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BEGIN:VEVENT
SUMMARY:Dang-Khoa Nguyen (University of Calgary)
DTSTART:20220926T180000Z
DTEND:20220926T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/1
DESCRIPTION:Title: Heig
ht gaps for coefficients of D-finite power series\nby Dang-Khoa Nguyen
(University of Calgary) as part of Lethbridge number theory and combinato
rics seminar\n\nLecture held in University of Lethbridge\, room M1040 (Mar
kin Hall).\n\nAbstract\nA power series $f(x_1\,\\ldots\,x_m)\\in \\mathbb{
C}[[x_1\,\\ldots\,x_m]]$ is said to be D-finite if all the partial derivat
ives of $f$\n span a finite dimensional vector space over\n the field $\\m
athbb{C}(x_1\,\\ldots\,x_m)$. For the univariate series $f(x)=\\sum a_nx^n
$\, this is equivalent to the condition that the sequence $(a_n)$ is P-rec
ursive meaning a non-trivial linear recurrence relation of the form:\n $$P
_d(n)a_{n+d}+\\cdots+P_0(n)a_n=0$$\n where the $P_i$'s are polynomials. In
this talk\, we consider D-finite power series with algebraic coefficients
and discuss the growth of the Weil height of these coefficients.\n \n \n
This is from a joint work with Jason Bell and Umberto Zannier in 2019 and
a more recent work in June 2022.\n
LOCATION:https://researchseminars.org/talk/NTC/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hugo Chapdelaine (Université Laval)
DTSTART:20221031T180000Z
DTEND:20221031T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/2
DESCRIPTION:Title: Comp
utation of Galois groups via permutation group theory\nby Hugo Chapdel
aine (Université Laval) as part of Lethbridge number theory and combinato
rics seminar\n\n\nAbstract\nIn this talk we will present a method to study
the Galois group of certain polynomials defined over $\\Q$.\nOur approach
is similar in spirit to some previous work of F. Hajir\, who studied\, mo
re than a decade ago\, the generalized Laguerre polynomials using a simila
r approach.\nFor example this method seems to be well suited to study the
Galois groups of Jacobi polynomials (a classical family of orthogonal poly
nomials with two parameters --- three if we include the degree). Given a p
olynomial $f(x)$ with rational coefficients of degree $N$ over $\\Q$\, the
idea consists in finding a good prime $p$ and look at the Newton polygon
of $f$ at $p$. Then combining the Galois theory of local field over $\\Q_p
$ and some classical results of the theory of permutation of groups we som
etimes succeed in showing that the Galois group of $f$ is not solvable or
even isomorphic to $A_N$ or $S_N$ ($N\\geq 5$).\n\nThe existence of a good
prime $p$ is subtle. In order to get useful results one would need to hav
e some "effective prime existence results". As an illustration\, we would
like to have an explicit constant $C$ (not too big) such that for any $N>C
$\, there exists a prime $p$ in the range $N < p < \\frac{3N}{2}$ such tha
t\ngcd$(p-1\,N)= 1 \\text{ or } 2$ (depending on the parity of $N$). Such
a result is not so easy to get when $N$ is divisible by many distinct and
small primes. We hope that such effective prime existence results are with
in the reach of the current techniques used in analytic number theory.\n
LOCATION:https://researchseminars.org/talk/NTC/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Desjardins (University of Toronto)
DTSTART:20221117T210000Z
DTEND:20221117T220000Z
DTSTAMP:20260422T212553Z
UID:NTC/3
DESCRIPTION:Title: Tors
ion points and concurrent lines on Del Pezzo surfaces of degree one\nb
y Julie Desjardins (University of Toronto) as part of Lethbridge number th
eory and combinatorics seminar\n\n\nAbstract\nThe blow up of the anticanon
ical base point on X\, a del Pezzo surface of degree 1\, gives rise to a r
ational elliptic surface E with only irreducible fibers. The sections of m
inimal height of E are in correspondence with the 240 exceptional curves o
n X. A natural question arises when studying the configuration of those cu
rves : \n\nIf a point of X is contained in "many" exceptional curves\, is
it torsion on its fiber on E?\n\nIn 2005\, Kuwata proved for del Pezzo sur
faces of degree 2 (where there is 56 exceptional curves) that if "many" eq
uals 4 or more\, then yes. In a joint paper with Rosa Winter\, we prove th
at for del Pezzo surfaces of degree 1\, if "many" equals 9 or more\, then
yes. Moreover\, we find counterexamples where a torsion point lies at the
intersection of 7 exceptional curves.\n
LOCATION:https://researchseminars.org/talk/NTC/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Dutour (University of Alberta)
DTSTART:20221128T190000Z
DTEND:20221128T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/4
DESCRIPTION:Title: Thet
a-finite pro-Hermitian vector bundles from loop groups elements\nby Ma
thieu Dutour (University of Alberta) as part of Lethbridge number theory a
nd combinatorics seminar\n\nLecture held in University of Lethbridge\, roo
m M1040 (Markin Hall).\n\nAbstract\nIn the finite-dimensional situation\,
Lie's third theorem provides a correspondence between Lie groups and Lie a
lgebras. Going from the latter to the former is the more complicated const
ruction\, requiring a suitable representation\, and taking exponentials of
the endomorphisms induced by elements of the group.\n\nAs shown by Garlan
d\, this construction can be adapted for some Kac-Moody algebras\, obtaine
d as (central extensions of) loop algebras. The resulting group is called
a loop group. One also obtains a relevant infinite-rank Chevalley lattice\
, endowed with a metric. Recent work by Bost and Charles provide a natural
setting\, that of pro-Hermitian vector bundles and theta invariants\, in
which to study these objects related to loop groups. More precisely\, we w
ill see in this talk how to define theta-finite pro-Hermitian vector bundl
es from elements in a loop group. Similar constructions are expected\, in
the future\, to be useful to study loop Eisenstein series for number field
s.\n\nThis is joint work with Manish M. Patnaik.\n
LOCATION:https://researchseminars.org/talk/NTC/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (University of California - Irvine)
DTSTART:20221205T190000Z
DTEND:20221205T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/5
DESCRIPTION:Title: Nega
tive moments of the Riemann zeta-function\nby Alexandra Florea (Univer
sity of California - Irvine) as part of Lethbridge number theory and combi
natorics seminar\n\n\nAbstract\nI will talk about recent work towards a co
njecture of Gonek regarding negative shifted moments of the Riemann zeta-f
unction. I will explain how to obtain asymptotic formulas when the shift i
n the Riemann zeta function is big enough\, and how we can obtain non-triv
ial upper bounds for smaller shifts. I will also discuss some applications
to the question of obtaining cancellation of averages of the Mobius funct
ion. Joint work with H. Bui.\n
LOCATION:https://researchseminars.org/talk/NTC/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debanjana Kundu (University of British Columbia)
