BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Shalom Eliahou (Universite du Littoral Cote d'Opale) DTSTART;VALUE=DATE-TIME:20200710T150000Z DTEND;VALUE=DATE-TIME:20200710T160000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200903T190000Z DTEND;VALUE=DATE-TIME:20200903T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200910T190000Z DTEND;VALUE=DATE-TIME:20200910T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200917T190000Z DTEND;VALUE=DATE-TIME:20200917T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200924T190000Z DTEND;VALUE=DATE-TIME:20200924T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201001T190000Z DTEND;VALUE=DATE-TIME:20201001T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201008T190000Z DTEND;VALUE=DATE-TIME:20201008T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201015T190000Z DTEND;VALUE=DATE-TIME:20201015T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201029T190000Z DTEND;VALUE=DATE-TIME:20201029T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201029T190000Z DTEND;VALUE=DATE-TIME:20201029T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201029T190000Z DTEND;VALUE=DATE-TIME:20201029T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201029T190000Z DTEND;VALUE=DATE-TIME:20201029T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201112T200000Z DTEND;VALUE=DATE-TIME:20201112T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201105T200000Z DTEND;VALUE=DATE-TIME:20201105T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201119T200000Z DTEND;VALUE=DATE-TIME:20201119T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201203T200000Z DTEND;VALUE=DATE-TIME:20201203T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201210T200000Z DTEND;VALUE=DATE-TIME:20201210T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20201217T200000Z DTEND;VALUE=DATE-TIME:20201217T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210128T200000Z DTEND;VALUE=DATE-TIME:20210128T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200204T200000Z DTEND;VALUE=DATE-TIME:20200204T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200204T200000Z DTEND;VALUE=DATE-TIME:20200204T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20200204T200000Z DTEND;VALUE=DATE-TIME:20200204T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210204T200000Z DTEND;VALUE=DATE-TIME:20210204T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210211T200000Z DTEND;VALUE=DATE-TIME:20210211T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210218T200000Z DTEND;VALUE=DATE-TIME:20210218T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210304T200000Z DTEND;VALUE=DATE-TIME:20210304T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210311T200000Z DTEND;VALUE=DATE-TIME:20210311T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210318T190000Z DTEND;VALUE=DATE-TIME:20210318T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210325T190000Z DTEND;VALUE=DATE-TIME:20210325T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210225T200000Z DTEND;VALUE=DATE-TIME:20210225T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210401T190000Z DTEND;VALUE=DATE-TIME:20210401T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210408T190000Z DTEND;VALUE=DATE-TIME:20210408T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20210930T190000Z DTEND;VALUE=DATE-TIME:20210930T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211007T190000Z DTEND;VALUE=DATE-TIME:20211007T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211014T190000Z DTEND;VALUE=DATE-TIME:20211014T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211021T190000Z DTEND;VALUE=DATE-TIME:20211021T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211028T190000Z DTEND;VALUE=DATE-TIME:20211028T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211104T190000Z DTEND;VALUE=DATE-TIME:20211104T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211111T200000Z DTEND;VALUE=DATE-TIME:20211111T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211202T200000Z DTEND;VALUE=DATE-TIME:20211202T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211209T200000Z DTEND;VALUE=DATE-TIME:20211209T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20211216T200000Z DTEND;VALUE=DATE-TIME:20211216T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220203T200000Z DTEND;VALUE=DATE-TIME:20220203T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220210T200000Z DTEND;VALUE=DATE-TIME:20220210T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220217T200000Z DTEND;VALUE=DATE-TIME:20220217T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220224T200000Z DTEND;VALUE=DATE-TIME:20220224T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220303T200000Z DTEND;VALUE=DATE-TIME:20220303T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220310T200000Z DTEND;VALUE=DATE-TIME:20220310T213000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220317T190000Z DTEND;VALUE=DATE-TIME:20220317T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220324T190000Z DTEND;VALUE=DATE-TIME:20220324T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220331T190000Z DTEND;VALUE=DATE-TIME:20220331T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220414T190000Z DTEND;VALUE=DATE-TIME:20220414T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220407T190000Z DTEND;VALUE=DATE-TIME:20220407T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220428T190000Z DTEND;VALUE=DATE-TIME:20220428T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220505T190000Z DTEND;VALUE=DATE-TIME:20220505T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220512T190000Z DTEND;VALUE=DATE-TIME:20220512T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220616T190000Z DTEND;VALUE=DATE-TIME:20220616T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220623T190000Z DTEND;VALUE=DATE-TIME:20220623T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220630T190000Z DTEND;VALUE=DATE-TIME:20220630T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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;VALUE=DATE-TIME:20220707T190000Z DTEND;VALUE=DATE-TIME:20220707T203000Z DTSTAMP;VALUE=DATE-TIME:20220703T091557Z 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 END:VCALENDAR