{\\sqrt{\\log \\log x}}\\right)\,\n \\end{equation}\n
improving on a result of Tao's. Tao's result relied on a different\n ap
proach (entropy decrement)\, requiring $H\\leq (\\log N)^{o(1)}$\n and le
ading to weaker bounds.\n\n We also prove the stronger statement\n that
Chowla's conjecture is true at almost all scales\n with an error term as
in (\\ref{eq:cruciator})\,\n improving on a result by Tao and Terav\\"ain
en.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20201001T190000Z
DTEND:20201001T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/6
DESCRIPTION:Title: Convergent and divergent series of integers with mi
ssing digits\nby Mel Nathanson (CUNY) as part of New York Number Theor
y Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Hopkins (Saint Peter's University)
DTSTART:20201008T190000Z
DTEND:20201008T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/8
DESCRIPTION:Title: Rank\, crank\, and mex: New connections between par
tition statistics\nby Brian Hopkins (Saint Peter's University) as part
of New York Number Theory Seminar\n\n\nAbstract\nAbout 100 years ago\, Ra
manujan proved certain patterns in the counts of integer partitions\, but
not in a way that fully ``explained'' them. A young Freeman Dyson wrote
in a somewhat cheeky 1944 article that a new notion he called the rank of
a partition explained some of the patterns of partition counts---without p
roving it---and that something called the crank should explain the rest---
without defining crank! Everything he proposed was eventually proven by o
thers to be correct. The new part of the story is recent work of the spea
ker and James Sellers that explains crank\, whose definition is somewhat t
ricky\, in terms of the minimal excluded part (``mex'') of integer partiti
ons. This allows us to improve and simplify a recent result in the Ramanuj
an Journal.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20201015T190000Z
DTEND:20201015T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/9
DESCRIPTION:Title: Dirichlet series of integers with missing digits\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\n
Abstract\nFor certain sequences $A$ of positive integers with missing $g$-
adic digits\, the Dirichlet series $F_A(s) = \\sum_{a\\in A} a^{-s}$ has a
bscissa of convergence $\\sigma_c < 1$. The number $\\sigma_c$ is compute
d. This generalizes and strengthens a classical theorem of Kempner on th
e convergence of the sum of the reciprocals of a sequence of integers with
missing decimal digits.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART:20201029T190000Z
DTEND:20201029T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/10
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\nby Mat
thias Beck (San Francisco State University) as part of New York Number The
ory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART:20201029T190000Z
DTEND:20201029T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/11
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\nby Mat
thias Beck (San Francisco State University) as part of New York Number The
ory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART:20201029T190000Z
DTEND:20201029T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/12
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\nby Mat
thias Beck (San Francisco State University) as part of New York Number The
ory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART:20201029T190000Z
DTEND:20201029T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/13
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\nby Mat
thias Beck (San Francisco State University) as part of New York Number The
ory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emma Bailey (CUNY Graduate Center)
DTSTART:20201112T200000Z
DTEND:20201112T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/14
DESCRIPTION:Title: L-functions and random matrix theory\nby Emma
Bailey (CUNY Graduate Center) as part of New York Number Theory Seminar\n\
n\nAbstract\nI will review the (conjectured but well evidenced) connection
between families of $L$-functions and characteristic polynomials of rando
m matrices. The canonical example connects the Riemann zeta function with
unitary matrices. I will then explain some recent results pertaining to va
rious moments of interest (both of characteristic polynomials and of $L$-f
unctions). Our work has further connections to log-correlated fields and
combinatorics. This is joint work with Jon Keating and Theo Assiotis.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART:20201105T200000Z
DTEND:20201105T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/15
DESCRIPTION:Title: Discrete energy and applications to Erdos type pro
blems\nby Alex Iosevich (University of Rochester) as part of New York
Number Theory Seminar\n\n\nAbstract\nWe are going to survey a simple conve
rsion mechanism that allows one to deduce certain quantitative discrete re
sults from their continuous analogs.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNYMinimal bases in additive number theory)
DTSTART:20201119T200000Z
DTEND:20201119T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/16
DESCRIPTION:Title: Minimal bases in additive number theory\nby Me
l Nathanson (CUNYMinimal bases in additive number theory) as part of New Y
ork Number Theory Seminar\n\n\nAbstract\nThe set $A$ of nonnegative intege
rs is an \\emph{asymptotic basis of order $h$} if every \n sufficiently la
rge integer can be represented as the sum of $h$ elements of $A$. \n An a
symptotic basis of order $h$ is \\emph{minimal} if no proper subset of $A$
\n is an asymptotic basis of order $h$. Minimal asymptotic bases are ext
remal objects \n in additive number theory\, and related to the conjecture
of Erd\\H os and Tur\\' an that \n the representation function of an asym
ptotic basis must be unbounded. \n This talk describes the construction o
f a new class of minimal asymptotic bases.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20201203T200000Z
DTEND:20201203T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/17
DESCRIPTION:Title: Sidon sets and perturbations\nby Mel Nathanson
(CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nA subset
$A$ of an additive abelian group is an $h$-Sidon set if every element in t
he $h$-fold sumset \n$hA$ has a unique representation as the sum of $h$ no
t necessarily distinct elements of $A$. \nLet $\\mathbf{F}$ be a field o
f characteristic 0 with a nontrivial absolute value\, \nand let $A = \\{a_
i :i \\in \\mathbf{N} \\}$ and $B = \\{b_i :i \\in \\mathbf{N} \\}$ be sub
sets of $\\mathbf{F}$.\nLet $\\varepsilon = \\{ \\varepsilon_i:i \\in \\
mathbf{N} \\}$\, where $\\varepsilon_i > 0$ for all $i \\in \\mathbf{N}$.
\nThe set $B$ is an $\\varepsilon$-perturbation of $A$ \nif $|b_i-a_i| <
\\varepsilon_i$ for all $i \\in \\mathbf{N}$.\nIt is proved that\, for eve
ry $\\varepsilon = \\{ \\varepsilon_i:i \\in \\mathbf{N} \\}$ with $\\v
arepsilon_i > 0$\, \nevery set $A = \\{a_i :i \\in \\mathbf{N} \\}$ has
an $\\varepsilon$-perturbation $B$ \nthat is an $h$-Sidon set. This resu
lt extends to sets of vectors \nin $\\mbF^n$.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20201210T200000Z
DTEND:20201210T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/18
DESCRIPTION:Title: Multiplicative representations of integers and Ram
sey's theorem\nby Mel Nathanson (CUNY) as part of New York Number Theo
ry Seminar\n\n\nAbstract\nLet $\\mathcal{B} = (B_1\,\\ldots\, B_h)$ be an
$h$-tuple of sets of positive integers. \nLet $g_{\\mathcal{B}}(n)$ count
the number of multiplicative representations of $n$ \nin the form $n = b_
1\\cdots b_h$\, \nwhere $b_i \\in B_i$ for all $i \\in \\{1\,\\ldots\, h\\
}$. \nIt is proved that $\\liminf_{n\\rightarrow \\infty} g_{\\mathcal{B}
}(n) \\geq 2$ \nimplies $\\limsup_{n\\rightarrow \\infty} g_{\\mathcal{B}}
(n) = \\infty$.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arindam Biswas (Technion\, Israel)
DTSTART:20201217T200000Z
DTEND:20201217T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/19
DESCRIPTION:Title: Direct and inverse problems related to minimal com
plements\nby Arindam Biswas (Technion\, Israel) as part of New York Nu
mber Theory Seminar\n\n\nAbstract\nMinimal complements of subsets of group
s have been popular objects of study in recent times. The notion was intro
duced by Nathanson in 2011. The past few years have seen a flurry of acti
vities focussing on the existence and nonexistence of minimal complement
s. In this talk\, we shall speak about the direct and the inverse problems
elated to minimal complements and discuss some of the recent results add
ressing some of these problems.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Kaplan (University of California\, Irvine)
DTSTART:20210128T200000Z
DTEND:20210128T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/20
DESCRIPTION:Title: Counting subrings of Z^n\nby Nathan Kaplan (Un
iversity of California\, Irvine) as part of New York Number Theory Seminar
\n\n\nAbstract\nHow many subgroups of $\\mathbb{Z}^n$ have index at most $
X$? How many of these subgroups are also subrings? We can give an asympt
otic answer to the first question by computing the ‘subgroup zeta functi
on’ of $\\mathbb{Z}^n$. For the second question\, we only know an asymp
totic answer for small $n$ because the ‘subring zeta function’ of $\\m
athbb{Z}^n$ is much harder to compute. It is not difficult to show that i
t is enough to understand the number of subrings of prime power index. Le
t $f_n(p^e)$ be the number of subrings of $\\mathbb{Z}^n$ with index $p^e$
. When $n$ and $e$ are fixed\, how does $f_n(p^e)$ vary as a function of
p? We will discuss the quotient $\\mathbb{Z}^n/L$ where $L$ is a `random
’ subgroup or subring of $\\mathbb{Z}^n$. We will also see connections
to counting orders in number fields.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART:20200204T200000Z
DTEND:20200204T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/21
DESCRIPTION:Title: The number of Sidon sets and an application to an
extremal problem for random sets of integers\nby Yoshiharu Kohayakawa
(University of Sao Paulo\, Brazil) as part of New York Number Theory Semin
ar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART:20200204T200000Z
DTEND:20200204T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/22
DESCRIPTION:Title: The number of Sidon sets and an application to an
extremal problem for random sets of integers\nby Yoshiharu Kohayakawa
(University of Sao Paulo\, Brazil) as part of New York Number Theory Semin
ar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART:20200204T200000Z
DTEND:20200204T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/23
DESCRIPTION:Title: The number of Sidon sets and an application to an
extremal problem for random sets of integers\nby Yoshiharu Kohayakawa
(University of Sao Paulo\, Brazil) as part of New York Number Theory Semin
ar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART:20210204T200000Z
DTEND:20210204T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/24
DESCRIPTION:Title: The number of Sidon sets and an extremal problem f
or random sets of integers\nby Yoshiharu Kohayakawa (University of Sao
Paulo\, Brazil) as part of New York Number Theory Seminar\n\n\nAbstract\n
A set of integers is a Sidon set if the pairwise sums of its elements are
all distinct. We discuss the number of Sidon sets contained in $[n]=\\{1\,
\\dots\,n\\}$. As an application\, we investigate random sets of integers
$R\\subset[n]$ of a given\ncardinality $m=m(n)$ and study $F(R)$\, the ty
pical maximal cardinality of a Sidon set contained in $R$. The behaviour
of $F(R)$ as $m=m(n)$ varies is somewhat unexpected\, presenting two poin
ts of ``phase transition.'' We shall also briefly discuss the case in whic
h the random set $R$ is\nan infinite random subset of the set of natural n
umbers\, according to\na natural model\; that is\, we shall discuss infini
te Sidon sets\ncontained in certain infinite random sets of integers. Fin
ally\, we shall mention extensions to $B_h$-sets. Joint work with D. Della
monica Jr.\, S. J. Lee\, C. G. Moreira\, V. R\\"odl\, and W. Samotij.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20210211T200000Z
DTEND:20210211T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/25
DESCRIPTION:Title: Sidon sets for linear forms\nby Mel Nathanson
(CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nLet $\\var
phi(x_1\,\\ldots\, x_h) = c_1 x_1 + \\cdots + c_h x_h $ be a linear form
\nwith coefficients in a field $\\mathbf{F}$\, and let $V$ be a vector spa
ce over $\\mathbf{F}$. \nA nonempty subset $A$ of $V$ is a \n$\\varphi$-
Sidon set if\, \nfor all $h$-tuples $(a_1\,\\ldots\, a_h) \\in A^h$ and $
(a'_1\,\\ldots\, a'_h) \\in A^h$\, \nthe relation \n$\\varphi(a_1\,\\ld
ots\, a_h) = \\varphi(a'_1\,\\ldots\, a'_h) \n$ implies $(a_1\,\\ldots\, a
_h) = (a'_1\,\\ldots\, a'_h)$. \nThere exist infinite Sidon sets for the
linear form $\\varphi$ if and only if the set of coefficients of $\\varphi
$ has distinct subset sums. \nIn a normed vector space with $\\varphi$-Si
don sets\, \nevery infinite sequence of vectors is \nasymptotic to a $\\va
rphi$-Sidon set of vectors.\nResults on $p$-adic perturbations of $\\varph
i$-Sidon sets of integers and bounds on the growth \nof $\\varphi$-Sidon s
ets of integers are also obtained.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Miller (Williams College)
DTSTART:20210218T200000Z
DTEND:20210218T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/26
DESCRIPTION:Title: How low can we go? Understanding zeros of L-functi
ons near the central point\nby Steve Miller (Williams College) as part
of New York Number Theory Seminar\n\n\nAbstract\nSpacings between zeros o
f $L$-functions occur throughout modern number theory\, \n such as in Che
byshev's bias and the class number problem. Montgomery and Dyson \n disc
overed in the 1970's that random matrix theory models these spacings. \n
The initial models are insensitive to finitely many zeros\, and thus miss
the behavior \n near the central point. This is the most arithmetically i
nteresting place\; for example\, \n the Birch and Swinnerton-Dyer conject
ure states that the rank of the Mordell-Weil group \n equals the order of
vanishing of the associated $L$-function there. To investigate the zeros
\n near the central point\, Katz and Sarnak developed a new statistic\, t
he $n$-level density\; \n one application is to bound the average order o
f vanishing at the central point for a given \n family of $L$-functions b
y an integral of a weight against some test function $\\phi$. After \n re
viewing early results in the subject and describing how these statistics a
re computed\, \n we discuss as time permits recent progress and ongoing w
ork on several questions. \n We describe the Excised Orthogonal Ensembles
and their success in explaining the \n observed repulsion of zeros near