DTSTART:20221003T180000Z
DTEND:20221003T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/6
DESCRIPTION:Title: Stud
ying Hilbert's 10th problem via explicit elliptic curves\nby Debanjana
Kundu (University of British Columbia) as part of Lethbridge number theor
y and combinatorics seminar\n\nLecture held in University of Lethbridge: M
1040 (Markin Hall).\n\nAbstract\nIn 1900\, Hilbert posed the following pro
blem: "Given a Diophantine equation with integer coefficients: to devise a
process according to which it can be determined in a finite number of ope
rations whether the equation is solvable in (rational) integers."\n\nBuild
ing on the work of several mathematicians\, in 1970\, Matiyasevich proved
that this problem has a negative answer\, i.e.\, such a general `process'
(algorithm) does not exist.\n\nIn the late 1970's\, Denef--Lipshitz formul
ated an analogue of Hilbert's 10th problem for rings of integers of number
fields. \n\nIn recent years\, techniques from arithmetic geometry have be
en used extensively to attack this problem. One such instance is the work
of García-Fritz and Pasten (from 2019) which showed that the analogue of
Hilbert's 10th problem is unsolvable in the ring of integers of number fie
lds of the form $\\mathbb{Q}(\\sqrt[3]{p}\,\\sqrt{-q})$ for positive propo
rtions of primes $p$ and $q$. In joint work with Lei and Sprung\, we impro
ve their proportions and extend their results in several directions. We ac
hieve this by using multiple elliptic curves\, and by replacing their Iwas
awa theory arguments by a more direct method.\n
LOCATION:https://researchseminars.org/talk/NTC/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elchin Hasanalizade (University of Lethbridge)
DTSTART:20221017T180000Z
DTEND:20221017T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/7
DESCRIPTION:Title: Sums
of Fibonacci numbers close to a power of $2$\nby Elchin Hasanalizade
(University of Lethbridge) as part of Lethbridge number theory and combina
torics seminar\n\nLecture held in University of Lethbridge: M1040 (Markin
Hall).\n\nAbstract\nThe Fibonacci sequence $(F_n)_{n \\geq 0}$ is the bina
ry recurrence sequence defined by $F_0 = F_1 = 1$ and\n$$\nF_{n+2} = F_{n+
1} + F_n \\text{ for all } n \\geq 0.\n$$\nThere is a broad literature on
the Diophantine equations involving the Fibonacci numbers. In this talk\,
we will study the Diophantine inequality\n$$\n| F_n + F_m - 2^a | < 2^{a/
2}\n$$\nin positive integers $n\, m$ and $a$ with $n \\geq m$. The main to
ols used are lower bounds for linear forms in logarithms due to Matveev an
d Dujella-Pethö version of the Baker-Davenport reduction method in Diopha
ntine approximation.\n
LOCATION:https://researchseminars.org/talk/NTC/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dave Morris (University of Lethbridge)
DTSTART:20221024T180000Z
DTEND:20221024T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/8
DESCRIPTION:Title: On v
ertex-transitive graphs with a unique Hamiltonian circle\nby Dave Morr
is (University of Lethbridge) as part of Lethbridge number theory and comb
inatorics seminar\n\nLecture held in University of Lethbridge: M1040 (Mark
in Hall).\n\nAbstract\nWe will discuss graphs that have a unique Hamiltoni
an cycle and are vertex-transitive\, which means there is an automorphism
that takes any vertex to any other vertex. Cycles are the only examples wi
th finitely many vertices\, but the situation is more interesting for infi
nite graphs. (Infinite graphs do not have ``Hamiltonian cycles''\, but the
re are natural analogues.) The case where the graph has only finitely many
ends is not difficult\, but we do not know whether there are examples wit
h infinitely many ends. This is joint work in progress with Bobby Miraftab
.\n
LOCATION:https://researchseminars.org/talk/NTC/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Solaleh Bolvardizadeh (University of Lethbridge)
DTSTART:20221121T190000Z
DTEND:20221121T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/9
DESCRIPTION:Title: On t
he Quality of the $ABC$-Solutions\nby Solaleh Bolvardizadeh (Universit
y of Lethbridge) as part of Lethbridge number theory and combinatorics sem
inar\n\nLecture held in University of Lethbridge: M1040 (Markin Hall).\n\n
Abstract\nThe quality of the triplet $(a\,b\,c)$\, where $\\gcd(a\,b\,c) =
1$\, satisfying $a + b = c$ is defined as\n$$\nq(a\,b\,c) = \\frac{\\max\
\{\\log |a|\, \\log |b|\, \\log |c|\\}}{\\log \\mathrm{rad}(|abc|)}\,\n$$\
nwhere $\\mathrm{rad}(|abc|)$ is the product of distinct prime factors of
$|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture st
ates that given $\\epsilon > 0$ the number of the $ABC$-solutions $(a\,b\,
c)$ with $q(a\,b\,c) \\geq 1 + \\epsilon$ is finite.\n\nIn the first part
of this talk\, under the $ABC$-conjecture\, we explore the quality of cert
ain families of the $ABC$-solutions formed by terms in Lucas and associate
d Lucas sequences. We also introduce\, unconditionally\, a new family of $
ABC$-solutions that has quality $> 1$.\n\nIn the remaining of the talk\, w
e prove a conjecture of Erd\\"os on the solutions of the Brocard-Ramanujan
equation\n$$\nn! + 1 = m^2\n$$\nby assuming an explicit version of the $A
BC$-conjecture proposed by Baker.\n
LOCATION:https://researchseminars.org/talk/NTC/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Douglas Ulmer (University of Arizona)
DTSTART:20230327T180000Z
DTEND:20230327T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/10
DESCRIPTION:Title: $p$
-torsion of Jacobians for unramified $\\mathbb{Z}/p\\mathbb{Z}$-covers of
curves\nby Douglas Ulmer (University of Arizona) as part of Lethbridge
number theory and combinatorics seminar\n\nLecture held in University of
Lethbridge: M1040 (Markin Hall).\n\nAbstract\nIt is a classical problem to
understand the set of Jacobians of curves\namong all abelian varieties\,
i.e.\, the image of the map $M_g\\to A_g$\nwhich sends a curve $X$ to its
Jacobian $J_X$. In characteristic $p$\,\n$A_g$ has interesting filtration
s\, and we can ask how the image of\n$M_g$ interacts with them. Concretel
y\, which groups schemes arise as\nthe p-torsion subgroup $J_X[p]$ of a Ja
cobian? We consider this\nproblem in the context of unramified $Z/pZ$ cov
ers $Y\\to X$ of curves\,\nasking how $J_Y[p]$ is related to $J_X[p]$. Tr
anslating this into a\nproblem about de Rham cohmology yields some results
using\nclassical ideas of Chevalley and Weil. This is joint work with Br
yden\nCais.\n
LOCATION:https://researchseminars.org/talk/NTC/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Males (University of Manitoba)
DTSTART:20230320T180000Z
DTEND:20230320T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/11
DESCRIPTION:Title: For
gotten conjectures of Andrews for Nahm-type sums\nby Joshua Males (Uni
versity of Manitoba) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: M1040 (Markin Hall).
