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X-WR-CALNAME:Claremont Center for the Mathematical Sciences
X-ORIGINAL-URL:https://colleges.claremont.edu/ccms
X-WR-CALDESC:Events for Claremont Center for the Mathematical Sciences
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221206T121500
DTEND;TZID=America/Los_Angeles:20221206T131000
DTSTAMP:20221130T053013Z
CREATED:20221130T053013Z
LAST-MODIFIED:20221130T053013Z
UID:10001713-1670328900-1670332200@colleges.claremont.edu
SUMMARY:Positive semigroups in lattices and totally real number fields (Lenny Fukshansky\, CMC)
DESCRIPTION:Let  L be a full-rank lattice in R^n and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases\, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L+\, and we investigate the distribution of the gaps of S(X) in L+\, i.e. points in L+ outside of S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of these sets\, for which we produce bounds in the spirit of Minkowski’s successive minima theorem. We apply these results to obtain analogous bounds for the successive minima with respect to Weil height of totally positive sub-semigroups of ideals in totally real number fields. Joint work with Siki Wang (CMC’22).
URL:https://colleges.claremont.edu/ccms/event/positive-semigroups-in-lattices-and-totally-real-number-fields-lenny-fukshansky-cmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221115T121500
DTEND;TZID=America/Los_Angeles:20221115T131000
DTSTAMP:20221102T220943Z
CREATED:20220823T003904Z
LAST-MODIFIED:20221102T220943Z
UID:10001704-1668514500-1668517800@colleges.claremont.edu
SUMMARY:Minimal Mahler measure in number fields (Kate Petersen\, University of Minnesota Duluth)
DESCRIPTION:The Mahler measure of a polynomial is the modulus of its leading term multiplied by the moduli of all roots outside the unit circle.  The Mahler measure of an algebraic number b\, M(b) is the Mahler measure of its minimal polynomial. By a result of Kronecker\, an algebraic number b satisfies M(b)=1 if and only if b is a root of unity. Famously\, Lehmer asked if there are algebraic numbers with Mahler measures arbitrarily close to 1 (but not equal to 1). We will investigate the minimal Mahler measure of a number field.  For a number field K this is the smallest Mahler measure of a non-torsion generator for K\, written M(K). There are known upper and lower bounds for M(K) in terms of the degree and discriminant of K.  Focusing on cubics\, we will discuss how these bounds correspond to other properties of the number field\, and the sharpness of these bounds.  This is joint work with Lydia Eldredge.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-kate-petersen-university-of-minnesota-duluth/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221108T121500
DTEND;TZID=America/Los_Angeles:20221108T131000
DTSTAMP:20221031T203245Z
CREATED:20220824T204820Z
LAST-MODIFIED:20221031T203245Z
UID:10001705-1667909700-1667913000@colleges.claremont.edu
SUMMARY:Factoring translates of polynomials (Arvind Suresh\, University of Arizona - Tucson)
DESCRIPTION:Given a degree d polynomial f(x) in Q[x]\, consider the subset S_f  of Q consisting of rational numbers t for which the translated polynomial f(x) – t factors completely in Q[x]. For example\, if f is linear or quadratic then S_f is always infinite\, but if degree of f is at least 3\, then interesting things can happen. In this talk\, we discuss a connection between the set S_f and the classical Prouhet–Tarry–Escott problem (which asks for integer solutions to certain symmetric family of equations)\, and we present two infinite families of polynomials f for which S_f is infinite (upon replacing Q with certain number fields). Time permitting\, we outline how these can then be used to produce algebraic curves over number fields having a record number of rational points (relative to their genus).
URL:https://colleges.claremont.edu/ccms/event/antc-talk-arvind-suresh-university-of-arizona-tucson/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221101T121500
DTEND;TZID=America/Los_Angeles:20221101T131000
DTSTAMP:20221031T180722Z
CREATED:20220906T211012Z
LAST-MODIFIED:20221031T180722Z
UID:10001711-1667304900-1667308200@colleges.claremont.edu
SUMMARY:A tale of two worlds: parking functions &  reduction algebras (Dwight Anderson Williams II\, Pomona)
DESCRIPTION:“A Tale of Two Cities” is a novel told in three books/parts. Here we describe three projects related both to published work and ongoing pieces: \nPROJECT 1: In the world of combinatorics\, parking functions are combinatorial objects arising from the situation of parking cars under a parking strategy. In this part of the talk\, we will refresh the notion of classical parking functions given by the classical parking rules/strategy. We will then state an interesting correspondence between certain classical parking functions and so-called ideal states of the famous Tower of Hanoi game. This work is to appear in The American Mathematical Monthly with the following co-authors: Y. Aguillon\, D. Alvarenga\, P.E. Harris\, S. Kotapati\, J.C. Martinez Mori\, C. Monroe\, Z. Saylor\, and C. Tieu. \nPROJECT 2: In the world of algebra\, we shed light on representation theory of Lie superalgebras by constructing reduction algebras. These algebras provide structures to study in their own right\, and we give an example in presenting the diagonal reduction algebra of $osp(1|2)$\, first described in a joint paper with Jonas T. Hartwig. \nPROJECT 3: Continuing down an algebraic pathway\, we summarize the general framework given by Zhelobenko to apply representation theory of reduction algebras as a method to solve equations. Fixing equations important to the study of physics has led to recent work with Jonas T. Hartwig and Erin Dolecheck\, as well\, Irmak Bukey.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-dwight-anderson-williams-ii-pomona/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221025T121500