the central point for families of $L$-functions\, \n and efforts to exten
d to other families. We discuss an alternative to the Katz-Sarnak \n expa
nsion for the $n$-level density which facilitate comparisons with random m
atrix theory\,\n and applications to improving the bounds on high vanishi
ng at the central point. \n This work is joint with numerous summer REU st
udents.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Mantilla Soler (Universidad Konrad Lorenz\, Bogota\, Col
ombia)
DTSTART:20210304T200000Z
DTEND:20210304T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/27
DESCRIPTION:Title: Arithmetic equivalence and classification of numbe
r fields via the integral trace\nby Guillermo Mantilla Soler (Univers
idad Konrad Lorenz\, Bogota\, Colombia) as part of New York Number Theory
Seminar\n\n\nAbstract\nTwo number fields are called arithmetically equival
ent if their Dedekind zeta functions coincide. Thanks to the work of R. Pe
rlis\, we know that much of the arithmetic information of a number field i
s encoded in its zeta function. By interpreting the Dedekind zeta function
as the Artin $L$-function attached to a certain Galois representation of
$G_{\\mathbb{Q}}$\, we see how all the information mentioned above can be
recovered in a very natural way. Moreover\, we will show how this approa
ch leads to new results. Going further\, we will see how from zeta functio
ns we can connect with trace forms and we will explore the classification
power of integral trace forms.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thai Hoang Le (University of Mississippi)
DTSTART:20210311T200000Z
DTEND:20210311T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/28
DESCRIPTION:by Thai Hoang Le (University of Mississippi) as part of New Yo
rk Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Bienvenue (Universite Claude Bernard Lyon)
DTSTART:20210318T190000Z
DTEND:20210318T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/29
DESCRIPTION:by Pierre Bienvenue (Universite Claude Bernard Lyon) as part o
f New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Hanson (University of Georgia)
DTSTART:20210325T190000Z
DTEND:20210325T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/30
DESCRIPTION:Title: Sum-product and convexity\nby Brandon Hanson (
University of Georgia) as part of New York Number Theory Seminar\n\n\nAbst
ract\nA recurring theme in number theory is that addition and multiplicati
on do not mix well. \n A combinatorial take on this theme is the Erdos-Sz
emeredi sum-product problem\, \n which says that a finite set of numbers
(in an appropriate field) must have either a large \n sumset or a large pr
oduct set. Depending on the field one is working in\, there are
\n different tools which are useful for attacking this problem. Over
the real numbers\, \n convexity is one such tool. In this talk\, I will d
iscuss the sum-product problem and its\n variants\, and progress that has
been made on it. I will then discuss some elementary \n methods of using
convexity to obtain some new results. This will all be based on recent \
n and ongoing work with P. Bradshaw\, O. Roche-Newton\, and M. Rudnev.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgis Petridis (University of Georgia)
DTSTART:20210225T200000Z
DTEND:20210225T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/31
DESCRIPTION:Title: Almost eventowns\nby Giorgis Petridis (Univers
ity of Georgia) as part of New York Number Theory Seminar\n\n\nAbstract\nL
et $n$ be an even positive integer. An eventown is a collection of subsets
of $\\{1\,\\ldots\,n\\}$ \n with the property that every two not necessar
ily distinct elements have even intersection. \n Berlekamp determined the
largest size of an even town in the 1960s\, answering \n a question of Er
d\\H{o}s. In line with other Erd\\H{o}s questions\, Ahmadi and Mohammadian
\n made a conjecture on the size of the largest size of an almost eventow
n: \n a family of subsets of $\\{1\, …\,n\\}$ with the property that amo
ng any three elements \n there are two with even intersection. In this tal
k we will prove the conjecture and \n mention other related results proved
in joint work with Ali Mohammadi.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Trudgian (UNSW Canberra at the Australian Defense Force Academ
y)
DTSTART:20210401T190000Z
DTEND:20210401T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/32
DESCRIPTION:Title: Verifying the Riemann hypothesis to a new height\nby Tim Trudgian (UNSW Canberra at the Australian Defense Force Academy
) as part of New York Number Theory Seminar\n\n\nAbstract\nSadly\, I won't
have time to prove the Riemann hypothesis in this talk. However\, I do ho
pe to outline recent work in a record partial-verification of RH. This is
joint work with Dave Platt\, in Bristol\, UK.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20210408T190000Z
DTEND:20210408T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/33
DESCRIPTION:Title: Inverse problems for Sidon sets\nby Mel Nathan
son (Lehman College (CUNY)) as part of New York Number Theory Seminar\n\n\
nAbstract\nThe Riemann zeta function is an important function in number th
eory. It captures \n arithmetic properties of the integers. Riemann zeta
values and multiple zeta values\, \n defined by Euler and Zagier\, can be
expressed in terms of iterated path integrals. \n Those iterated integral
s a quite special. They have a very good meaning in terms \n of algebraic
geometry. More precisely\, the underlying algebraic variety is the Deligne
-Mumford comactification of the moduli space of curves of genus zero. I w
ill explain intuitively what that means. \n\n If we adjoin $\\sqrt{2}$ or
$i$ to the integers\, then the corresponding zeta functions are called De
dekind zeta functions. My main interest in this area is related to the De
dekind \n zeta functions. I express them in terms of a higher dimensional
iterated integrals\, \n which I call iterated integrals on membranes. Usin
g this tool\, one can define multiple \n Dedekind zeta values as a number
theoretic analogue of multiple zeta values and \n relate them to algebrai
c geometry and motives.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20210930T190000Z
DTEND:20210930T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/34
DESCRIPTION:Title: Egyptian fractions and the Muirhead-Rado inequalit
y\nby Mel Nathanson (Lehman College (CUNY)) as part of New York Number
Theory Seminar\n\n\nAbstract\nFibonacci proved that a greedy algorithm co
nstructs a representation of a positive rational number as the sum of a fi
nite number of Egyptian fractions. Sylvester used a greedy approximation
algorithm to construct an increasing sequence of positive integers $a_1\,
a_2\, \\ldots$ such that $\\sum_{i=1}^n 1/a_i < 1$ and\, if $b_1\, \\ldot
s\, b_n$ is any increasing sequence of positive integers such that $\\sum
_{i=1}^n 1/a_i \\leq \\sum_{i=1}^n 1/b_i < 1$\, then $a_i = b_i$ for all $
i = 1\,\\ldots\, n$. This result (conjectured by Kellogg and proved\, or
believed to have been proved\, by several mathematicians) extends to Egypt
ian fraction approximations of other positive rational numbers. The proof
uses an application of the Muirhead inequality first observed by Soundara
rajan.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20211007T190000Z
DTEND:20211007T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/35
DESCRIPTION:Title: Problems and results on Egyptian fractions\nby
Mel Nathanson (Lehman College (CUNY)) as part of New York Number Theory S
eminar\n\n\nAbstract\nSome problems related to the theorem that Sylvester'
s sequence (defined recursively by $a_0=1$\, $a_{n+1} = 1 +\\prod_{i=1}^n
a_i $) gives the best underapproximation to 1.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Lebowitz-Lockard (Philadelphia)
DTSTART:20211014T190000Z
DTEND:20211014T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/36
DESCRIPTION:Title: Binary Egyptian fractions\nby Noah Lebowitz-Lo
ckard (Philadelphia) as part of New York Number Theory Seminar\n\n\nAbstra
ct\nDefine a ``unit fraction" as a fraction with numerator $1$. We say tha
t an ``Egyptian fraction representation" of a number is a sum of distinct
unit fractions. In this talk\, we discuss the history of these representat
ions\, starting with their origins on an ancient Egyptian papyrus. In part
icular\, we look at several recent results related to binary Egyptian frac
tions\, which are sums of two unit fractions. Most of these results relate
to how often a given rational number has a binary Egyptian fraction repre
sentation.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART:20211021T190000Z
DTEND:20211021T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/37
DESCRIPTION:Title: Limiting behavior of resistances in triangular gra
phs\nby Russell Jay Hendel (Towson University) as part of New York Num
ber Theory Seminar\n\n\nAbstract\nCertain electric circuit can be perceive
d as undirected graphs whose edges are 1-ohm resistances. \nOhm's law al
lows calculation of equivalent single resistances \nbetween two arbi
trary points on the electric circuit. \nFor graphs embeddable in the plane
\, there are four functions that allow the implementation of Ohm's la
w and \ncalculation of equivalent resistances. \nConsequently\, no k
nowledge of electrical engineering is needed for this talk. \nIt is a ta
lk about interesting properties of graphs \nwhose edges have specific
resistances and \nwhich allow reduction to other graphs. \nInteresting res
ults are possible when the underlying graph \nbelongs to certain famili
es. For example \nthe resistance between two corners \n(degree-two verti
ces) of a graph on $n$\nedges consisting of $n-2$ triangles arranged in
a line is \n$\\frac{n-1}{5}+ \\frac{4}{5} \\frac{F_{n-1}}{L_{n-1}}$\nwith
$F$ and $L$ representing the Fibonacci and Lucas numbers respectively\n
\nThis presentation explores \ntriangular graphs of $n$ rows of equilatera
l triangles. \nThese triangular graphs were mentioned in passing \nin one
paper with a conjecture on the equivalent resistance between \ntwo corners
. In this presentation we present new computation methods\, \nallowing rev
iewing more data. It turns out that the \nlimiting behavior of these $n$-r
ow triangular grids \n(as $n$ goes to infinity) has unexpected simply desc
ribed behavior: \nThe sides of individual triangles are conjectured to \n
asymptotically equal products of basically \nfractional linear transformat
ions and $e^{-1}.$ \nWe also introduce new proof methods based on a simple
\n$verification$ method.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20211028T190000Z
DTEND:20211028T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/38
DESCRIPTION:Title: Some results in elementary number theory\nby M
el Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstrac
t\nVariations on Euler's totient function and associated arithmetic identi
ties.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajmain Yamin (CUNY Graduate Center)
DTSTART:20211104T190000Z
DTEND:20211104T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/39
DESCRIPTION:Title: Complete regular dessins\nby Ajmain Yamin (CUN
Y Graduate Center) as part of New York Number Theory Seminar\n\n\nAbstract
\nA map is an embedding of a graph into a topological surface such that th
e complement of the image is a union of topological disks. A regular map
is one that exhibits the maximal amount of symmetry\, that is\, the automo
rphism group of the map acts transitively on flags. In 1985\, James and Jo
nes classified complete regular maps\, i.e. regular maps where the underly
ing graph is complete. The first goal of my talk is to give a brief overvi
ew of this story and in particular review Biggs' construction of complete
regular maps as Cayley maps associated to finite fields. \n\n Given any ma
p\, one obtains a dessin by taking the bipartification of the underlying g
raph and embedding that into the surface. Dessins associated to complete r
egular maps will be called \\emph{complete regular dessins} in my talk. Af
ter reviewing the basic theory of dessins\, I will introduce the main ques
tion of my talk: can one obtain an explicit model for the Riemann surface
underlying a complete regular dessin as an algebraic curve over $\\mathbb{
\\overline{Q}}$? What about the its Belyi function as a rational map down
to $\\mathbb{P}^1(\\mathbb{C})$? In this talk I will explain how to obtain
such an affine model for the complete regular dessin $K_5$ embedded in th
e torus. In the process\, we will be led to consider airithmetic in the G
aussian integers\, uniformization of elliptic curves\, Galois theory of fu
nction fields and Weierstrass $\\wp$ functions.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laszlo Toth (University of Pecs\, Hungary)
DTSTART:20211111T200000Z
DTEND:20211111T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/40
DESCRIPTION:Title: Menon's identity: proofs\, generalizations and ana
logs\nby Laszlo Toth (University of Pecs\, Hungary) as part of New Yo
rk Number Theory Seminar\n\n\nAbstract\nMenon's identity states that for e
very positive integer $n$ one has \n$\\sum (a-1\,n) = \\varphi(n) \\tau(n)
$\, where $a$ runs through a reduced residue system (mod $n$)\, \n$(a-1\,n
)$ stands for the greatest common divisor of $a-1$ and $n$\,\n$\\varphi(n)
$ is Euler's totient function and $\\tau(n)$ is the number of divisors of
$n$. It is named after Puliyakot Kesava Menon\, \nwho proved it in 1965. M
enon's identity has been the subject of many research papers\, also in the
last years.\n\nIn this talk I will present different methods to prove thi
s identity\, and will point out those that I could not identify in the lit
erature. \nThen I will survey the directions to obtain generalizations and
analogs. I will also present some of my own general identities.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Cohen (MIT)
DTSTART:20211202T200000Z
DTEND:20211202T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/41
DESCRIPTION:Title: An optimal inverse theorem for tensors over large
fields\nby Alex Cohen (MIT) as part of New York Number Theory Seminar\
n\n\nAbstract\nA degree $k$ tensor $T$ over a finite field $\\mathbf{F}_q$
can be viewed as a multilinear function $\\mathbf{F}_q^n \\times \\dots
\\times \\mathbf{F}_q^n \\to \\mathbf{F}_q.$\n The analytic rank of $T$ ta
kes a value between $0$ and $n$\, and is small if the output distribution
is far from uniform---in some sense\, it is a measure of how randomly $T$
behaves. On the other hand\, the partition rank of $T$ is small if $T$ can
be decomposed into a few highly structured pieces. It is not hard to show
that the analytic rank is less than the partition rank---or in other word
s\, if $T$ is highly structured\, then it does not \n behave randomly. In
2008 Green and Tao proved a qualitative inverse theorem stating that the
partition rank is bounded by some (large) function of the analytic rank.