\n\nAbstract\nIn his famous '86 paper\, Andrews made several conjectures o
n\nthe function $\\sigma(q)$ of Ramanujan\, including that it has\ncoeffic
ients (which count certain partition-theoretic objects) whose\nsup grows i
n absolute value\, and that it has infinitely many Fourier\ncoefficients t
hat vanish. These conjectures were famously proved by\nAndrews-Dyson-Hicke
rson in their '88 Invent. paper\, and the function\n$\\sigma$ has been rel
ated to the arithmetic of $\\mathbb{Z}[\\sqrt{6}]$\nby Cohen (and extensio
ns by Zwegers)\, and is an important first\nexample of quantum modular for
ms introduced by Zagier.\n\nA closer inspection of Andrews' '86 paper reve
als several more\nfunctions that have been a little left in the shadow of
their sibling\n$\\sigma$\, but which also exhibit extraordinary behaviour.
In an\nongoing project with Folsom\, Rolen\, and Storzer\, we study the f
unction\n$v_1(q)$ which is given by a Nahm-type sum and whose coefficients
\ncount certain differences of partition-theoretic objects. We give\nexpla
nations of four conjectures made by Andrews on $v_1$\, which\nrequire a bl
end of novel and well-known techniques\, and reveal that\n$v_1$ should be
intimately linked to the arithmetic of the imaginary\nquadratic field $\\m
athbb{Q}[\\sqrt{-3}]$.\n
LOCATION:https://researchseminars.org/talk/NTC/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay (Centro de Investigación en Matemáticas (CIMAT)\
, Guanajuato)
DTSTART:20230206T190000Z
DTEND:20230206T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/12
DESCRIPTION:Title: Gen
eralized valuations and idempotization of schemes\nby Cristhian Garay
(Centro de Investigación en Matemáticas (CIMAT)\, Guanajuato) as part of
Lethbridge number theory and combinatorics seminar\n\nLecture held in Uni
versity of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nClassical valuati
on theory has proved to be a valuable tool in number theory\, algebraic ge
ometry and singularity theory. For example\, one can enrich spectra of rin
gs with new points coming from valuations defined on them and taking value
s in totally ordered abelian groups.\n\n\n\nTotally ordered groups are exa
mples of idempotent semirings\, and generalized valuations appear when we
replace totally ordered abelian groups with more general idempotent semiri
ngs. An important example of idempotent semiring is the tropical semifield
. \n\n\nAs an application of this set of ideas\, we show how to associate
an idempotent version of the structure sheaf of a scheme\, which behaves p
articularly well with respect to idempotization of closed subschemes.\n\n\
nThis is a joint work with Félix Baril Boudreau.\n
LOCATION:https://researchseminars.org/talk/NTC/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renate Scheidler (University of Calgary)
DTSTART:20230313T180000Z
DTEND:20230313T190000Z
DTSTAMP:20260422T212553Z
UID:NTC/13
DESCRIPTION:Title: Ori
enteering on Supersingular Isogeny Volcanoes Using One Endomorphism\nb
y Renate Scheidler (University of Calgary) as part of Lethbridge number th
eory and combinatorics seminar\n\nLecture held in University of Lethbridge
: M1040 (Markin Hall).\n\nAbstract\nElliptic curve isogeny path finding ha
s many applications in number theory and cryptography. For supersingular c
urves\, this problem is known to be easy when one small endomorphism or th
e entire endomorphism ring are known. Unfortunately\, computing the endomo
rphism ring\, or even just finding one small endomorphism\, is hard. How
difficult is path finding in the presence of one (not necessarily small) e
ndomorphism? We use the volcano structure of the oriented supersingular is
ogeny graph to answer this question. We give a classical algorithm for pat
h finding that is subexponential in the degree of the endomorphism and lin
ear in a certain class number\, and a quantum algorithm for finding a smoo
th isogeny (and hence also a path) that is subexponential in the discrimin
ant of the endomorphism. A crucial tool for navigating supersingular orien
ted isogeny volcanoes is a certain class group action on oriented elliptic
curves which generalizes the well-known class group action in the setting
of ordinary elliptic curves.\n
LOCATION:https://researchseminars.org/talk/NTC/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youness Lamzouri (Institut Élie Cartan de Lorraine (IECL) of the
Université de Lorraine in Nancy)
DTSTART:20230109T190000Z
DTEND:20230109T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/14
DESCRIPTION:Title: A w
alk on Legendre paths\nby Youness Lamzouri (Institut Élie Cartan de L
orraine (IECL) of the Université de Lorraine in Nancy) as part of Lethbri
dge number theory and combinatorics seminar\n\nLecture held in University
of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nThe Legendre symbol is on
e of the most basic\, mysterious and extensively studied objects in number
theory. It is a multiplicative function that encodes information about wh
ether an integer is a square modulo an odd prime $p$. The Legendre symbol
was introduced by Adrien-Marie Legendre in 1798\, and has since found coun
tless applications in various areas of mathematics as well as in other fie
lds including cryptography. In this talk\, we shall explore what we call `
`Legendre paths''\, which encode information about the values of the Legen
dre symbol. The Legendre path modulo $p$ is defined as the polygonal path
in the plane formed by joining the partial sums of the Legendre symbol mod
ulo $p$. In particular\, we will attempt to answer the following questions
as we vary over the primes $p$: how are these paths distributed? how do t
heir maximums behave? and what proportion of the path is above the real ax
is? Among our results\, we prove that these paths converge in law\, in the
space of continuous functions\, to a certain random Fourier series constr
ucted using Rademakher random multiplicative functions. Part of this work
is joint with Ayesha Hussain.\n\nThis talk is part of the PIMS Distinguish
