BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Dang-Khoa Nguyen (University of Calgary) DTSTART;VALUE=DATE-TIME:20220926T180000Z DTEND;VALUE=DATE-TIME:20220926T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221031T180000Z DTEND;VALUE=DATE-TIME:20221031T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221117T210000Z DTEND;VALUE=DATE-TIME:20221117T220000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221128T190000Z DTEND;VALUE=DATE-TIME:20221128T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221205T190000Z DTEND;VALUE=DATE-TIME:20221205T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221003T180000Z DTEND;VALUE=DATE-TIME:20221003T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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 $\\mathbb{Q}(\\sqrt{p}\,\\sqrt{-q})$ 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;VALUE=DATE-TIME:20221017T180000Z DTEND;VALUE=DATE-TIME:20221017T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221024T180000Z DTEND;VALUE=DATE-TIME:20221024T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20221121T190000Z DTEND;VALUE=DATE-TIME:20221121T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230327T180000Z DTEND;VALUE=DATE-TIME:20230327T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230320T180000Z DTEND;VALUE=DATE-TIME:20230320T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230206T190000Z DTEND;VALUE=DATE-TIME:20230206T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230313T180000Z DTEND;VALUE=DATE-TIME:20230313T190000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230109T190000Z DTEND;VALUE=DATE-TIME:20230109T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230123T163000Z DTEND;VALUE=DATE-TIME:20230123T173000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230116T163000Z DTEND;VALUE=DATE-TIME:20230116T173000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230130T190000Z DTEND;VALUE=DATE-TIME:20230130T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230227T163000Z DTEND;VALUE=DATE-TIME:20230227T173000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230306T190000Z DTEND;VALUE=DATE-TIME:20230306T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230206T221000Z DTEND;VALUE=DATE-TIME:20230206T234500Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230209T221000Z DTEND;VALUE=DATE-TIME:20230209T234500Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230403T163000Z DTEND;VALUE=DATE-TIME:20230403T173000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230213T190000Z DTEND;VALUE=DATE-TIME:20230213T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230220T190000Z DTEND;VALUE=DATE-TIME:20230220T200000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230919T200000Z DTEND;VALUE=DATE-TIME:20230919T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20230926T200000Z DTEND;VALUE=DATE-TIME:20230926T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z 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;VALUE=DATE-TIME:20231003T200000Z DTEND;VALUE=DATE-TIME:20231003T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/27 DESCRIPTION:by Kübra Benli (University of Lethbridge) as part of Lethbrid ge number theory and combinatorics seminar\n\nLecture held in University o f Lethbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/27/ END:VEVENT BEGIN:VEVENT SUMMARY:TBA DTSTART;VALUE=DATE-TIME:20231010T200000Z DTEND;VALUE=DATE-TIME:20231010T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/28 DESCRIPTION:by TBA as part of Lethbridge number theory and combinatorics s eminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n Abstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Hiva Gheisari (University of Lethbridge) DTSTART;VALUE=DATE-TIME:20231017T200000Z DTEND;VALUE=DATE-TIME:20231017T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/29 DESCRIPTION:by Hiva Gheisari (University of Lethbridge) as part of Lethbri dge number theory and combinatorics seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Yu-Ru Liu (University of Waterloo) DTSTART;VALUE=DATE-TIME:20231025T200000Z DTEND;VALUE=DATE-TIME:20231025T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/30 DESCRIPTION:by Yu-Ru Liu (University of Waterloo) as part of Lethbridge nu mber theory and combinatorics seminar\n\nLecture held in University of Let hbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Joy Morris (University of Lethbridge) DTSTART;VALUE=DATE-TIME:20231031T200000Z DTEND;VALUE=DATE-TIME:20231031T210000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/31 DESCRIPTION:by Joy Morris (University of Lethbridge) as part of Lethbridge number theory and combinatorics seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Abbas Maarefparvar (University of Lethbridge) DTSTART;VALUE=DATE-TIME:20231107T210000Z DTEND;VALUE=DATE-TIME:20231107T220000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/32 DESCRIPTION:by Abbas Maarefparvar (University of Lethbridge) as part of Le thbridge number theory and combinatorics seminar\n\nLecture held in Univer sity of Lethbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Sreerupa Bhattacharjee (University of Lethbridge) DTSTART;VALUE=DATE-TIME:20231121T210000Z DTEND;VALUE=DATE-TIME:20231121T220000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/34 DESCRIPTION:by Sreerupa Bhattacharjee (University of Lethbridge) as part o f Lethbridge number theory and combinatorics seminar\n\nLecture held in Un iversity of Lethbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Ha Tran (Concordia University of Edmonton) DTSTART;VALUE=DATE-TIME:20231128T210000Z DTEND;VALUE=DATE-TIME:20231128T220000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/35 DESCRIPTION:by Ha Tran (Concordia University of Edmonton) as part of Lethb ridge number theory and combinatorics seminar\n\nLecture held in Universit y of Lethbridge: M1060 (Markin Hall).\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/35/ END:VEVENT BEGIN:VEVENT SUMMARY:TBA DTSTART;VALUE=DATE-TIME:20231205T210000Z DTEND;VALUE=DATE-TIME:20231205T220000Z DTSTAMP;VALUE=DATE-TIME:20230925T235516Z UID:NTC/36 DESCRIPTION:by TBA as part of Lethbridge number theory and combinatorics s eminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n Abstract: TBA\n LOCATION:https://researchseminars.org/talk/NTC/36/ END:VEVENT END:VCALENDAR