We prove an \n optimal inverse theorem: Analytic rank and partition rank
are equivalent up to linear factors (over large enough fields). This theo
rem allows us to explain any lack of randomness in $T$ by the presence of
structure. Our techniques are very different from the usual methods in t
his area. We rely on algebraic geometry rather than additive combinatoric
s. This is joint work with Guy Moshkovitz.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunping Jiang (Queens College (CUNY))
DTSTART:20211209T200000Z
DTEND:20211209T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/42
DESCRIPTION:Title: Ergodic theory motivated by Sarnak's conjecture in
number theory\nby Yunping Jiang (Queens College (CUNY)) as part of Ne
w York Number Theory Seminar\n\n\nAbstract\nSarnak's conjecture brings tog
ether number theory\, ergodic theory\, and dynamical systems. \n Motivated
by this conjecture\, we started a study in ergodic theory about orders of
oscillating \n sequences and minimally mean attractable (MMA) and minimal
ly mean-L-stable (MMLS) flows. \n The Mobius function in number theory giv
es an example of oscillating sequences of order $d$ \n for all $d>0$. From
the dynamical systems point of view\, we found another class of examples
\n of oscillating sequences of order $d$ for all $d>0$. All equicontinuou
s flows are MMA and MMLA. \n I will talk about two non-trivial examples of
MMA and MMLS flows that are not equicontinuous. \n One is a Denjoy counte
rexample in circle homeomorphisms and the other is an infinitely \n renorm
alizable one-dimensional map. I will show that all oscillation sequences o
f order 1\n are linearly disjoint with (or meanly orthogonal to) MMA and
MMLA flows. Thus\, we confirm \n Sarnak's conjecture for a large class of
zero topological entropy flows. For oscillating sequences \n of order $d>1
$\, I will show that they are linearly disjoint from all affine distal flo
ws on the \n $d$-torus. One of the consequences is that Sarnak's conjectur
e holds for all zero topological \n entropy affine flows on the $d$-torus
and some nonlinear zero topological entropy flows \n on the $d$-torus. I w
ill also review some current developments after our work on this topic \n
about flows with the quasi-discrete spectrum and the Thue-Morse sequence\,
which has zero \n topological entropy and small Gowers norms and thus is
a higher-order oscillating sequence.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guy Moshkovitz (Baruch College (CUNY))
DTSTART:20211216T200000Z
DTEND:20211216T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/43
DESCRIPTION:Title: An optimal inverse theorem for tensors over large
fields II\nby Guy Moshkovitz (Baruch College (CUNY)) as part of New Yo
rk Number Theory Seminar\n\n\nAbstract\nWe will give more details about ou
r recent proof\, joint with Alex Cohen\, showing that the partition rank a
nd the analytic rank of tensors are equal up to a constant\, over finite f
ields of every characteristic and of mildly large size (independent of the
number of variables). Proving the equivalence between these two quantitie
s is a central question in additive combinatorics\, the main question in t
he "bias implies low rank" line of work\, and corresponds to the first non
-trivial case of the Polynomial Gowers Inverse conjecture.\n\nThe talk wil
l be a continuation of Alex Cohen's talk from December 2nd\, though I will
aim for it to be mostly self-contained.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220203T200000Z
DTEND:20220203T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/44
DESCRIPTION:Title: Best underapproximation by Egyptian fractions\
nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAb
stract\nAn increasing sequence $(x_i)_{i=1}^n$ of positive integers is an
$n$-term Egyptian \nunderapproximation of $\\theta \\in (0\,1]$ if $\\sum
_{i=1}^n \\frac{1}{x_i} < \\theta$.\nA greedy algorithm constructs an $n$
-term underapproximation of $\\theta$. For some but not all numbers $\\th
eta$\, the greedy algorithm gives a unique best $n$-term underapproximati
on for all $n \\geq 1$. An infinite set of rational numbers is constructe
d for which the greedy underapproximations are best\, and numbers for whi
ch the greedy algorithm is not best are also studied.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220210T200000Z
DTEND:20220210T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/45
DESCRIPTION:Title: A chapter on the theory of equations: Descartes\,
Budan-Fourier\, and Sturm\nby Mel Nathanson (CUNY) as part of New York
Number Theory Seminar\n\n\nAbstract\nA discussion of the theorems of Desc
artes\, Budan-Fourier\, and Sturm on the number of positive solutions a po
lynomials equation in an interval $(a\,b]$. This is in preparation for a
discussion of Tarski's extension of Sturm's theorem and the Tarski-Seiden
berg decidability result.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josiah Sugarman (CUNY Graduate Center)
DTSTART:20220217T200000Z
DTEND:20220217T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/46
DESCRIPTION:Title: The spectrum of the quaquaversal operator is real<
/a>\nby Josiah Sugarman (CUNY Graduate Center) as part of New York Number
Theory Seminar\n\n\nAbstract\nIn the mid 90s Conway and Radin introduced t
he Quaquaversal Tiling. It is a hierarchical tiling of three dimensional s
pace that exhibits statistical rotational symmetry\, in the sense that the
distribution of tiles chosen uniformly at random from a large sphere has
a nearly uniform distribution of orientations. Any hierarchical tiling has
an associated operator whose spectrum can be analyzed to study the distri
bution of orientations in a large sample. Radin and Conway showed that 1 h
as multiplicity 1 in the spectrum of this operator to show that the operat
or exhibited statistical rotational symmetry. By numerically analyzing the
spectrum of this operator Draco\, Sadun\, and Wieren found eigenvalues ve
ry close to 1 and concluded that the rate with which the distribution appr
oaches uniformity is fairly slow\, mentioning that a galactic scale sample
of a material with this crystal structure at the molecular level would ex
hibit noticeable anisotropy. Bourgain and Gamburd proved\, on the other ha
nd\, that a certain class of operators including this one have a nonzero g
ap between 1 and the second largest eigenvalue\, concluding that the distr
ibution must approach uniformity at an exponential rate.\n\nIn this talk I
will introduce hierarchical tilings\, discuss results similar to those ab
ove\, and prove that the spectrum of this operator is real. Answering a qu
estion of Draco\, Sadun\, and Wieren.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220224T200000Z
DTEND:20220224T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/47
DESCRIPTION:Title: The Budan-Fourier theorem and multiplicity matrice
s of polynomials\nby Mel Nathanson (CUNY) as part of New York Number T
heory Seminar\n\n\nAbstract\nThe Budan-Fourier theorem gives an upper boun
d for the number of zeros \n (with multiplicity) of a polynomial $f(x)$ of
degree $n$ in the interval $(a\,b]$ \n in terms of the number of sign va
riations in the vector of derivatives \n$D_f(\\lambda) = \\left( f(\\lamb
da)\, f'(\\lambda)\, f''(\\lambda)\,\\ldots\, f^{(n)}(\\lambda) \\right)$
\n at $\\lambda=a$ and $\\lambda=b$. \n One proof of the Budan-Fourier th
eorem considers the multiplicity vector \n $M_f(\\lambda) = \\left( \\mu_
0(\\lambda)\, \\mu_1(\\lambda)\, \\ldots\, \\mu_n(\\lambda) \\right)$\, \n
where $\\mu_j(\\lambda)$ is the multiplicity of $\\lambda$ as a root \n of
the $j$th derivative $f^{(j)}(x)$. \n The inverse problem asks: What vect
ors are the multiplicity vectors of polynomials\, \n and\, given a multipl
icity vector\, what are the associated polynomials? \n The simultaneous s
tudy of multiplicities of real numbers $\\lambda_1\,\\ldots\, \\lambda_m$
leads to \n multiplicity matrices and their associated polynomials.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220303T200000Z
DTEND:20220303T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/48
DESCRIPTION:Title: Multiplicity matrices for polynomials\nby Mel
Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\n
Let $f(x)$ be a polynomial of degree $n$ and let $f^{(j)}(x)$ be the $j$th
derivative of $f(x)$.\n Let $\\Lambda = (\\lambda_1\,\\ldots\, \\lambda_m
)$ be a strictly increasing sequence of real numbers. \n For $i \\in \\{1
\,\\ldots\, m\\}$ and $j \\in \\{0\,1\,\\ldots\, n\\}$\, \nlet $ \\mu_{i\
,j}$ be the multiplicity of $\\lambda_i$ as a root \n of the polynomial $f
^{(j)}(x)$. For $i \\in \\{1\,\\ldots\, m\\}$ and $j \\in \\{0\,1\,\\ldot
s\, n\\}$\, \nlet $ \\mu_{i\,j}$ be the \n multiplicity of $\\lambda_i$
as a root of the polynomial $f^{(j)}(x)$. \nThe multiplicity matrix of $f$
\n with respect to $\\lambda_1\,\\ldots\, \\lambda_m$\nis the $m \\times
(n+1)$ matrix \n$\nM_f(\\Lambda) = \n\\begin{matrix} \\mu_{i\,j} \n\\e
nd{matrix}.\n$\n The problem is to describe the matrices are multiplicity
matrices of polynomials.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Princeton University)
DTSTART:20220310T200000Z
DTEND:20220310T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/49
DESCRIPTION:Title: Multiplicity matrices and zeros of polynomials
\nby Noah Kravitz (Princeton University) as part of New York Number Theory
Seminar\n\n\nAbstract\nEarlier this week\, Nathanson introduced the notio
n of the derivative matrix associated with a polynomial and a finite tuple
of points. He established several properties of derivative matrices and
proposed a number of appealing open problems. I will discuss Nathanson's
setup\, the solutions to a few of his problems\, and partial progress on n
atural follow-up questions.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David and Gregory Chudnovsky (NYU Tandon School of Engineering)
DTSTART:20220317T190000Z
DTEND:20220317T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/50
DESCRIPTION:Title: How to break step\nby David and Gregory Chudno
vsky (NYU Tandon School of Engineering) as part of New York Number Theory
Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Itay Londner (Weizmann Institute of Science\, Israel)
DTSTART:20220324T190000Z
DTEND:20220324T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/51
DESCRIPTION:Title: Tiling the integers with translates of one tile: t
he Coven-Meyerowitz tiling conditions\nby Itay Londner (Weizmann Insti
tute of Science\, Israel) as part of New York Number Theory Seminar\n\n\nA
bstract\nIt is well known that if a finite set of integers A tiles the int
egers by translations\, then the translation set must be periodic\, so th
at the tiling is equivalent to a factorization $A+B=Z_M$ of a finite cycl
ic group. Coven and Meyerowitz (1998) proved that when the tiling period $
M$ has at most two distinct prime factors\, each of the sets A and B can b
e replaced by a highly ordered "standard" tiling complement. It is not kno
wn whether this behavior persists for all tilings with no restrictions on
the number of prime factors of $M$. In joint work with Izabella Laba (UBC
)\, we proved that this is true for all sets tiling the integers with peri
od $M=(pqr)^2$. In my talk I will discuss this problem and introduce some
ideas from the proof.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Darij Grinberg (Drexel University)
DTSTART:20220331T190000Z
DTEND:20220331T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/52
DESCRIPTION:Title: From the Vandermonde determinant to generalized fa
ctorials to greedoids and back\nby Darij Grinberg (Drexel University)
as part of New York Number Theory Seminar\n\n\nAbstract\nA classical resul
t in elementary number theory says that the\nproduct of the pairwise \n di
fferences between any given $n + 1$ integers\nis divisible by the product
of the pairwise \n differences between $0\, 1\,\n...\, n$. In the late 90s
\, Manjul Bhargava developed this much further\n into a theory of "general
ized factorials\," in particular giving a\nquasi-algorithm for finding \n
the gcd of the products of the pairwise\ndifferences between any $n + 1$
integers in $S$\, \n where $n$ is a given number\nand $S$ is a given set o
f integers.\nIn this talk\, I will explain \n why this is actually a combi
natorial\nquestion in disguise\, and how to answer it in full \n generalit
y (joint\nwork with Fedor Petrov). The general setting is a finite set $E$
\nequipped \n with weights (every element of $E$ has a weight) and distanc
es\n(any two distinct elements \n of $E$ have a distance)\, where the dist
ances\nsatisfy the ultrametric triangle inequality. \n The question is the
n to\nfind a subset of $E$ of given size that has maximum perimeter \n (i.