ed Speaker Series. The registration link is only valid for this talk.\n
LOCATION:https://researchseminars.org/talk/NTC/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antonella Perucca (University of Luxembourg)
DTSTART:20230123T163000Z
DTEND:20230123T173000Z
DTSTAMP:20260422T212553Z
UID:NTC/15
DESCRIPTION:Title: Rec
ent advances in Kummer theory\nby Antonella Perucca (University of Lux
embourg) as part of Lethbridge number theory and combinatorics seminar\n\n
\nAbstract\nKummer theory is a classical theory about radical extensions o
f fields in the case where suitable roots of unity are present in the base
field. Motivated by problems close to Artin's primitive root conjecture\,
we have investigated the degree of families of general Kummer extensions
of number fields\, providing parametric closed formulas. We present a seri
es of papers that are in part joint work with Christophe Debry\, Fritz Hö
rmann\, Pietro Sgobba\, and Sebastiano Tronto.\n
LOCATION:https://researchseminars.org/talk/NTC/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neelam Kandhil (The Institute of Mathematical Sciences (IMSc)\, Ch
ennai)
DTSTART:20230116T163000Z
DTEND:20230116T173000Z
DTSTAMP:20260422T212553Z
UID:NTC/16
DESCRIPTION:Title: On
linear independence of Dirichlet L-values\nby Neelam Kandhil (The Inst
itute of Mathematical Sciences (IMSc)\, Chennai) as part of Lethbridge num
ber theory and combinatorics seminar\n\n\nAbstract\nIt is an open question
of Baker whether the Dirichlet L-values at 1 with fixed modulus are linea
rly\nindependent over the rational numbers. The best-known result is due t
o Baker\, Birch and Wirsing\, which affirms\nthis when the modulus of the
associated Dirichlet character is co-prime to its Euler's phi value. In th
is talk\,\nwe will discuss an extension of this result to any arbitrary fa
mily of moduli. The interplay between the\nresulting ambient number fields
brings new technical issues and complications hitherto absent in the cont
ext of\na fixed modulus. We will also investigate the linear independence
of such values at integers greater than 1.\n
LOCATION:https://researchseminars.org/talk/NTC/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oussama Hamza (University of Western Ontario)
DTSTART:20230130T190000Z
DTEND:20230130T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/17
DESCRIPTION:Title: Fil
trations\, arithmetic and explicit examples in an equivariant context\
nby Oussama Hamza (University of Western Ontario) as part of Lethbridge nu
mber theory and combinatorics seminar\n\nLecture held in M1040 (Markin Hal
l).\n\nAbstract\nPro-$p$ groups arise naturally in number theory as quotie
nts of absolute Galois groups over number fields. These groups are quite m
ysterious. During the 60's\, Koch gave a presentation of some of these quo
tients. Furthermore\, around the same period\, Jennings\, Golod\, Shafarev
ich and Lazard introduced two integer sequences $(a_n)$ and $(c_n)$\, clos
ely related to a special filtration of a finitely generated pro-p group $G
$\, called the Zassenhaus filtration. These sequences give the cardinality
of $G$\, and characterize its topology. For instance\, we have the well-k
nown Gocha's alternative (Golod and Shafarevich): There exists an integer
$n$ such that $a_n=0$ (or $c_n$ has a polynomial growth) if and only if $G
$ is a Lie group over $p$-adic fields.\n\nIn 2016\, Minac\, Rogelstad and
Tan inferred an explicit relation between $a_n$ and $c_n$. Recently (2022)
\, considering geometrical ideas of Filip and Stix\, Hamza got more precis
e relations in an equivariant context: when the automorphism group of $G$
admits a subgroup of order a prime $q$ dividing $p-1$.\n\nIn this talk\, w
e present equivariant relations inferred by Hamza (2022) and give explicit
examples in an arithmetical context.\n
LOCATION:https://researchseminars.org/talk/NTC/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florent Jouve (Université de Bordeaux)
DTSTART:20230227T163000Z
DTEND:20230227T173000Z
DTSTAMP:20260422T212553Z
UID:NTC/18
DESCRIPTION:Title: Flu
ctuations in the distribution of Frobenius automorphisms in number field e
xtensions\nby Florent Jouve (Université de Bordeaux) as part of Lethb
ridge number theory and combinatorics seminar\n\n\nAbstract\nGiven a Galoi
s extension of number fields $L/K$\, the Chebotarev Density Theorem assert
s that\, away from ramified primes\, Frobenius automorphisms equidistribut
e in the set of conjugacy classes of ${\\rm Gal}(L/K)$. In this talk we re
port on joint work with D. Fiorilli in which we study the variations of th
e error term in Chebotarev’s Theorem as $L/K$ runs over certain families
of extensions. We shall explain some consequences of this analysis: regar
ding first "Linnik type problems" on the least prime ideal in a given Frob
enius set\, and second\, the existence of unconditional "Chebyshev biases"
in the context of number fields. Time permitting we will mention joint wo
rk with R. de La Bretèche and D. Fiorilli in which we go one step further
and study moments of the distribution of Frobenius automorphisms.\n
LOCATION:https://researchseminars.org/talk/NTC/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART:20230306T190000Z
DTEND:20230306T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/19
DESCRIPTION:Title: A n
orm refinement of Bezout's Lemma\, and quaternion orders\nby John Voig
ht (Dartmouth College) as part of Lethbridge number theory and combinatori
cs seminar\n\nLecture held in University of Lethbridge: M1040 (Markin Hall
).\n\nAbstract\nGiven coprime integers a\,b\, the classical identity of Be
zout provides\nintegers u\,v such that au-bv = 1. We consider refinements
to this\nidentity\, where we ask that u\,v are norms from a quadratic ext
ension.\nWe then find ourselves counting optimal embeddings of a quadratic
\norder in a quaternion order\, for which we give explicit formulas in\nma
ny cases. This is joint work with Donald Cartwright and Xavier\nRoulleau.