e.\, sum\nof weights of elements plus their pairwise distances). It turns
out\nthat all such \n subsets form a "strong greedoid" -- a type of set sy
stem\nparticularly adapted to optimization. \n Even better\, this greedoid
is a\n"Gaussian elimination greedoid" -- which\, roughly speaking\, \n me
ans that\nthe problem reduces to linear algebra.\nIf time allows\, I will
briefly discuss \n another closely related\ngreedoid coming from a rather
similar problem in phylogenetics. \n (This\nis mostly due to Manson\, Moul
ton\, Semple and Steel.)\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgis Petridis (University of Georgia)
DTSTART:20220414T190000Z
DTEND:20220414T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/54
DESCRIPTION:Title: On a question of Yufei Zhao on the interface of co
mbinatorial geometry\nby Giorgis Petridis (University of Georgia) as p
art of New York Number Theory Seminar\n\n\nAbstract\nLet $A$ be a finite s
et of integers and consider the lines determined by pairs of points of $P
= \\{(a\,a^2) : a \\in A\\}$. The sum set of $A$ is the set of slopes of t
hese lines and the product set of $A$ is the set of $y$-intercepts. We kno
w from the celebrated sum-product theorem of Erd\\H{o}s and Szemer\\'edi t
hat at least one of these sets is much larger than $|A|$. Geometrically\,
this observation can be phrased as follows: infinity cannot both be close
to the minimum. Motivated by this observation\, Yufei Zhao asked if this
is a manifestation of a more general phenomenon. The goal of the talk is t
o answer this in the affirmative. Joint work with O. Roche-Newton\, M. Ru
dnev and A. Warren.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220407T190000Z
DTEND:20220407T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/55
DESCRIPTION:Title: Exponential automorphisms and a problem of Myciels
ki\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\
n\n\nAbstract\nAn exponential automorphism of $\\mathbf{C}$ is a function
$\\alpha: \\mathbf{C} \\rightarrow \\mathbf{C}$ such that \n$\\alpha(z +
w) = \\alpha(z) + \\alpha(w)$\nand\n$\\alpha\\left( e^z \\right) = e^{\\al
pha(z)}$\nfor all $z\, w \\in \\C$. \nMycielski asked if $\\alpha(\\log 2)
= \\log 2$ and if $\\alpha(2^{1/k}) = 2^{1/k}$ for $k = 2\, 3\, 4$.\nThis
paper solves these problems.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220428T190000Z
DTEND:20220428T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/56
DESCRIPTION:Title: Multiplicity interpolation and the theorems of Des
cartes and Budan-Fourier\nby Mel Nathanson (CUNY) as part of New York
Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220505T190000Z
DTEND:20220505T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/57
DESCRIPTION:Title: Polynomials and the Budan-Fourier theorem\nby
Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\nAbstract
: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220512T190000Z
DTEND:20220512T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/58
DESCRIPTION:Title: van der Waerden's proof of Sturm's theorem\nby
Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstr
act\nContinuation of series of talks on classical results for counting the
number of real roots of polynomials with real coefficients.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven J. Miller (Williams College)
DTSTART:20220616T190000Z
DTEND:20220616T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/59
DESCRIPTION:Title: Benford's Law: Why the IRS might care about the 3x
+1 problem and zeta(s)\nby Steven J. Miller (Williams College) as part
of New York Number Theory Seminar\n\n\nAbstract\nMany systems exhibit a d
igit bias. For example\, the first digit base 10 of the \n Fibonacci numbe
rs or of $2^n$ equals 1 about 30\\% of the time\; the IRS uses this digit
bias to detect fraudulent corporate tax returns. This phenomenon\, \n know
n as Benford's Law\, was first noticed by observing which pages of log tab
les \n were most worn from age -- it's a good thing there were no calculat
ors 100 years ago! \n We'll discuss the general theory and application\,
talk about some fun examples \n (ranging from the $3x+1$ problem to the Ri
emann zeta function to fragmentation \n problems\, as time permits)\, and
see how the irrationality type of numbers often \n enter into the analysis
(through error terms in equidistribution theorems).\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220623T190000Z
DTEND:20220623T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/60
DESCRIPTION:Title: Arithmetic functions and fixed points of powers of
permutations\nby Mel Nathanson (CUNY) as part of New York Number Theo
ry Seminar\n\n\nAbstract\nLet $\\sigma$ be a permutation of a finite or i
nfinite set $X$\, \nand let $F_X\\left( \\sigma^k\\right)$ count the numbe
r of fixed points of \nthe $k$th power of $\\sigma$.\nThis paper describes
how the sequence $\\left(F_X\\left( \\sigma^k\\right) \\right)_{k=1}^{\\i
nfty}$ \ndetermines the conjugacy class of the permutation $\\sigma$. \n
We also describe the arithmetic functions that are fixed point sequences o
f permutations.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220630T190000Z
DTEND:20220630T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/61
DESCRIPTION:Title: Continuity of the roots of a polynomial\nby Me
l Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract
\nLet $K$ be an algebraically closed field with an absolute value. We giv
e an elementary \n (high school algebra) proof of the classical result tha
t the roots of a polynomial \n with coefficients in $K$ are continuous fu
nctions of the coefficients of the polynomial. \n \n Joint work with Davi
d Ross.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David A. Ross (University of Hawaii)
DTSTART:20220707T190000Z
DTEND:20220707T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/62
DESCRIPTION:Title: Yet another proof that the roots of a polynomial d
epend continuously on the coefficients\nby David A. Ross (University o
f Hawaii) as part of New York Number Theory Seminar\n\n\nAbstract\nThe roo
ts of a complex polynomial depend continuously on the coefficients\; that
is\, an infinitesimal perturbation of the coefficients results in an infin
itesimal perturbation of the roots. I'll give a short\, straightforward
proof of this using infinitesimals.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220714T190000Z
DTEND:20220714T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/63
DESCRIPTION:Title: A nonstandard proof of continuity of affine variet
ies\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar
\n\n\nAbstract\nExtending the classical result \nthat the roots of a polyn
omial with coefficients in $\\mathbf{C}$ are continuous functions \nof the
coefficients of the polynomial\, nonstandard analysis is used to prove th
at \nif $\\mathcal{F} = \\{f_{\\lambda} :\\lambda \\in \\Lambda\\}$ \nis
a set of polynomials in $\\C[t_1\,\\ldots\, t_n]$ and if \n $^*\\mathcal{
G} = \\{g_{\\lambda} :\\lambda \\in \\Lambda\\}$ \n is a set of polynomia
ls in $^*\\mathbf{C}_0[t_1\,\\ldots\, t_n]$ \n such that $g_{\\lambda}$ is
an infinitesimal deformation of $f_{\\lambda}$ \n for all $\\lambda \\in
\\Lambda$\, \n then the nonstandard affine variety $^*V_0(\\mathcal{G})$
\n is an infinitesimal deformation of the affine variety $V(\\mathcal{F})$
.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island (CUNY))
DTSTART:20220721T190000Z
DTEND:20220721T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/64
DESCRIPTION:Title: On the size of finite Sidon sets\nby Kevin O'B
ryant (College of Staten Island (CUNY)) as part of New York Number Theory
Seminar\n\n\nAbstract\nIn 2021\, Balogh-Furedi-Roy proved that any Sidon s
et with $k$ elements has diameter at least $k^2-1.996 k^{3/2}$\, provided
that $k$ is sufficiently large. We give a method logically simpler than
the BFR one\, though trading on the same phenomena\, but substantively mor
e involved computationally. The diameter of a $k$ element Sidon set is at
least $k^2 - 1.99405 k^{3/2}$.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Khan (Johns Hopkins University)
DTSTART:20220728T190000Z
DTEND:20220728T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/65
DESCRIPTION:Title: Optimal chaotic dynamics\, the checkmap and the 2-
sector RSS model\nby Ali Khan (Johns Hopkins University) as part of Ne
w York Number Theory Seminar\n\n\nAbstract\nThis talk reports on joint wor
k with Deng\, Fujio and Rajan on optimal chaotic dynamics in mathematical
economics revolving around the check-map and a model due to Robinson-Srini
vasan-Solow – the RSS model. I hope to emphasize number-theoretic consid
erations\, and touch on earlier work of Nathanson (1976 PAMS)\, and more c
onjecturally with two papers of Lagarias and co-authors (J. London Math. S
oc. 1993\; Ill. J. Math. 1994) on the asymmetric tent map. Geometry as a
n engine of analysis will also be emphasized.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220804T190000Z
DTEND:20220804T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/66
DESCRIPTION:Title: Patterns in the iteration of an arithmetic functio
n\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n
\n\nAbstract\nLet $\\Omega$ be a set of positive integers and let $S:\\Ome
ga \\rightarrow \\Omega$\n be an arithmetic function. Let $V = (v_i)_{i=1
}^n$ be a finite sequence of positive integers. \nAn integer $m \\in \\Om
ega$ has increasing-decreasing pattern $V$ with respect to $S$ if\, \nfor
odd integers $i \\in \\{1\,\\ldots\, n\\}$\, \n\\[\nS^{v_1+ \\cdots + v_
{i-1}}(m) < S^{v_1+ \\cdots + v_{i-1}+1}(m) < \\cdots < S^{v_1+ \\cdots +
v_{i-1}+v_{i}}(m)\n\\]\nand\, for even integers $i \\in \\{2\,\\ldots\, n
\\}$\, \n\\[\nS^{v_1+ \\cdots + v_{i-1}}(m) > S^{v_1+ \\cdots +v_{i-1}+1}
(m) > \\cdots > S^{v_1+ \\cdots +v_{i-1}+v_i}(m).\n\\]\nThe arithmetic fu
nction $S$ is wildly increasing-decreasing if\, \nfor every finite sequen
ce $V$ of positive integers\, there exists an integer $m \\in \\Omega$ \n
such that $m$ has increasing-decreasing pattern $V$ with respect to $S$.
\nThis paper gives a new proof that the Collatz function \nis wildly incre
asing-decreasing.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20220929T190000Z
DTEND:20220929T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/67
DESCRIPTION:Title: Poincare's Positivstellensatz\nby Mel Nathanso
n (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nPoincare
's proof of Poincare's theorem (H. Poincare\, Sur les equations algebriq
ues\, Comptes Rendus 97 (1883)\, 1418--1419).\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (CUNY\, College of Staten Island and The Graduate C
enter)
DTSTART:20221027T190000Z
DTEND:20221027T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/68
DESCRIPTION:Title: Finite Sidon sets\nby Kevin O'Bryant (CUNY\, C
ollege of Staten Island and The Graduate Center) as part of New York Numbe
r Theory Seminar\n\n\nAbstract\nA finite Sidon set is a set $A = \\{ a_1 <
a_2 < ... < a_k \\}$ with all the sums $a_i+a_j$ with $i \\leq j$ differe
nt. We will review the history of Sidon sets before turning \n our attenti
on to recent progress bounding the diameter of $A$ in terms of the size\n
of $A$. This talk is suitable for tourists and newcomers to additive numbe
r theory.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20221117T200000Z
DTEND:20221117T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/69
DESCRIPTION:Title: Von Neumann's decomposition of intervals into coun
tably infinitely many pairwise disjoint and congruent subsets\nby Mel
Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\n
An exposition of von Neumann's paper\, ``Die Zerlegung eines Intervalles
in abzahlbar viele kongruente Teilmengen'' (Fund. Math. 11 (1928)\, 230--2
38)\, of which Freeman Dyson wrote\, ``In another corner of [Johnny von Ne
umann's] garden\, there is a little flower all by itself\, a short paper .
.. [that] solves a problem raised by the Polish mathematician Hugo Steinha
us....''\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART:20230209T200000Z
DTEND:20230209T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/71
DESCRIPTION:Title: Finite point configurations and complexity\nby
Alex Iosevich (University of Rochester) as part of New York Number Theory
Seminar\n\n\nAbstract\nWe are going to discuss connections between the no
tion of the Vapnik-Chervonenkis dimension and some classical Erdos-type p
roblems.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhi-Wei Sun (Nanjing University\, P. R. China)
DTSTART:20230216T200000Z
DTEND:20230216T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/72
DESCRIPTION:Title: New results on power residues modulo primes\nb
y Zhi-Wei Sun (Nanjing University\, P. R. China) as part of New York Numbe
r Theory Seminar\n\n\nAbstract\nIn this talk we introduce some new results
on power residues modulo primes.\n\nLet $p$ be an odd prime\, and let $a$
be an integer not divisible by $p$.\nWhen $m$ is a positive integer with
$p\\equiv 1\\pmod{2m}$ and $2$ is an $m$th power residue modulo $p$\,\nthe
speaker determines the value of the product $\\prod_{k\\in R_m(p)}(1+\\ta
n\\pi\\frac{ak}p)$\, where\n$R_m(p)=\\{03$ be a prime.
\nLet $b\\in\\mathbb Z$ and $\\varepsilon\\in\\{\\pm1\\}$.\nJoint with Q.-
.H. Hou and H. Pan\, we prove that\n$\\left|\\left\\{N_p(a\,b):\\ 1\\{ax^2+b\\}_p$\, and\n$\\{m\\}_p$ with $m
\\in\\mathbb Z$ is the least nonnegative residue of $m$ modulo $p$.\n\nWe
will also mention some open conjectures.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hung Viet Chu (University of Illinois)
DTSTART:20230223T200000Z
DTEND:20230223T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/73
DESCRIPTION:Title: Underapproximation by Egyptian fractions and the w
eak greedy algorithm\nby Hung Viet Chu (University of Illinois) as par
t of New York Number Theory Seminar\n\n\nAbstract\nNathanson recently stud
ied the greedy underapproximation algorithm which\, given $\\theta\\in (0\
,1]$\, produces a sequence of positive integers $(a_n)_{n=1}^\\infty$ such
that $\\sum_{n=1}^\\infty 1/a_n = \\theta$. The algorithm is ``greedy" in
the sense that at each step\, $a_n$ is chosen to be the smallest positive
integer such that \n$$\\frac{1}{a_n} \\ <\\ \\theta-\\sum_{i=1}^{n-1}\\fr
ac{1}{a_i}.$$\n\n\nWe introduce the weak greedy underapproximation algorit
hm (WGUA)\, which follows the ``greedy choice up to a constant." In partic
ular\, for each $\\theta$\, the WGUA produces two sequences of positive in
tegers $(a_n)$ and $(b_n)$ such that \n\na) $\\sum_{n=1}^\\infty 1/b_n = \
\theta$\;\n\nb) $1/a_{n+1} < \\theta - \\sum_{i=1}^{n}1/b_i < 1/(a_{n+1}-1
)$ for all $n\\geqslant 1$\;\n\nc) there exists $t\\geqslant 1$ such that
$b_n/a_n \\leqslant t$ infinitely often.\n\nA sequence of positive integer
s $(b_n)_{n=1}^\\infty$ is called a weak greedy underapproximation of $\\t
heta$ if $\\sum_{n=1}^{\\infty}1/b_n = \\theta$.\nWe investigate when a gi
ven weak greedy underapproximation $(b_n)$ can be produced by the WGUA. Fu
rthermore\, we show that for any increasing $(a_n)$ with $a_1\\geqslant 2$
and $a_n\\rightarrow\\infty$\, there exist $\\theta$ and $(b_n)$ such tha
t a) and b) are satisfied\; whether c) is also satisfied depends on the se
quence $(a_n)$. Finally\, we address the uniqueness of $\\theta$ and $(b_n
)$ and apply our framework to specific sequences.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20230302T200000Z
DTEND:20230302T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/74
DESCRIPTION:Title: Sinkhorn limits for generalized doubly stochastic
matrices and tensors\nby Mel Nathanson (CUNY) as part of New York Num
ber Theory Seminar\n\n\nAbstract\nSinkhorn's theorem asserts that if $A$ i
s a square matrix with positive coordinates\, then there \nexist (essentia
lly unique) positive diagonal matrices $X$ and $Y$ such that $XAY$ is doub
ly stochastic. \nMenon applied Brouwer's fixed point theorem to prove thi
s result. This talk will describe Menon's method and an extension to high
er dimensional tensors.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Glaudo (Institute for Advanced Study)
DTSTART:20230330T190000Z
DTEND:20230330T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/75
DESCRIPTION:Title: Can you determine a set from its subset sums?\
nby Federico Glaudo (Institute for Advanced Study) as part of New York Num
ber Theory Seminar\n\n\nAbstract\nLet $A$ be a multiset with elements in a
n abelian group. \nLet $FS(A)$ be its subset sums \n multiset\, i.e.\, th
e multiset containing the $2^{|A|}$ sums of all subsets of $A$. \n Given
$FS(A)$\, can you determine $A$? \n\n\n If the abelian group is $\\Z$\,
one can see that the two multisets $A=\\{-2\, 1\, 1\\}$ \n and $A'=\\{-1\,
-1\,2\\}$ satisfy $FS(A)=FS(A')$\; notice that one is obtained from the ot
her \n by changing signs to the elements. We will see that this is the onl
y obstruction and so\, \n up to the sign of the elements\, $FS(A)$ determi
nes $A$ in $\\Z$.\n\nIn a general abelian group the situation is much more
involved and we will see that the \n answer depends intimately on the o
rders of the torsion elements of the group.\n \n The core of the
proof relies on a delicate study of the structure of cyclotomic units \n
and on an inversion formula for a novel discrete Radon transform on fin
ite abelian groups.\n \n This is a joint work with Andrea Ciprietti.