\n
LOCATION:https://researchseminars.org/talk/NTC/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay (Centro de Investigación en Matemáticas (CIMAT)\
, Guanajuato)
DTSTART:20230206T221000Z
DTEND:20230206T234500Z
DTSTAMP:20260422T212553Z
UID:NTC/20
DESCRIPTION:Title: An
invitation to the algebraic geometry over idempotent semirings (Lecture 1
of 2)\nby Cristhian Garay (Centro de Investigación en Matemáticas (C
IMAT)\, Guanajuato) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: B716 (University Hall
).\n\nAbstract\nIdempotent semirings have been relevant in several branche
s of applied mathematics\, like formal languages and combinatorial optimiz
ation.\n\n\nThey were brought recently to pure mathematics thanks to its l
ink with tropical geometry\, which is a relatively new branch of mathemati
cs that has been useful in solving some problems and conjectures in classi
cal algebraic geometry. \n\n\nHowever\, up to now we do not have a proper
algebraic formalization of what could be called “Tropical Algebraic Geom
etry”\, which is expected to be the geometry arising from idempotent sem
irings. \n\n\nIn this mini course we aim to motivate the necessity for suc
h theory\, and we recast some old constructions in order theory in terms o
f commutative algebra of semirings and modules over them.\n
LOCATION:https://researchseminars.org/talk/NTC/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay (Centro de Investigación en Matemáticas (CIMAT)\
, Guanajuato)
DTSTART:20230209T221000Z
DTEND:20230209T234500Z
DTSTAMP:20260422T212553Z
UID:NTC/21
DESCRIPTION:Title: An
invitation to the algebraic geometry over idempotent semirings (Lecture 2
of 2)\nby Cristhian Garay (Centro de Investigación en Matemáticas (C
IMAT)\, Guanajuato) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: B716 (University Hall
).\n\nAbstract\nIdempotent semirings have been relevant in several branche
s of applied mathematics\, like formal languages and combinatorial optimiz
ation.\n\n\nThey were brought recently to pure mathematics thanks to its l
ink with tropical geometry\, which is a relatively new branch of mathemati
cs that has been useful in solving some problems and conjectures in classi
cal algebraic geometry. \n\n\nHowever\, up to now we do not have a proper
algebraic formalization of what could be called “Tropical Algebraic Geom
etry”\, which is expected to be the geometry arising from idempotent sem
irings. \n\n\nIn this mini course we aim to motivate the necessity for suc
h theory\, and we recast some old constructions in order theory in terms o
f commutative algebra of semirings and modules over them.\n
LOCATION:https://researchseminars.org/talk/NTC/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Andrés Helfgott (University of Göttingen/Institut de Math
ématiques de Jussieu)
DTSTART:20230403T163000Z
DTEND:20230403T173000Z
DTSTAMP:20260422T212553Z
UID:NTC/22
DESCRIPTION:Title: Exp
ansion\, divisibility and parity\nby Harald Andrés Helfgott (Universi
ty of Göttingen/Institut de Mathématiques de Jussieu) as part of Lethbri
dge number theory and combinatorics seminar\n\nLecture held in University
of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nWe will discuss a graph t
hat encodes the divisibility properties of integers by primes. We prove th
at this graph has a strong local expander property almost everywhere. We t
hen obtain several consequences in number theory\, beyond the traditional
parity barrier\, by combining our result with Matomaki-Radziwill. For inst
ance: for lambda the Liouville function (that is\, the completely multipli
cative function with $\\lambda(p) = -1$ for every prime)\, $(1/\\log x) \\
sum_{n\\leq x} \\lambda(n) \\lambda(n+1)/n = O(1/\\sqrt(\\log \\log x))$\,
which is stronger than well-known results by Tao and Tao-Teravainen. We a
lso manage to prove\, for example\, that $\\lambda(n+1)$ averages to $0$ a
t almost all scales when $n$ restricted to have a specific number of prime
divisors $\\Omega(n)=k$\, for any "popular" value of $k$ (that is\, $k =
\\log \\log N + O(\\sqrt(\\log \\log N)$) for $n \\leq N$).\n
LOCATION:https://researchseminars.org/talk/NTC/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly Emmrich (Colorado State University)
DTSTART:20230213T190000Z
DTEND:20230213T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/23
DESCRIPTION:Title: The
principal Chebotarev density theorem\nby Kelly Emmrich (Colorado Stat
e University) as part of Lethbridge number theory and combinatorics semina
r\n\nLecture held in University of Lethbridge: M1040 (Markin Hall).\n\nAbs
tract\nLet K/k be a finite Galois extension. We define a principal version
of the Chebotarev density theorem which represents the density of prime i
deals of k that factor into a product of principal prime ideals in K. We f
ind explicit equations to express the principal density in terms of the in
variants of K/k and give an effective bound which can be used to verify th
e non-splitting of the Hilbert exact sequence.\n
LOCATION:https://researchseminars.org/talk/NTC/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No talk - Reading Week
DTSTART:20230220T190000Z
DTEND:20230220T200000Z
DTSTAMP:20260422T212553Z
UID:NTC/24
DESCRIPTION:by No talk - Reading Week as part of Lethbridge number theory
and combinatorics seminar\n\nLecture held in University of Lethbridge: M10
40 (Markin Hall).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTC/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Verret (University of Auckland\, New Zealand)
DTSTART:20230919T200000Z
DTEND:20230919T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/25
DESCRIPTION:Title: Ver
tex-transitive graphs with large automorphism groups\nby Gabriel Verre
t (University of Auckland\, New Zealand) as part of Lethbridge number theo
ry and combinatorics seminar\n\nLecture held in University of Lethbridge:
M1060 (Markin Hall).\n\nAbstract\nMany results in algebraic graph theory c
an be viewed as upper bounds on the size of the automorphism group of grap
hs satisfying various hypotheses. These kinds of results have many applica
tions. For example\, Tutte's classical theorem on 3-valent arc-transitive
graphs led to many other important results about these graphs\, including
enumeration\, both of small order and in the asymptotical sense. This natu
rally leads to trying to understand barriers to this type of results\, nam
ely graphs with large automorphism groups. We will discuss this\, especial
ly in the context of vertex-transitive graphs of fixed valency. We will hi
ghlight the apparent dichotomy between graphs with automorphism group of p
olynomial (with respect to the order of the graph) size\, versus ones with
exponential size.\n
LOCATION:https://researchseminars.org/talk/NTC/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sedanur Albayrak (University of Calgary)
DTSTART:20230926T200000Z
DTEND:20230926T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/26
DESCRIPTION:Title: Qua
ntitative Estimates for the Size of an Intersection of Sparse Automatic Se
ts\nby Sedanur Albayrak (University of Calgary) as part of Lethbridge
number theory and combinatorics seminar\n\nLecture held in University of L
ethbridge: M1060 (Markin Hall).\n\nAbstract\nIn 1979\, Erdős conjectured
that for $k \\geq 9$\, $2^k$ is not the sum of distinct powers of $3$. Tha
t is\, the set of powers of two (which is $2$-automatic) and the $3$-autom
atic set consisting of numbers\nwhose ternary expansions omit $2$ has fini
te intersection. In the theory of automata\, a theorem of Cobham (1969) sa
ys that if $k$ and $\\ell$ are two multiplicatively independent natural nu
mbers then a subset of the natural numbers that is both $k$- and $\\ell$-a
utomatic is eventually periodic. A multidimensional extension was later gi
ven by Semenov (1977). Motivated by Erdős' conjecture and in light of Cob
ham’s theorem\, we give a quantitative version of the Cobham-Semenov the
orem for sparse automatic sets\, showing that the intersection of a sparse
$k$-automatic subset of $\\mathbb{N}^d$ and a sparse $\\ell$-automatic su
bset of $\\mathbb{N}^d$ is finite. Moreover\, we give effectively computab
le upper bounds on the size of the intersection in terms of data from the
automata that accept these sets.\n
LOCATION:https://researchseminars.org/talk/NTC/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kübra Benli (University of Lethbridge)
DTSTART:20231003T200000Z
DTEND:20231003T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/27
DESCRIPTION:Title: Sum
s of proper divisors with missing digits\nby Kübra Benli (University
of Lethbridge) as part of Lethbridge number theory and combinatorics semin
ar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n\nAb
stract\nIn 1992\, Erdős\, Granville\, Pomerance\, and Spiro conjectured t
hat if $\\mathcal{A}$ is a set of integers with asymptotic density zero th
en the preimage of $\\mathcal{A}$ under $s(n)$\, sum-of-proper-divisors fu
nction\, also has asymptotic density zero. In this talk\, we will discuss
the verification of this conjecture when $\\mathcal{A}$ is taken to be the
set of integers with missing digits (also known as ellipsephic integers)
by giving a quantitative estimate on the size of the set $s^{-1}(\\mathcal
{A})$. This is joint work with Giulia Cesana\, Cécile Dartyge\, Charlotte
Dombrowsky and Lola Thompson.\n
LOCATION:https://researchseminars.org/talk/NTC/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (Washington University in St. Louis)
DTSTART:20231012T200000Z
DTEND:20231012T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/28
DESCRIPTION:Title: Bas
ic reductions of abelian varieties\nby Wanlin Li (Washington Universit
y in St. Louis) as part of Lethbridge number theory and combinatorics semi
nar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n\nA
bstract\nGiven an abelian variety $A$ defined over a number field\, a conj
ecture attributed to Serre states\nthat the set of primes at which $A$ adm
its ordinary reduction is of positive density. This conjecture had been pr
oved for elliptic curves (Serre\, 1977)\, abelian surfaces (Katz 1982\, Sa
win 2016) and certain higher dimensional abelian varieties (Pink 1983\, Fi
te 2021\, etc).\n\nIn this talk\, we will discuss ideas behind these resul
ts and recent progress for abelian varieties with non-trivial endomorphism
s\, including the case where $A$ has almost complex multiplication by an a
belian CM field\, based on joint work with Cantoral-Farfan\, Mantovan\, Pr
ies\, and Tang.\n\nApart from ordinary reduction\, we will also discuss th
e set of primes at which an abelian variety admits basic reduction\, gener
alizing a result of Elkies on the infinitude of supersingular primes for e
lliptic curves. This is joint work with Mantovan\, Pries\, and Tang.\n
LOCATION:https://researchseminars.org/talk/NTC/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhenchao Ge (University of Waterloo)
DTSTART:20231017T200000Z
DTEND:20231017T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/29
DESCRIPTION:Title: A W
eyl-type inequality for irreducible elements in function fields\, with app
lications\nby Zhenchao Ge (University of Waterloo) as part of Lethbrid
ge number theory and combinatorics seminar\n\nLecture held in University o
f Lethbridge: M1060 (Markin Hall).\n\nAbstract\nWe establish a Weyl-type e
stimate for exponential sums over irreducible elements in function fields.
As an application\, we generalize an equidistribution theorem of Rhin. Ou
r estimate works for polynomials with degree higher than the characteristi
c of the field\, a barrier to the traditional Weyl differencing method. In
this talk\, we briefly introduce Lê-Liu-Wooley’s original argument for
ordinary Weyl sums (taken over all elements)\, and how we generalize it t
o estimate bilinear exponential sums with general coefficients. This is jo
int work with Jérémy Campagne (Waterloo)\, Thái Hoàng Lê\n(Mississipp
i) and Yu-Ru Liu (Waterloo).\n
LOCATION:https://researchseminars.org/talk/NTC/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu-Ru Liu (University of Waterloo)
DTSTART:20231025T200000Z
DTEND:20231025T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/30
DESCRIPTION:Title: Fer
mat vs Waring: an introduction to number theory in function fields\nby
Yu-Ru Liu (University of Waterloo) as part of Lethbridge number theory an
d combinatorics seminar\n\nLecture held in University of Lethbridge: M1040
(Markin Hall).\n\nAbstract\nLet $\\Z$ be the ring of integers\, and let $
\\mathbb{F}_p[t]$ be the ring of polynomials in one variable defined over
the finite field $\\mathbb{F}_p$ of $p$ elements. Since the characteristic
of $\\Z$ is $0$\, while that of $\\mathbb{F}_p[t]$ is the positive prime
number $p$\, it is a striking theme in arithmetic that these two rings fai
thfully resemble each other. The study of the similarity and difference be
tween $\\Z$ and $\\mathbb{F}_p[t]$ lies in the field that relates number f
ields to function fields. In this talk\, we will investigate some Diophant
ine problems in the settings of $\\Z$ and $\\mathbb{F}_p[t]$\, including F
ermat's Last Theorem and Waring's problem.\n
LOCATION:https://researchseminars.org/talk/NTC/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joy Morris (University of Lethbridge)
DTSTART:20231031T200000Z
DTEND:20231031T210000Z
DTSTAMP:20260422T212553Z
UID:NTC/31
DESCRIPTION:Title: Eas
y Detection of (Di)Graphical Regular Representations\nby Joy Morris (U
niversity of Lethbridge) as part of Lethbridge number theory and combinato
rics seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Ha
ll).\n\nAbstract\nGraphical and Digraphical Regular Representations (GRRs
and DRRs) are a concrete way to visualise the regular action of a group\,
using graphs. More precisely\, a GRR or DRR on the group $G$ is a (di)grap
h whose automorphism group is isomorphic to the regular action of $G$ on i
tself by right-multiplication.\n\nFor a (di)graph to be a DRR or GRR on $G
$\, it must be a Cayley (di)graph on $G$. Whenever the group $G$ admits an
automorphism that fixes the connection set of the Cayley (di)graph setwis
e\, this induces a nontrivial graph automorphism that fixes the identity v