\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART:20230309T200000Z
DTEND:20230309T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/76
DESCRIPTION:Title: Recursions\, closed forms\, and characteristic pol
ynomials of the circuit array\nby Russell Jay Hendel (Towson Universit
y) as part of New York Number Theory Seminar\n\n\nAbstract\nOne modern gra
ph metric represents an electrical circuit with a graph whose edges are r
eplaced with resistors and the so-called resistance distance between the n
odes is determined by calculating the electrical resistance in the circuit
. Electrical circuit theory provides functions that allow ``reduction'' of
one circuit to another circuit where the resistance distance between cert
ain vertices is preserved. Recently there has been study of a graph\, repr
esentable in the Cartesian plan as an n-grid\, n rows of upright equilater
al triangles\, all of whose edges are labeled one. It is possible to reduc
e the n-grid to an (n-1)-grid with resistance preserving operations. The c
ollections of successive reductions has many interesting properties. In th
is talk we continue to study a ``slice'' of this collection of grids repre
sented by the Circuit Array\, an infinite array of rational functions. We
show that certain closed forms\, recursions\, and characteristic polynomia
ls (annihilators) emerge. One surprising result is that the annihilators o
f the numerators and denominators of the underlying rational functions exc
lusively have roots which are integral powers of 9.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Rajan
DTSTART:20230420T190000Z
DTEND:20230420T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/77
DESCRIPTION:Title: A diophantine problem on products of three consecu
tive integers\nby Ashvin Rajan as part of New York Number Theory Semin
ar\n\n\nAbstract\nWe prove that (3\,4\,5) is the only triple of consecutiv
e positive integers whose product when doubled also factors as a product o
f three consecutive integers. Joint work with Francois Ramaroson.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sayak Sengupta (SUNY-Binghamton)
DTSTART:20230504T190000Z
DTEND:20230504T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/78
DESCRIPTION:Title: Locally nilpotent polynomials over Z\nby Sayak
Sengupta (SUNY-Binghamton) as part of New York Number Theory Seminar\n\n\
nAbstract\nLet $K$ be a number field and $\\mathcal{O}_K$ be the ring of i
ntegers of $K$. For a polynomial $u(x)$ in $\\mathcal{O}_K[x]$ and $r\\in\
\mathcal{O}_K$\, we can construct a dynamical sequence $u(r)\,u^{(2)}(r)\,
\\ldots$. Let $P(u^{(n)}(r)):=\\{\\mathfrak{p}\\in \\text{MSpec}(\\mathcal
{O}_K)~|~u^{(n)}(r)\\in \\mathfrak{p}\,\\text{for some }n\\in\\mathbb{N} \
\}$. For which polynomials $u(x)$ and $r\\in \\mathcal{O}_K$ do we expect
to have $P(u^{(n)}(r))=\\text{MSpec}(\\mathcal{O}_K)$? If we hit 0 somewhe
re in the above sequence\, then we obviously have the equality. If we do n
ot hit zero for any iteration then the question becomes very interesting.
In this talk\, we will define such polynomials for a general number field
$K$ and then we will look at some results in the particular case of $K=\\m
athbb{Q}.$ This talk is based on a preprint of my ongoing work\, which is
available in arXiv under the same name as the title.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Shallit (University of Waterloo)
DTSTART:20230427T190000Z
DTEND:20230427T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/79
DESCRIPTION:Title: Doing additive number theory with logic and automa
ta\nby Jeffrey Shallit (University of Waterloo) as part of New York Nu
mber Theory Seminar\n\n\nAbstract\nThe classical tools of the additive num
ber theorist include analytic\nand combinatorial methods\, such as the cir
cle method and the sieve method.\nIn this talk I will present another meth
od\, based on logic and automata\ntheory\, that can sometimes be used to p
rove results in additive number\ntheory in relatively simple ways. No exp
erience with finite automata is \nassumed.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20240201T200000Z
DTEND:20240201T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/80
DESCRIPTION:Title: Finitely many implies infinitely many (for polyno
mials in infinitely many variables)\nby Mel Nathanson (CUNY) as part o
f New York Number Theory Seminar\n\n\nAbstract\nMany mathematical statemen
ts have the following form: Let $X$ be an infinite set of equations. If
every finite subset of the equations has a common solution\, then the inf
inite set of equations has a common solution. A result of this type wil
l be described for certain infinite sets of polynomial equations in infin
itely many variables. \n\n This is joint work with David Ross.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ross (University of Hawaii)
DTSTART:20240208T200000Z
DTEND:20240208T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/81
DESCRIPTION:Title: Finitely many implies infinitely many\, part 3: th
e nonstandard version\nby David Ross (University of Hawaii) as part of
New York Number Theory Seminar\n\n\nAbstract\nIn a pair of recent seminar
s\, Mel Nathanson has discussed proofs\, using the Tychonoff Theorem\, for
existence of solutions to infinite sets of equations in infinitely many v
ariables. In at least one case the proof was an adaptation of an argument
using nonstandard analysis. In this talk I'll try to explain such nonstand
ard arguments\, hopefully making them intelligible to mathematicians who h
aven't seen nonstandard methods before.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (Wits and Oxford)
DTSTART:20240215T200000Z
DTEND:20240215T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/82
DESCRIPTION:Title: Positive integers $k$ such that $3^k+1\\equiv 0\\p
mod {3k+1}$\nby Florian Luca (Wits and Oxford) as part of New York Num
ber Theory Seminar\n\n\nAbstract\nIn my talk we will look at positive inte
gers $k$ such that $3^k+1\\equiv 0\\pmod {3k+1}$. We show that there are i
nfinitely many such. They are all odd and composite and they have a counti
ng function that is much smaller than the primes. This is work in progress
.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sayak Sengupta (Binghamton University (SUNY))
DTSTART:20240222T200000Z
DTEND:20240222T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/83
DESCRIPTION:Title: Nilpotent and infinitely nilpotent integer sequenc
es\nby Sayak Sengupta (Binghamton University (SUNY)) as part of New Yo
rk Number Theory Seminar\n\n\nAbstract\nWe say that an integer sequence $\
\{r_n\\}_{n\\ge 0}$ has a generating polynomial $u(x)$ over $\\mathbb{Z}$
if for every positive integer $n$ one has $u^{(n)}(r_0)=r_n$. In addition\
, if such a sequence satisfies the condition that $r_n=0$ for some positiv
e integer $n$ (respectively\, $r_n=0$ for infinitely many positive integer
s $n$)\, then we say that $\\{r_n\\}_{n\\ge 0}$ is a nilpotent sequence (r
espectively\, $\\{r_n\\}_{n\\ge 0}$ is an infinitely nilpotent sequence).
In this talk we will provide (and discuss) some important characteristics
of nilpotent and infinitely nilpotent sequences.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Senia Sheydvasse (Bates College)
DTSTART:20240229T200000Z
DTEND:20240229T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/84
DESCRIPTION:Title: Hidden structures in families of Ulam sequences\nby Senia Sheydvasse (Bates College) as part of New York Number Theory S
eminar\n\n\nAbstract\nStanislaw Ulam defined the original Ulam sequence as
follows: Start with 1\,2\, and then each subsequent term is the next smal
lest integer that is the sum of two distinct prior terms in exactly one wa
y. (The next few terms are 1\,2\,3\,4\,6\,8\,...) There is now a veritable
zoo of "Ulam-like" sequences and sets\, most of which share the main trai
t of the original: there is clear numerical evidence that there is an unde
rlying structure\, but for the most part we can prove almost nothing. (As
a simple example: computation of trillions of terms of the Ulam sequence s
trongly suggests that it grows linearly. The best known bound is that it c
an't grow faster than exponentially fast.) One of the few partial results
that we can prove concerns what has been termed the Rigidity Conjecture. T
he original proofs surrounding this were model-theoretic in nature---what
we shall show is that there is a completely constructive proof using a new
variation of Ulam sequences\, and the hints toward a broader solution tha
t this offers.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Sellers (University of Minnesota - Duluth)
DTSTART:20240307T200000Z
DTEND:20240307T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/85
DESCRIPTION:Title: Surprising connections between integer partitions
statistics: The crank\, minimal excludant\, and partition fixed points
\nby James Sellers (University of Minnesota - Duluth) as part of New York
Number Theory Seminar\n\n\nAbstract\nA {\\it partition} of an integer $n$
is a finite sequence of positive integers $p_1\\geq p_2\\geq \\dots \\geq
p_k$ such that $n=p_1+p_2+\\dots + p_k.$ We let $p(n)$ denote the number
of partitions of $n$. For example\, $p(4) = 5$ because there are five pa
rtitions of the integer $n=4$: \n\n$$4\, \\ \\ 3+1\, \\ \\ 2+2\, \\ \\ 2+
1+1\, \\ \\ 1+1+1+1$$\n\nIn 1919\, just one year before his death\, Ramanu
jan discovered and proved some unexpected\, and truly amazing\, divisibili
ty properties for the function $p(n).$ Since then\, several mathematician
s have studied $p(n)$ from different perspectives\, trying to better under
stand these divisibility properties\, especially from a combinatorial pers
pective. In the process\, numerous ``statistics'' have been defined on pa
rtitions\, including the rank and crank of a partition. In this talk\, I
will discuss this history in more detail\, and then I will transition to s
ome relatively new partition statistics\, including the {\\it missing excl
udant} (or {\\it mex}) of a partition. I will discuss unexpected connecti
ons between this mex statistic and the crank\, and then we will transition
to some very recent work of Blecher and Knopfmacher on partition fixed po
ints which\, unbeknownst to them\, is very closely connected to the crank
and mex statistics. We will close by generalizing this concept of partiti
on fixed points and show how this new family of functions naturally connec
ts with generalized versions of the aforementioned partition statistics.
\n\nThis is joint work with Brian Hopkins\, St. Peter's University.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David and Gregory Chudnovsky (New York University)
DTSTART:20240314T190000Z
DTEND:20240314T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/86
DESCRIPTION:Title: The telephone gossip problem: An hommage to Richar
d Bumby\nby David and Gregory Chudnovsky (New York University) as part
of New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island (CUNY))
DTSTART:20240411T190000Z
DTEND:20240411T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/87
DESCRIPTION:Title: $B_h$-sets\nby Kevin O'Bryant (College of Stat
en Island (CUNY)) as part of New York Number Theory Seminar\n\n\nAbstract\
nFix a positive integer $h$. A $B_h$-set is a set of natural numbers that
does not contain $x_i\,y_i$ with $x_1+\\cdots +x_h=y_1+\\cdots +y_h$\, exc
ept for the trivial solutions where $x_1\,\\dots\,x_h$ is a rearrangement
of $x_1\,\\dots\,x_h$. The primary challenge is to make the $k$-th largest
element of a $B_h$-set as small as possible. This talk will contain the s
tate of the art for this problem\, with special attention to how the probl
em changes as $h$ grows.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20240328T190000Z
DTEND:20240328T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/88
DESCRIPTION:Title: Landau's converse to Holder's inequality\, and oth
er inequalities\nby Mel Nathanson (CUNY) as part of New York Number Th
eory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leo Schaefer (University of Gottingen)
DTSTART:20240418T190000Z
DTEND:20240418T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/89
DESCRIPTION:Title: Telling apart coarsifications of the integers\
nby Leo Schaefer (University of Gottingen) as part of New York Number Theo
ry Seminar\n\n\nAbstract\nWe introduce an invariant for coarse groups that
is able to differentiate some coarsifications of the integers up to isomo
rphism. In particular\, we will see that coarsifications coming from pro-$
Q$ topologies (and therefore also the $p$-adic topologies) are not isomorp
hic.\n Partial results for metrics stemming from Cayley graphs are also
obtained\, but there remain open questions in this regard.\n \n Thi
s is joint work with Federico Vigolo.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Borisov (Binghamton University)
DTSTART:20240502T190000Z
DTEND:20240502T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/90
DESCRIPTION:Title: Locally integer polynomial functions\nby Alexa
nder Borisov (Binghamton University) as part of New York Number Theory Sem
inar\n\n\nAbstract\nA locally integer polynomial function on a subset $X$
of $\\mathbb Z$ is a function $f: X\\to \\mathbb Z$ such that its restric
tion to every finite subset is given by a polynomial in $\\mathbb Z[x]$. I
hope to convince you that these objects are interesting and deserve furth
er study. The talk will be based on my recent preprint https://arxiv.org/
abs/2401.17955 and on further work in progress on a rather mysterious ana
logy between locally integer polynomial functions on infinite $X$ and com
plex analytic functions. Several open questions will be proposed\, highli
ghting how little appears to be known about these seemingly elementary ob
jects.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Quentin Dubroff (Rutgers University)
DTSTART:20240509T190000Z
DTEND:20240509T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/91
DESCRIPTION:Title: The Erdos distinct subset sums problem\nby Que
ntin Dubroff (Rutgers University) as part of New York Number Theory Semina
r\n\n\nAbstract\nA conjecture of Erd\\H{o}s from the 1930s states that an
y set of $n$ positive integers with distinct subset sums contains an eleme
nt larger than $c2^n$ for some fixed constant c. I'll give (at least) thre
e proofs of the weaker result that any such set contains an element larger
than $c2^n/\\sqrt{n}$. Two of these proofs will achieve a bound with the
best-known constant $c = \\sqrt{2/\\pi}$\, which seems to be a significant
sticking point. I'll highlight similarities and differences between the p
roofs\, which use a wide range of tools such as isoperimetric inequalities
\, Minkowski's theorem in the geometry of numbers\, and the Berry-Esseen q
uantitative central limit theorem.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (University of Witwatersrand)
DTSTART:20240516T190000Z
DTEND:20240516T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/92
DESCRIPTION:Title: On transcendence of Sturmian and Arnoux-Rauzy word
s\nby Florian Luca (University of Witwatersrand) as part of New York N
umber Theory Seminar\n\n\nAbstract\nWe consider numbers of the form $\\alp
ha={\\displaystyle{\\sum_{n=0}^{\\infty} \\frac{u_n}{\\beta^n}}}$\, where
$(u_n)$ \nis an infinite word over a finite alphabet and $\\beta$ is a co
mplex number of absolute\nvalue greater than one. We present a combinatori
al criterion on $u$\, called\nechoing\, that implies that $\\alpha$ is tra
nscendental whenever $\\beta$ is algebraic. We\nshow that every Sturmian w
ord is echoing\, as is the Tribonacci word\, a leading\nexample of an Arno
ux-Rauzy word. We give an application of our\ntranscendence results to the
theory of dynamical systems\, showing that for\na contracted rotation on
the unit circle with algebraic slope\, its limit set is\neither finite or
consists exclusively of transcendental elements other than its\nendpoints
$0$ and $1$. This confirms a conjecture of Bugeaud\, Kim\, Laurent\,\nand
Nogueira.\n\nJoint work with P. Kebis\, A. Scoones\, J. Ouaknine and J. Wo
rrell.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20240905T190000Z
DTEND:20240905T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/93
DESCRIPTION:Title: Shnirel'man density and the Dyson transform\nb
y Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbst
ract\nA ``hot topic'' in the 1930s and 1940s was Khinchin's $\\alpha+\\bet
a$ conjecture for the Shnirel'man density of the sum of two sets of intege
rs. This was solved by Henry B. Mann in 1942. The following year Emil A
rtin and Peter Scherk published a refinement of his proof. In 1945\, Fre
eman Dyson introduced the ``Dyson transform'' of an $n$-tuple of sets of p
ositive integers and extended Mann's result to rank $r$ sums of $n$ sets o
f integers. The goal of this talk to simplify Dyson's method and generali
ze his result.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Xu (NYU Courant)
DTSTART:20240912T190000Z
DTEND:20240912T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/94
DESCRIPTION:Title: Two stories about multiplicative energy\nby Ma
x Xu (NYU Courant) as part of New York Number Theory Seminar\n\n\nAbstract
\nThe multiplicative energy $E_{\\times}(A)$ of a given set $A$ is defined
to be the number \n of solutions to the equation \n$a_1a_2 = a_3a_4$\,\nw
here all $a_i$ are in $A$. \n We show two recent applications of studying
multiplicative energy. \n The first application is to study conjectures of
Elekes and Ruzsa on the size \n of product sets of arithmetic progressio
ns. \n The second story is about a recent popular topic\, random multiplic
ative functions\, \n and we show how multiplicative energy is involved. \
n The talk is based on joint work with Yunkun Zhou and K. Soundararajan\,
respectively.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island and CUNY Graduate Center)
DTSTART:20241017T190000Z
DTEND:20241017T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/95
DESCRIPTION:Title: Visualizing the sum-product conjecture\nby Kev
in O'Bryant (College of Staten Island and CUNY Graduate Center) as part of
New York Number Theory Seminar\n\n\nAbstract\nThe Erdos sum-product conje
cture states that\, for every $\\epsilon>0$\, there is $k_0$ such that if
$A$ is any finite set of positive integers with $|A|>k_0$\, \nthen $|(A+A)
\\cup(AA)| > |A|^{2-\\epsilon}$. In other words\, for sufficiently large s
ets either the sumset or the product set will be nearly as large as concei
vable. We survey progress on this conjecture\, and provide a visual repres
entation of progress and counterexamples. There will be a few beautiful pr
oofs (not the speaker's)\, several interesting examples\, and scores of st
riking pictures.{\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman Callege and CUNY Graduate Center)
DTSTART:20240919T190000Z
DTEND:20240919T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/96
DESCRIPTION:Title: Sums of lattice points\, ordered groups\, and the
Hahn embedding theorem\nby Mel Nathanson (Lehman Callege and CUNY Grad
uate Center) as part of New York Number Theory Seminar\n\n\nAbstract\nExte
nsion of Shnirel'man's theorem to sums of sets of nonnegative lattice poin
ts and to other additive problems associated with ordered groups.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman Callege and CUNY Graduate Center)
DTSTART:20240926T190000Z
DTEND:20240926T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/97
DESCRIPTION:Title: Addition theorems in partially ordered groups
\nby Mel Nathanson (Lehman Callege and CUNY Graduate Center) as part of Ne
w York Number Theory Seminar\n\n\nAbstract\nShnirel'man's inequality and S
hnirel'man's basis theorem are fundamental \nresults about sums of sets of
positive integers in additive number theory. \nIt is proved that these
results are inherently order-theoretic \nand extend to partially ordered a
belian and nonabelian groups. \nOne abelian application is an addition th
eorem \nfor sums of sets of $n$-dimensional lattice points.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ross (University of Hawai'i)
DTSTART:20241031T190000Z
DTEND:20241031T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/98
DESCRIPTION:Title: Egyptian fractions on groups\nby David Ross (U
niversity of Hawai'i) as part of New York Number Theory Seminar\n\n\nAbstr
act\nIn 1956 Sierpinski published several results about the structure of t
he set of Egyptian fractions. A few years ago Nathanson extended these res
ults to more general sets of real numbers\, and independently I showed tha
t nonstandard methods make it possible to simplify and extend Sierpinski's
results. In this talk I'll describe a further generalization\, to certai
n subsets of ordered topological groups.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Senia Sheydvasser (Bates College)
DTSTART:20241114T200000Z
DTEND:20241114T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/99
DESCRIPTION:Title: Distribution of Ulam words\nby Senia Sheydvass
er (Bates College) as part of New York Number Theory Seminar\n\n\nAbstract
\nLet 0\,1 denote the generators of the free semigroup on two generators.
We say that a word is 'Ulam' if it is either 0 or 1\, or it can be written
as the concatenation of two smaller (distinct) Ulam words in exactly one
way. This is a nonabelian analog of Ulam sequences\, defined by Bade et al
. in 2020. In this talk\, we will discuss a few new conjectures and result
s about the distribution of Ulam words---we will see that there is a natur
al corresponding integer sequence\, and so it makes sense to ask questions
about density\, equidistribution modulo $N$\, and so on.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20241024T190000Z
DTEND:20241024T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/100
DESCRIPTION:Title: Orderable groups and inverse theorems\nby Mel
Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\
nThis talk will review some basic facts about orderable groups and the ext
ension \nof Freiman's $3k+4$ theorem from the integers to orderable groups
.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan M. Keith (Monash University\, Australia)
DTSTART:20241107T200000Z
DTEND:20241107T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/101
DESCRIPTION:Title: Asymptotic density and related set functions\
nby Jonathan M. Keith (Monash University\, Australia) as part of New York
Number Theory Seminar\n\n\nAbstract\nAsymptotic density is a convenient me
asure of the size of a set of natural numbers\, which has uses in many mat
hematical contexts including statistics\, ergodic processes\, complex anal
ysis and number theory. In this talk\, I'll briefly review the history and
uses of asymptotic density and related set functions\, then discuss how s
uch functions can be used to induce pseudometrics on the power set of the
natural numbers $\\mathcal{P}(\\mathbb{N})$\, and why this is mathematical
ly useful. I'll present some new theorems about the completeness of such p
seudometrics\, and characterisations of closed sets in these topologies.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nick Fischer (INSAIT\, Sofia University\, Bulgaria)
DTSTART:20241121T200000Z
DTEND:20241121T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/102
DESCRIPTION:Title: Recognizing sumsets is NP-complete\nby Nick F
ischer (INSAIT\, Sofia University\, Bulgaria) as part of New York Number T
heory Seminar\n\n\nAbstract\nSumsets are central objects in additive combi
natorics. In 2007\, Granville asked whether one can efficiently recognize
whether a given set $S$ is a sumset\, i.e. whether there is a set $A$ such
that $A+A=S$. Granville suggested an algorithm that takes exponential tim
e in the size of the given set\, but can we do polynomial or even linear t
ime? This basic computational question is indirectly asking a fundamental
structural question: do the special characteristics of sumsets allow them
to be efficiently recognizable? In this paper\, we answer this question ne
gatively by proving that the problem is NP-complete. Specifically\, our re
sults hold for integer sets and over any finite field.\n\nJoint work with
Amir Abboud\, Ron Safier\, and Nathan Wallheimer.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grazyna Horbaczewska and Sebastian Lindner (University of Lodz\, P
oland)
DTSTART:20241212T200000Z
DTEND:20241212T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/103
DESCRIPTION:Title: On permutations preserving density\nby Grazyn
a Horbaczewska and Sebastian Lindner (University of Lodz\, Poland) as part
of New York Number Theory Seminar\n\n\nAbstract\nIn the 71st problem incl
uded in the Scottish Book\, Stanis{\\l}aw Ulam asks about the characterist
ics of permutations that preserve the density of subsets of natural number
s. An overview of partial answers to this problem will be presented.\n\n\n
M. Blumlinger\, N. Obata\, \\emph{Permutations preserving Ces\\`aro mean\
, densities of natural numbers and uniform distribution of sequences}\, An
nales de l'institut Fourier 41 (3) (1991)\, 665-678.\n\nJ. Coquet\, \\emph
{Permutations des entiers et reparition des suites}\, in:Analytic and Elem
entary Number Theory\, Marseille\, 1983\, Publ. Math. Orsay 86 (1986)\, 25
-39. \n \n M.B. Nathanson\, R. Parikh\, \\emph{Density of sets of natural
numbers and the Levy group}\, J. Number Theory 124 (2007)\, 151-158.\n \n
\n N. Obata\, \\emph{A note on certain permutation groups in the infinite-
dimensional rotation group}\, Nagoya Math. J. 109 (1988)\, 91-107\n \nM. S
leziak\, M. Ziman\, \\emph{Levy group and density measures} Journal of Num
ber Theory 128 (2008)\, 3005–3012\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20241219T193000Z
DTEND:20241219T210000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/104
DESCRIPTION:Title: Richard Bumby memorial session\nby Mel Nathan
son (CUNY) as part of New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Rowland (Hofstra University)
DTSTART:20250123T200000Z
DTEND:20250123T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/105
DESCRIPTION:Title: The exact values of the entries of a Sinkhorn lim
it\nby Eric Rowland (Hofstra University) as part of New York Number Th
eory Seminar\n\n\nAbstract\nThe Sinkhorn limit of a positive square matrix
is obtained by scaling the rows so each row sum is 1\, then scaling the c
olumns so each column sum is 1\, then scaling the rows again\, then the co
lumns again\, and so on. It has been used for almost 90 years in applicati
ons ranging from predicting telephone traffic to machine learning. But unt
il recently\, nothing was known about the exact values of its entries. In
2020\, Nathanson determined the Sinkhorn limit of a $2 \\times 2$ matrix.