ertex\, which means that the (di)graph is not a DRR or GRR. Checking wheth
er or not there is any group automorphism that fixes a particular connecti
on set can be done very quickly and easily compared with checking whether
or not any nontrivial graph automorphism fixes some vertex\, so it would b
e nice to know if there are circumstances under which the simpler test is
enough to guarantee whether or not the Cayley graph is a GRR or DRR. I wil
l present a number of results on this question.\n\nThis is based on joint
work with Dave Morris and with Gabriel Verret.\n
LOCATION:https://researchseminars.org/talk/NTC/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbas Maarefparvar (University of Lethbridge)
DTSTART:20231107T210000Z
DTEND:20231107T220000Z
DTSTAMP:20260422T212553Z
UID:NTC/32
DESCRIPTION:Title: Som
e Pólya fields of small degrees\nby Abbas Maarefparvar (University of
Lethbridge) as part of Lethbridge number theory and combinatorics seminar
\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n\nAbst
ract\nHistorically\, the notion of Pólya fields dates back to some works
of George Pólya and Alexander Ostrowski\, in 1919\, on entire functions w
ith integervalues at integers\; a number field $K$ with ring of integers
$\\mathcal{O}_K$ is called a Pólya field whenever the $\\mathcal{O}_K$-m
odule $\\{ f \\in K[X] : f(\\mathcal{O}_K ) \\subseteq \\mathcal{O}_K \\}
$ admits an $\\mathcal{O}_K$-basis with exactly one member from each degre
e. Pólya fields can be thought of as a generalization of number fields wi
th class number one\, and their classification of a specific degree has be
come recently an active research subject in algebraic number theory. In th
is talk\, I will present some criteria for $K$ to be a Pólya field. Then
I will give some results concerning Pólya fields of degrees $2\, 3$\, and
$6$.\n
LOCATION:https://researchseminars.org/talk/NTC/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sreerupa Bhattacharjee (University of Lethbridge)
DTSTART:20231121T210000Z
DTEND:20231121T220000Z
DTSTAMP:20260422T212553Z
UID:NTC/34
DESCRIPTION:Title: A S
urvey of Büthe's Method for Estimating Prime Counting Functions\nby S
reerupa Bhattacharjee (University of Lethbridge) as part of Lethbridge num
ber theory and combinatorics seminar\n\nLecture held in University of Leth
bridge: M1060 (Markin Hall).\n\nAbstract\nThis talk will begin with a stud
y on explicit bounds for $\\psi(x)$ starting with the work of Rosser in 19
41. It will also cover various improvements over the years including the w
orks of Rosser and Schoenfeld\, Dusart\, Faber-Kadiri\, Platt-Trudgian\, B
üthe\, and Fiori-Kadiri-Swidinsky. In the second part of this talk\, I wi
ll provide an overview of my master's thesis which is a survey on `Estimat
ing $\\pi(x)$ and Related Functions under Partial RH Assumptions' by Jan B
üthe. This article provides the best known bounds for $\\psi(x)$ for smal
l values of~$x$ in the interval $[e^{50}\,e^{3000}]$. A distinctive featur
e of this paper is the use of Logan's function and its Fourier Transform.
I will be presenting the main theorem in Büthe's paper regarding estimate
s for $\\psi(x)$ with other necessary results required to understand the p
roof.\n
LOCATION:https://researchseminars.org/talk/NTC/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ha Tran (Concordia University of Edmonton)
DTSTART:20231128T210000Z
DTEND:20231128T220000Z
DTSTAMP:20260422T212553Z
UID:NTC/35
DESCRIPTION:Title: The
Size Function For Imaginary Cyclic Sextic Fields\nby Ha Tran (Concord
ia University of Edmonton) as part of Lethbridge number theory and combina
torics seminar\n\nLecture held in University of Lethbridge: M1060 (Markin
Hall).\n\nAbstract\nThe size function $h^0$ for a number field is analogou
s to the dimension of the\nRiemann-Roch spaces of divisors on an algebraic
curve. Van der Geer and Schoof conjectured\nthat $h^0$ attains its maximu
m at the trivial class of Arakelov divisors if that field is Galois over\n
$\\mathbb{Q}$ or over an imaginary quadratic field. This conjecture was pr
oved for all number fields with the unit group of rank $0$ and $1$\, and a
lso for cyclic cubic fields which have unit group of rank\ntwo. In this ta
lk\, we will discuss the main idea to prove that the conjecture also holds
for\ntotally imaginary cyclic sextic fields\, another class of number fie
lds with unit group of rank\ntwo. This is joint work with Peng Tian and Am
y Feaver.\n
LOCATION:https://researchseminars.org/talk/NTC/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hadi Kharaghani (University of Lethbridge)
DTSTART:20240124T204500Z
DTEND:20240124T214500Z
DTSTAMP:20260422T212553Z
UID:NTC/36
DESCRIPTION:Title: Pro
jective Planes and Hadamard Matrices\nby Hadi Kharaghani (University o
f Lethbridge) as part of Lethbridge number theory and combinatorics semina
r\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n\nAbs
tract\nIt is conjectured that there is no projective plane of order 12. Ba
lanced splittable\nHadamard matrices were introduced in 2018. In 2023\, it
was shown that a projective\nplane of order 12 is equivalent to a balance
d multi-splittable Hadamard matrix of\norder 144. There will be an attempt
to show the equivalence in a way that may\nrequire little background.\n
LOCATION:https://researchseminars.org/talk/NTC/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Félix Baril Boudreau (University of Lethbridge)
DTSTART:20240229T204500Z
DTEND:20240229T214500Z
DTSTAMP:20260422T212553Z
UID:NTC/37
DESCRIPTION:Title: The
Distribution of Logarithmic Derivatives of Quadratic L-functions in Posit
ive Characteristic\nby Félix Baril Boudreau (University of Lethbridge
) as part of Lethbridge number theory and combinatorics seminar\n\nLecture
held in University of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nTo ea
ch square-free monic polynomial $D$ in a fixed polynomial ring $\\mathbb{F
}_q[t]$\, we can associate a real quadratic character $\\chi_D$\, and then
a Dirichlet $L$-function $L(s\,\\chi_D)$. We compute the limiting distrib
ution of the family of values $L'(1\,\\chi_D)/L(1\,\\chi_D)$ as $D$ runs t
hrough the square-free monic polynomials of $\\mathbb{F}_q[t]$ and establi
sh that this distribution has a smooth density function. Time permitting\,
we discuss connections of this result with Euler-Kronecker constants and
ideal class groups of quadratic extensions. This is joint work with Amir A
kbary.\n
LOCATION:https://researchseminars.org/talk/NTC/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sho Suda (National Defense Academy of Japan)
DTSTART:20240313T194500Z
DTEND:20240313T204500Z
DTSTAMP:20260422T212553Z
UID:NTC/38
DESCRIPTION:Title: On
extremal orthogonal arrays\nby Sho Suda (National Defense Academy of J
apan) as part of Lethbridge number theory and combinatorics seminar\n\nLec
ture held in University of Lethbridge: M1060 (Markin Hall).\n\nAbstract\nA
n orthogonal array with parameters $(N\,n\,q\,t)$ ($OA(N\,n\,q\,t)$ for sh
ort) is an $N\\times n$ matrix with entries from the alphabet $\\{1\,2\,..