Shortly after that\, Ekhad and Zeilberger determined the Sinkhorn limit of
a symmetric $3 \\times 3$ matrix. Recently\, Jason Wu and I determined th
e Sinkhorn limit of a general $3 \\times 3$ matrix. The result suggests th
e form for $n \\times n$ matrices\; in particular\, the entries seem to be
algebraic numbers with (generic) degree $\\binom{2 n - 2}{n - 1}$. This d
egree has a combinatorial interpretation as the number of minors of an $(n
- 1) \\times (n - 1)$ matrix.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Princeton University)
DTSTART:20250130T200000Z
DTEND:20250130T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/106
DESCRIPTION:Title: Relative sizes of iterated sumsets\nby Noah K
ravitz (Princeton University) as part of New York Number Theory Seminar\n\
n\nAbstract\nNathanson recently posed the following natural question about
the possible relative sizes of iterated sumsets: Given permutations $\\si
gma_1\,\\ldots\,\\sigma_H \\in \\mathfrak{S}_n$\, can one find finite subs
ets $A_1\,\\ldots\, A_n \\subseteq \\mathbb{Z}$ such that for each $1 \\le
q h \\leq H$\, the quantities $|hA_1|\,\\ldots\,|hA_n|$ have the same rela
tive order as $\\sigma_h$? We will describe several constructions that p
rovide affirmative answers to this and related questions.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salvatore Tringali (School of Mathematical Sciences\, Hebei Normal
University\, China)
DTSTART:20250206T200000Z
DTEND:20250206T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/107
DESCRIPTION:Title: A survey of power semigroups\nby Salvatore Tr
ingali (School of Mathematical Sciences\, Hebei Normal University\, China)
as part of New York Number Theory Seminar\n\n\nAbstract\nThe term ``power
semigroups'' is loosely used to refer to an assorted class of (commutati
ve and non-commutative) semigroups that\, among other things\, provide a n
atural algebraic framework for the formulation or reformulation of many in
triguing questions in additive combinatorics and related fields. \n\nI wil
l provide an overview of power semigroups\, with emphasis on additive-theo
retic aspects of their study\, recent developments\, and open problems.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20250213T200000Z
DTEND:20250213T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/108
DESCRIPTION:Title: New problems in additive number theory\nby Me
l Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract
\nIn the study of sums of finite sets of integers\, most attention has bee
n paid to sets with small sumsets (Freiman's theorem and related work) and
to sets with large sumsets (Sidon sets and $B_h$ sets). The focus of thi
s talk is on the full range of sizes of h-fold sums of a set of k integers
. Many new results and open problems will be presented.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Tait (Villanova University)
DTSTART:20250306T200000Z
DTEND:20250306T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/109
DESCRIPTION:Title: Cardinalities of g-difference sets\nby Michae
l Tait (Villanova University) as part of New York Number Theory Seminar\n\
n\nAbstract\nWhat is the minimum/maximum size of a set $A$ of integers tha
t has the property that every integer in $\\{1\,2\,\\cdots\, n\\}$ can be
written in at least/at most $g$ ways as a difference of elements of $A$? F
or the first question\, we show that the limit of this minimum size divide
d by $\\sqrt{n}$ exists and is nonzero\, answering a question of Kravitz.
For the second question\, we give an asymptotic formula for the maximum si
ze. We also consider the same problems but in the setting of a vector spac
e over a finite field. We will end the talk by discussing open problems an
d connections to coding theory. \n\nThis is joint work with Eric Schmutz.\
n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ross (University of Hawai'i)
DTSTART:20250227T200000Z
DTEND:20250227T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/110
DESCRIPTION:Title: A nonstandard construction of a B_h set\nby D
avid Ross (University of Hawai'i) as part of New York Number Theory Semina
r\n\n\nAbstract\nIn last week's seminar\, Mel Nathanson showed that given
an integer $h>1$and a finite set $A$ of positive reals that are linearly i
ndependent over the rationals\, $A$ can be transformed into a $B_h$-Sidon
set by multiplying its elements by a sufficiently large integer q and rou
nding off. I'll give an extremely short proof of this using nonstandard a
nalysis. (Unfortunately\, in contrast to Mel's argument\, this proof give
s no estimate for ``large enough".)\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Moreno (CUNY)
DTSTART:20250227T203000Z
DTEND:20250227T210000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/111
DESCRIPTION:Title: Multi-quadratic extensions of the rational field<
/a>\nby Carlos Moreno (CUNY) as part of New York Number Theory Seminar\n\n
\nAbstract\nWe will discuss briefly some elementary properties of quadrati
c extensions of the field of rationals. Some relations to quadratic recipr
ocity law and cyclotomy will also be mentioned.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20250313T190000Z
DTEND:20250313T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/112
DESCRIPTION:Title: Problems and results in combinatorial additive nu
mber theory\nby Mel Nathanson (CUNY) as part of New York Number Theory
Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Norbert Hegyvari (E\\"otv\\"os University and R\\'enyi Institute\,
Budapest)
DTSTART:20250403T190000Z
DTEND:20250403T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/113
DESCRIPTION:Title: Inverse results in combinatorial number theory\nby Norbert Hegyvari (E\\"otv\\"os University and R\\'enyi Institute\, B
udapest) as part of New York Number Theory Seminar\n\n\nAbstract\nThe inve
rse problems can be roughly described as follows: \n Given an additive
(or multiplicative) structure\, the task is to determine \n the structu
re of the original set. The celebrated Freiman theorem is an \n inverse
question par excellence. \n In this talk we consider inverse problems of
the subset sums in both the \n finite and infinite cases. For any addit
ive set $X$\, \n\\[ \nFS(X):= \\left\\{ \\sum_{i=1}^\\infty\\varepsilon_
ix_i: \\ x_i\\in X\, \\ \\varepsilon_i\n\\in \\{0\,1\\}\, \\ \\sum_{i=1}^\
\infty\\varepsilon_i<\\infty \\right\\}\n\\] \n denotes the set of subset
sums.\nThe finite case is due to Erd\\H os and Szemer\\'edi\, \n who as
ked the question: For which integer $t$ does there exist a set $X$ \n
with $n$ elements for which $|FS(X)|=t$? S. Burr introduced the analog
ous \n problem for the infinite case. \n\n In the rest of the talk we
discuss related questions in\nhigher dimensions.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James A. Sellers (University of Minnesota Duluth)
DTSTART:20250320T190000Z
DTEND:20250320T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/114
DESCRIPTION:Title: Arithmetic properties of $d$--fold partition diam
onds\nby James A. Sellers (University of Minnesota Duluth) as part of
New York Number Theory Seminar\n\n\nAbstract\nIn this talk\, we introduce
new combinatorial objects called $d$--fold partition diamonds\, which gene
ralize both the classical partition function (for unrestricted integer par
titions) and the partition diamonds of Andrews\, Paule\, and Riese. We con
sider two counting functions related to these combinatorial objects\, the
second of which we call ``Schmidt type'' $d$--fold partition diamonds\, wh
ich have counting function $s_d(n)$. After finding the generating function
for $s_d(n)$\, we identify a surprising connection to a well--known famil
y of polynomials. This allows us to develop elementary proofs of infinitel
y many Ramanujan--like congruences satisfied by $s_d(n)$ for various value
s of $d$\, including the following family: for all $d\\geq 1$ and all $n\\
geq 0\,$ $s_d(2n+1) \\equiv 0 \\pmod{2^d}.$\n\nThe talk will be self--cont
ained and accessible to all.\n\nThis is joint work with Dalen Dockery\, Ma
rie Jameson\, and Samuel Wilson (all of the University of Tennessee).\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20250327T190000Z
DTEND:20250327T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/115
DESCRIPTION:Title: Compression of sumsets of sets of lattice points<
/a>\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n
\nAbstract\nLet $\\mathcal{R}_{\\Z^n}(h\,k)$ be the set of all integers $t
$ \nsuch that there exists a subset $A$ of $\\Z^n$ \n with $|A|=k$ and $|h
A|=t$. \nIt is an open problem to compute a number \n $N = N(h\,k)$ such t
hat there exists \n$A' \\subseteq \\Z^n$ with $|A'|= k$\, $|hA'| = t$\, \n
and $\\|a\\|_1 \\leq N$ \nfor all $a \\in A$. It is shown how to ``compr
ess'' a widely disbursed set \n of lattice points while preserving the $h$
-fold sumset size. \nThis is a step toward the goal.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Bloom (University of Manchester\, UK)
DTSTART:20250410T190000Z
DTEND:20250410T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/116
DESCRIPTION:Title: Control in additive combinatorics and its applica
tions\nby Thomas Bloom (University of Manchester\, UK) as part of New
York Number Theory Seminar\n\n\nAbstract\nBounds for the third moment of t
he convolution have played an important role recently in additive combinat
orics\, most notably in the study of the sum-product problem and the growt
h of convex sets of real numbers. In this talk I will give a unified overv
iew of how and why such third moment bounds are useful in additive combina
torics\, including recent advances in the sum-product problem\, growth of
convex sets\, and the Balog-Szemeredi-Gowers theorem.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART:20250424T190000Z
DTEND:20250424T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/117
DESCRIPTION:Title: A family of Sequences Generalizing the Thue–Mor
se and Rudin-Shapiro Sequences\nby Russell Jay Hendel (Towson Universi
ty) as part of New York Number Theory Seminar\n\n\nAbstract\nFor $m \\ge 1
\,$ let $P_m =1^m\,$ the binary string of $m$ ones. Further define the inf
inite sequence $s_m$ by \n $s_{m\,n} = 1$ iff the number of (possibly ove
rlapping) occurrences of $P_m$ in the binary representation of $n$ is odd\
, $n \\ge 0.$ For $m=1\,2$ respectively $s_m$ is the Thue-Morse and Rudin
-Shapiro sequences. This paper shows: (i) $s_m$ is automatic\; (ii) the
minimal\, DFA (deterministic finite automata) accepting $s_m$ has $2m$ st
ates\; (iii) it suffices to use prefixes of length $2^{m-1}$ to distinguis
h all sequences in the 2-kernel of $s_m$\; and (iv) the characteristic fun
ction of the length $2^{m-1}$ prefix of the 2-kernel sequences of $s_m$ ca
n be formulated using the Vile and Jacobstahl sequences. The proofs exploi
t connections between string operations on binary strings and the numbers
they represent. Both Mathematica and Walnut are employed for exploratory a
nalysis of patterns. In particular\, we generalize the famous result about
the Thue-Morse sequence that the orders of squares in the sequence are of
the form $(2^i)_{i \\ge 0} \\cup 3 \\times (2^i)_{i \\ge 0.}$\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Rajan (Baltimore)
DTSTART:20250501T190000Z
DTEND:20250501T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/118
DESCRIPTION:Title: On a cubic Diophantine equation\nby Ashvin R
ajan (Baltimore) as part of New York Number Theory Seminar\n\n\nAbstract\n
We develop a method to find all the integral solutions of the cubic Diopha
ntine equation \n ${H_k}:y^3 - y = k(x^3 - x)$\, for certain positive int
egers $k$ that are not perfect cubes\, and illustrate our method by comple
tely solving \n this Diophantine equation for the cases $k =2$\, $k=3$\,
and $k =6$. \n Our approach relies on lower bounds for all rational appr
oximations to $\\sqrt[3]{k}$ \n that were obtained by M. Bennett\, D. Eas
ton\, and P. Voutier\, who build \n on a fundamental approach to finding
such estimates devised by A. Baker. \n This is joint work with Terutake Ab
e and Francois Ramaroson.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20250918T190000Z
DTEND:20250918T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/119
DESCRIPTION:Title: Hilbert polynomials and growth of sumsets\nby
Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstr
act\nGraded and multi-graded rings and modules and applications to additiv
e number theory.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20251009T190000Z
DTEND:20251009T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/120
DESCRIPTION:Title: Dickson's lemma and perfect numbers\nby Mel N
athanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nA
n exposition of Dickson's classic 1913 paper and its relation to Groebner
bases and Hilbert's basis theorem.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20251023T190000Z
DTEND:20251023T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/121
DESCRIPTION:Title: Triangular gaps in the most frequent sizes of the
iterated sumsets of sets of four natural numbers\nby Steven Senger (M
issouri State University) as part of New York Number Theory Seminar\n\n\nA
bstract\nWe discuss the triangular gaps observed experimentally in the mos
t popular sizes of the $h$-fold iterated sumset\, $hA\,$ when $A$ is a ran
domly chosen four-element subset of the first $q$ natural numbers\, for $q
$ much larger than $h.$ In particular\, we quantify frequent and infrequen
t sumset sizes\, and rigorously show that for any $h>2\,$ the first two g
aps between the largest frequent sumset sizes must be 1 and 3. We also ou
tline a method for showing that this pattern must continue.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20251030T190000Z
DTEND:20251030T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/122
DESCRIPTION:Title: Sumset sizes in abelian groups\nby Mel Nathan
son (CUNY) as part of New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruce Reznick (University of Illinois - Urbana)
DTSTART:20251113T200000Z
DTEND:20251113T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/123
DESCRIPTION:Title: The Stern sequence\nby Bruce Reznick (Univers
ity of Illinois - Urbana) as part of New York Number Theory Seminar\n\n\nA
bstract\nThe Stern sequence is defined by $s(0) = 0\, s(1) = 1$\, and for
$k > 0\, s(2k) = s(k)$\, and \n$ s(2k+1) = s(k) + s(k+1)$. It was defined
in 1858 and has many applications to and from number theory\, digital repr
esentations\, graph theory\, geometry\, analysis\, and probability. This
talk has cameo appearances by Eisenstein\, Minkowski\, Einstein\, de Rham\
, and Dijkstra. Stern himself was Gauss' first PhD student and led a very
interesting life. He proved that every positive rational $p/q$ can be wri
tten uniquely as $s(n)/s(n+1)$ well before Cantor\, where the binary expan
sion of $n$ is related to the simple continued fraction representation of
$p/q$. Others showed\, in effect\, that every positive rational can also b
e written uniquely as $s(m)/s(2^r-m)$ for odd $m$\, where $m$ and $n$ hav
e an unexpected relation. Also\, $s(n)$ is the number of ways to write $n-
1 = \\sum a_k 2^k$\, where $a_k$ is in $\\{0\,1\,2\\}$. De Rham used the S
tern sequence to define a convex curve on which points of zero flatness
and points of infinite flatness are dense. This talk will be accessible t
o first-year graduate students.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zach McGuirk (Bard College High School and Early College)
DTSTART:20251016T190000Z
DTEND:20251016T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/124
DESCRIPTION:Title: Free resolutions and relations-between-relations<
/a>\nby Zach McGuirk (Bard College High School and Early College) as part
of New York Number Theory Seminar\n\n\nAbstract\nThe fundamental question
driving this theory is: Given a module M over \n a ring R\, how can we u
nderstand its "structure" through its relations and \n relations-between-r
elations? This leads naturally to free resolutions\, which \n encode all t
he algebraic constraints defining M. If\, instead of a module\, we were \n
working with a vector space over a field\, then we would have a basis to
work with. \n However\, for an R-module\, there might not be a basis\, i.e
. there might be non-trivial \n relations between different generators\, a
nd then there might be relations between \n those relations. Hilbert's Sy
zygy Theorem states that this process of \n relations-between-relations mu
st eventually terminate.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART:20251204T200000Z
DTEND:20251204T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/125
DESCRIPTION:Title: Is number theory a science?\nby Carl Pomeranc
e (Dartmouth College) as part of New York Number Theory Seminar\n\n\nAbstr
act\nI often wonder why mathematics is considered\none of the sciences\, i
n fact\, even the queen of science.\nIs it really so?\nThe substance of sc
ience is comprised of theories where we\nthink we understand various proce
sses and we run experiments\nto either verify our understanding or perhaps
revise it.\nThere is an element of exactly this process with number theor
y\,\nand perhaps other branches of mathematics as well. In this\ntalk we'
ll visit various number theoretic conjectures from\nthis point of view\, f
rom the Riemann Hypothesis\, to the\ntwin prime conjecture\, and the distr
ibution of perfect numbers.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Borisov (SUNY - Binghamton)
DTSTART:20251106T200000Z
DTEND:20251106T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/126
DESCRIPTION:Title: A structure sheaf for Kirch topology on N\nby
Alexander Borisov (SUNY - Binghamton) as part of New York Number Theory S
eminar\n\n\nAbstract\nKirch topology on $\\mathbb N$ goes back to a 1969 p
aper of Kirch. It can be defined by a basis of open sets that consists of
all infinite arithmetic progressions $a+d\\mathbb N_0$\, such that $\\gcd(
a\,d)=1$ and $d$ is square-free. It is Hausdorff\, connected\, and locally
connected. One can hope that in the classical imperfect analogy betwee
n arithmetic and geometry this can serve as an arithmetic analog of the us
ual topology on $\\mathbb C$. However\, the usual topology on $\\mathbb
C$ comes with a structure sheaf of complex-analytic functions. As far as
I know\, no analog for Kirch topology has been proposed before me. I belie
ve that I have stumbled upon just such a thing\, more by accident than by
a conscious effort: locally LIP functions. These are functions from Kirc
h-open sets to $\\mathbb Z$ such that for every point in the domain there
is a Kirch-open neighborhood on which the function is "locally integer p
olynomial" (LIP): its interpolation polynomial on every finite set has in
teger coefficients. I will explain why this seems to be a natural object\,
what I know about it\, and what I hope to achieve. Some of the material
of this talk will be based on my latest paper: https://people.math.bingh
amton.edu/borisov/research.html\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James A. Sellers (University of Minnesota - Duluth)
DTSTART:20251211T200000Z
DTEND:20251211T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/127
DESCRIPTION:Title: $m$-convolutive sequences through the lens of int
eger partition functions\nby James A. Sellers (University of Minnesota
- Duluth) as part of New York Number Theory Seminar\n\n\nAbstract\nIn 200
2\, Andrews\, Lewis\, and Lovejoy introduced the combinatorial objects cal
led \\emph{partitions with designated summands}\, which are partitions whe
re exactly one part of each size in the partition is marked. For example\,
$5'+2+2'+2+1'+1$ is a partition of $13$ with designated summands where we
have marked the only part of size five\, the second part of size two\, an
d the first part of size one. \n\nIn the same paper\, Andrews\, Lewis\, a
nd Lovejoy also considered partitions with designated summands in which al
l parts are odd\, and they denoted the number of such objects of weight $n
$ by the function $PDO(n)$. As has been recognized by several authors ove
r the last two decades\, the $PDO$ function satisfies a very curious prope
rty which we call $2$-convolutivity:\n$$\n \\sum_{n\\ge 0} PDO(2n)q^n = \\
left ( \\sum_{n\\ge 0} PDO(n)q^n \\right )^2\n$$\nThat is to say\, the sub
sequence $(PDO(2n))_{n\\ge 0}$ is the convolution of the sequence $(PDO(n)
)_{n\\ge 0}$ with itself. \n\nOur work (which is joint with Shane Chern
(Vienna)\, Shishuo Fu (Chongqing)\, and Dennis Eichhorn (UC Irvine)) evol
ved out of our attempts to \\emph{combinatorially} understand the $2$-conv
olutivity of $PDO(n)$. Thus far\, such a combinatorial proof remains elusi
ve. However\, our endeavors led us to consider a more general concept of c
onvolutivity which will be discussed in the talk\, with the hope that we m
ight identify other restricted integer partition functions that are themse
lves convolutive.\n\nIndeed\, an exhaustive search of the Online Encyclope
dia of Integer Sequences during the summer 2025 identified a handful of $2
$- and $3$-convolutive sequences. We were able to prove the respective con
volutive property of each such sequence via generating function manipulati
ons\, and we also provided combinatorial proofs for the $2$-convolutive pr
operty for three of the sequences identified. \n\nIn this talk\, I will s
hare details of many of the results that we obtained\, and I will close wi
th several questions for future study.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Fel (Technion\, Israel)
DTSTART:20251218T200000Z
DTEND:20251218T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/128
DESCRIPTION:Title: Frobenius problem in numerical semigroups $S({\\b
f d}^m)$\nby Leonid Fel (Technion\, Israel) as part of New York Number
Theory Seminar\n\n\nAbstract\nContent :\n\\begin{itemize}\n\\item Definit
ion and notations\n\\item Frobenius problem -- basic facts\n\\item Special
numerical semigroups $S({\\bf d}^m)$\n\\item Symmetric numerical semigrou
ps $S({\\bf d}^m)$\n\\item Semigroups $S({\\bf d}^3)$ and minimal relation
s\n\\item Semigroup rings $k\\left[S({\\bf d}^m)\\right]$\n\\item Gorenste
in rings and complete intersection\n\\item Identities for degrees of syzyg
ies in $S({\\bf d}^m)$\n\\item Applying identities\n\\item Weak asymptotic
s of Frobenius numbers $F\\left({\\bf d}^3\n\\right)$\n\\item Conjectures
and questions\n\\end{itemize}\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isaac Rajogopal (MIT)
DTSTART:20260212T200000Z
DTEND:20260212T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/129
DESCRIPTION:Title: Possible sizes of sumsets\nby Isaac Rajogopal
(MIT) as part of New York Number Theory Seminar\n\n\nAbstract\nNathanson
introduced the range of cardinalities of $h$-fold sumsets \n $ \\mathcal{
R}(h\,k):= \\{|hA|:A \\subseteq \\mathbb{Z} \\text{ and }|A| = k\\}. $ \n
Following a remark of Erdos and Szemeredi that determined the form of $\\
mathcal{R}(h\,k)$ when $h=2$\, Nathanson asked what the form of $\\mathcal
{R}(h\,k)$ is for arbitrary $h\, k \\in \\mathbb{N}$. For $h \\in \\mathb
b{N}$\, we prove there is some constant $k_h \\in \\mathbb{N}$ such that i
f $k > k_h$\, then $\\mathcal{R}(h\,k)$ is the entire interval $\\left[hk
-h+1\,\\binom{h+k-1}{h}\\right]$ except for a specified set of $\\binom{h-
1}{2}$ numbers. Moreover\, we show that one can take $k_3 = 2$.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20260219T200000Z
DTEND:20260219T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/130
DESCRIPTION:Title: Diversity\, equity\, and inclusion for problems i
n additive number theory\nby Mel Nathanson (CUNY) as part of New York
Number Theory Seminar\n\n\nAbstract\nThis talk will survey the diversity o
f problems in additive number theory\, observe that equity requires the co
nsideration of less currently popular problems\, and argue for their inclu
sion in the additive canon. Of particular interest will be problems about
the sizes of sumsets of finite sets of integers and problems about the in
tersection of sumsets.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruce Reznick (University of Illinois at Urbana-Champaign)
DTSTART:20260226T200000Z
DTEND:20260226T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/131
DESCRIPTION:Title: Equal sums of two cubes of quadratic forms\nb
y Bruce Reznick (University of Illinois at Urbana-Champaign) as part of Ne
w York Number Theory Seminar\n\n\nAbstract\nWe give a complete description
of all solutions to the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$\nfor qua
dratic forms $f_j \\in \\mathbb C[x\,y]$ and show how\, roughly two thirds
of the time\, it can be \nextended to three equal sums of pairs of cubes
. We also count\nthe number of ways a sextic $p \\in \\mathbb C[x\,y]$ can
be written as a sum of two cubes. The\nextreme example is $p(x\,y) = xy(x
^4-y^4)$\, which has six such representations. There are name-drops\nof Eu
ler and Ramanujan.\n\nThis talk is based on the paper of the same name\, w
hich appeared in the\n{\\it International Journal of Number Theory} {\\bf
17} (2021)\, pp.761-786\, MR4254775.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shalom Eliahou (Universit\\'e du Littoral C\\^ote d'Opale\, Calais
\, France)
DTSTART:20260326T190000Z
DTEND:20260326T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/132
DESCRIPTION:Title: On Wilf's conjecture for numerical semigroups
\nby Shalom Eliahou (Universit\\'e du Littoral C\\^ote d'Opale\, Calais\,
France) as part of New York Number Theory Seminar\n\n\nAbstract\nA numeric
al semigroup $S$ is a cofinite submonoid of $\\mathbb{N}$. This means that
$S$ is stable under addition\, contains $0$\, and has finite complement i
n $\\mathbb{N}$. Some important numbers attached to $S$ are its genus $g =
\\text{card}(\\mathbb{N} \\setminus S)$\, its conductor $c=\\max(\\mathbb
{Z} \\setminus S)+1$ and its minimal number $n$ of generators. Half a cent
ury ago\, Herbert Wilf came up with a very clever conjectural upper bound
on the genus $g$ in terms of $c$ and $n$\, namely \n$$g \\le c(1-1/n).$$\n
In this talk\, we will provide a brief overview of the current status of W
ilf's conjecture and discuss connections with additive combinatorics\, gra
ph theory and commutative algebra.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Larsen (MIT)
DTSTART:20260305T200000Z
DTEND:20260305T213000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/133
DESCRIPTION:Title: Additive bases and minimal subbases\nby Danie
l Larsen (MIT) as part of New York Number Theory Seminar\n\n\nAbstract\nWe
construct an additive basis $A$ of order 2 whose representation function
satisfies \n$r_A(n) > \\varepsilon \\log n$ for all sufficiently large $n$
\, yet which contains no minimal asymptotic subbasis. This confirms a con
jecture of Erdos and Nathanson from 1979. More generally\, we discuss a s
trategy for constructing asymptotic bases which do not have any minimal as
ymptotic subbases and satisfy other desired properties.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20260312T190000Z
DTEND:20260312T203000Z
DTSTAMP:20260422T215257Z
UID:New_York_Number_Theory_Seminar/134
DESCRIPTION:Title: Sumset intersection problems\nby Mel Nathanso
n (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nLet $\n
= \\{1\,2\,3\,\\ldots\\}$ be the set of positive integers. Let $A$ be a su
bset of an additive abelian semigroup $S$ and let $hA$ be the $h$-fold su
mset of $A$. The following question is considered:\nLet $(A_q)_{q=1}^{\\
infty}$ be a strictly decreasing sequence of sets in\n$S$ and let $A = \\
bigcap_{q=1}^{\\infty} A_q$. \nDescribe the set $\\mathcal{H}(A_q)$ of
positive integers such that\n\\[\nhA = \\bigcap_{q=1}^{\\infty} hA_q.\n\\]
\nA sample result: If $A_q$ is a set of positive integers for all $q$\,
then $\\mathcal{H}(A_q)=\n$.\n\nHere are some nice open problems. \n\n1.
For a given set $X$\, does there exist a strictly decreasing sequence\n$(
A_q)_{q=1}^{\\infty} $ of sets of integers such that $\\mathcal{H}(A_q)= X
$. \n\n2. For given sets $A$ and $X$\, does there exist a strictly de
creasing sequence\n$(A_q)_{q=1}^{\\infty} $ of sets of integers such that
$A = \\bigcap_{q=1}^{\\infty} A_q$\nand $\\mathcal{H}(A_q)= X$. \n\n3. Do
es there exist a set $Y$ of positive integers such that $\\mathcal{H}(A_q)
\\neq Y$\nfor every strictly decreasing sequence $(A_q)_{q=1}^{\\infty}$
of sets of integers?\n\n4. For a given set $A$\, let $\\mathcal{H}^*(A)$ b
e the set of all sets $X$ such that $\\mathcal{H}(A_q) = X$ for some str
ictly decreasing sequence $(A_q)_{q=1}^{\\infty}$ with $A = \\bigcap_{q=1}
^{\\infty} A_q$.\n\n5. Compute $\\mathcal{H}^*(A)$ for $A = \\{0\\}$.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
134/
END:VEVENT
END:VCALENDAR