.\,q\\}$ such that in any its $t$ columns\, all possible row vectors of le
ngth $t$ occur equally often. \nRao showed the following lower bound on $N
$ for $OA(N\,n\,q\,2e)$: \n\\[\nN\\geq \\sum_{k=0}^e \\binom{n}{k}(q-1)^k\
, \n\\]\nand an orthogonal array is said to be complete or tight if it ach
ieves equality in this bound. \nIt is known by Delsarte (1973) that for co
mplete orthogonal arrays $OA(N\,n\,q\,2e)$\, the number of Hamming distanc
es between distinct two rows is $e$. \nOne of the classical problems is to
classify complete orthogonal arrays. \n\nWe call an orthogonal array $OA
(N\,n\,q\,2e-1)$ extremal if the number of Hamming distances between disti
nct two rows is $e$. \nIn this talk\, we review the classification proble
m of complete orthogonal arrays with our contribution to the case $t=4$ an
d show how to extend it to extremal orthogonal arrays. \nMoreover\, we giv
e a result for extremal orthogonal arrays which is a counterpart of a resu
lt in block designs by Ionin and Shrikhande in 1993.\n
LOCATION:https://researchseminars.org/talk/NTC/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ertan Elma (University of Lethbridge)
DTSTART:20240131T204500Z
DTEND:20240131T214500Z
DTSTAMP:20260422T212553Z
UID:NTC/39
DESCRIPTION:Title: A D
iscrete Mean Value of the Riemann Zeta Function and its Derivatives\nb
y Ertan Elma (University of Lethbridge) as part of Lethbridge number theor
y and combinatorics seminar\n\nLecture held in University of Lethbridge: M
1060 (Markin Hall).\n\nAbstract\nIn this talk\, we will discuss an estimat
e for a discrete mean value of the Riemann zeta function and its derivativ
es multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis\,
we obtain a lower bound for the 2kth moment of all the derivatives of the
Riemann zeta function evaluated at its nontrivial zeros. This is based on
a joint work with Kübra Benli and Nathan Ng.\n
LOCATION:https://researchseminars.org/talk/NTC/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samprit Ghosh (University of Calgary)
DTSTART:20240207T204500Z
DTEND:20240207T214500Z
DTSTAMP:20260422T212553Z
UID:NTC/40
DESCRIPTION:Title: Mom
ents of higher derivatives related to Dirichlet L-functions\nby Sampri
t Ghosh (University of Calgary) as part of Lethbridge number theory and co
mbinatorics seminar\n\nLecture held in University of Lethbridge: M1060 (Ma
rkin Hall).\n\nAbstract\nThe distribution of values of Dirichlet $L$-funct
ions $L(s\, \\chi)$ for variable $\\chi$ has been studied extensively and
has a vast literature. Moments of higher derivatives has been studied as
well\, by Soundarajan\, Sono\, Heath-Brown etc. However\, the study of th
e same for the logarithmic derivative $L’(s\, \\chi)/ L(s\, \\chi)$ is m
uch more recent and was initiated by Ihara\, Murty etc. In this talk we wi
ll discuss higher derivatives of the logarithmic derivative and present so
me new results related to their distribution and moments at $s=1$.\n
LOCATION:https://researchseminars.org/talk/NTC/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbas Maarefparvar (University of Lethbridge)
DTSTART:20240214T204500Z
DTEND:20240214T214500Z
DTSTAMP:20260422T212553Z
UID:NTC/41
DESCRIPTION:Title: Hil
bert Class Fields and Embedding Problems\nby Abbas Maarefparvar (Unive
rsity of Lethbridge) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).
\n\nAbstract\nThe class number one problem is one of the central subjects
in algebraic number theory that turns back to the time of Gauss. This prob
lem has led to the classical embedding problem which asks whether or not a
ny number field K can be embedded in a finite extension L with class numbe
r one. Although Golod and Shafarevich gave a counterexample for the classi
cal embedding problem\, yet one may ask about the embedding in 'Polya fiel
ds'\, a special generalization of class number one number fields. The latt
er is the 'new embedding problem' investigated by Leriche in 2014.\nIn thi
s talk\, I briefly review some well-known results in the literature on the
embedding problems. Then\, I will present the 'relativized' version of th
e new embedding problem studied in a joint work with Ali Rajaei.\n
LOCATION:https://researchseminars.org/talk/NTC/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Fiori (University of Lethbridge)
DTSTART:20240306T204500Z
DTEND:20240306T214500Z
DTSTAMP:20260422T212553Z
UID:NTC/42
DESCRIPTION:Title: Tig
ht approximation of sums over zeros of L-functions\nby Andrew Fiori (U
niversity of Lethbridge) as part of Lethbridge number theory and combinato
rics seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Ha
ll).\n\nAbstract\nIn various contexts explicit formula’s relate sums ove
r primes (eg: numbers or ideals) to sums over zeros of some corresponding
L-function(s). The aim of this talk is to explain how we tightly approxima
te these sums over zeros in the context where one has zero free regions an
d zero density results for the corresponding L-function(s) and how we use
this to get essentially best possible bounds for the error term in the pri
me number theorem.\n\nThis talk discusses joint work with Habiba Kadiri an
d Joshua Swidinsky as well as ongoing work with Mikko Jaskari and Nizar Bo
u Ezz.\n
LOCATION:https://researchseminars.org/talk/NTC/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Dijols (University of British Columbia)
DTSTART:20240320T194500Z
DTEND:20240320T204500Z
DTSTAMP:20260422T212553Z
UID:NTC/44
DESCRIPTION:Title: Par
abolically induced representations of p-adic $G_2$ distinguished by $SO_4$
\nby Sarah Dijols (University of British Columbia) as part of Lethbrid
ge number theory and combinatorics seminar\n\nLecture held in University o
f Lethbridge: M1060 (Markin Hall).\n\nAbstract\nI will explain how the Geo
metric Lemma allows us to classify parabolically induced representations o
f the p-adic group $G_2$ distinguished by $SO_4$. In particular\, I will d
escribe a new approach\, in progress\, where we use the structure of the p
-adic octonions and their quaternionic subalgebras to describe the double
coset space $P \\setminus G_2 / SO_4$\, where $P$ stands for the maximal p
arabolic subgroups of $G_2$.\n
LOCATION:https://researchseminars.org/talk/NTC/44/
END:VEVENT
END:VCALENDAR