DTEND;TZID=America/Los_Angeles:20221025T131000
DTSTAMP:20221012T181637Z
CREATED:20220906T160323Z
LAST-MODIFIED:20221012T181637Z
UID:10001709-1666700100-1666703400@colleges.claremont.edu
SUMMARY:Properties of redistricting Markov chains (Sarah Cannon\, CMC)
DESCRIPTION:Markov chains have become widely-used to generate random political districting plans. These random districting plans can be used to form a baseline for comparison\, and any proposed districting plans that differ significantly from this baseline can be flagged as potentially gerrymandered. However\, very little is rigorously known about these Markov chains – Are they irreducible? What is their mixing time? For some\, even the stationary distribution remains unknown. I will present recent work that answers some of these questions\, which uses tools from probability\, computational geometry\, and more.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-sarah-cannon-cmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221011T121500
DTEND;TZID=America/Los_Angeles:20221011T131000
DTSTAMP:20221004T211454Z
CREATED:20220825T192011Z
LAST-MODIFIED:20221004T211454Z
UID:10001706-1665490500-1665493800@colleges.claremont.edu
SUMMARY:On the geometry of lattice extensions (Max Forst\, CGU)
DESCRIPTION:Given a lattice L\, an extension of L is a lattice M of strictly greater rank so that L is equal to the intersection of the subspace spanned by L with M. In this talk\, we will discus constructions of such lattice extensions with particular geometric invariants of M\, such as the determinant\, covering radius and successive minima related to the analogous invariants of L. Joint work with Lenny Fukshansky.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-max-forst-cgu/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20221004T121500
DTEND;TZID=America/Los_Angeles:20221004T131000
DTSTAMP:20221003T234340Z
CREATED:20220829T210323Z
LAST-MODIFIED:20221003T234340Z
UID:10001707-1664885700-1664889000@colleges.claremont.edu
SUMMARY:Recent developments on the slice rank polynomial method with applications (Mohamed Omar\, HMC)
DESCRIPTION:The slice rank polynomial method\, motivated by groundbreaking work of Croot\, Lev and Pach and refined by Tao\, has opened the door to the resolution of many problems in extremal combinatorics. We survey these results and discuss contributions in several of the speaker’s recent papers.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-mohamed-omar-hmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20220927T121500
DTEND;TZID=America/Los_Angeles:20220927T131000
DTSTAMP:20220922T053209Z
CREATED:20220906T160640Z
LAST-MODIFIED:20220922T053209Z
UID:10001710-1664280900-1664284200@colleges.claremont.edu
SUMMARY:Spinning switches on a wreath product (Peter Kagey\, HMC)
DESCRIPTION:This talk discusses a puzzle called “Spinning Switches\,” based on a problem popularized by Martin Gardner in his February 1979 column of “Mathematical Games”. This puzzle can be generalized to a two-player game on a finite wreath products. This talk will provide a classification of several families of these generalized puzzles\, including a full classification in the case of Abelian groups.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-peter-kagey-hmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20220920T121500
DTEND;TZID=America/Los_Angeles:20220920T131000
DTSTAMP:20220906T231455Z
CREATED:20220811T002022Z
LAST-MODIFIED:20220906T231455Z
UID:10001694-1663676100-1663679400@colleges.claremont.edu
SUMMARY:Arithmetical structures (Luis Garcia Puente\, Colorado College)
DESCRIPTION:An arithmetical structure on a finite\, connected graph G without loops is given by an assignment of positive integers to the vertices such that\, at each vertex\, the integer there is a divisor of the sum of the integers at adjacent vertices\, counted with multiplicity if the graph is not simple. Alternatively\,  an arithmetical structure on G is a pair  of positive integer vectors (d\,r) such that  Mr = 0\, where M = diag(d) – A  is a square matrix whose diagonal entries are given by the vector d\, and whose off-diagonal elements are given by the negative adjacency matrix of G. Arithmetical structures were first introduced by Lorenzini in 1989; matrices of the form (diag(d) – A) arise in algebraic geometry as intersection matrices of degenerating curves.  However\, they also naturally appear in the context of algebraic graph theory as matrices of the form  (diag(d) – A)  generalize the Laplacian matrix of a graph.\n\nIn this talk\, I will give an introduction to the topic. We will discuss some combinatorial\, structural and computational aspects of arithmetical structures. In particular\, we will count the number of distinct arithmetical structures on certain graph families such as path\, cycle\, complete and bident graphs. For paths\, we will show that arithmetical structures are enumerated by the Catalan numbers. For cycles\, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1\,n-1).  We will also discuss results about the associated critical group of an arithmetical structure\, i.e.\,  the cokernel of the matrix M.   This talk will be accessible to undergraduate students with some knowledge of linear algebra and discrete mathematics.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-luis-garcia-puente-colorado-college/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20220913T121500
DTEND;TZID=America/Los_Angeles:20220913T131000
DTSTAMP:20220906T231347Z
CREATED:20220902T001706Z
LAST-MODIFIED:20220906T231347Z
UID:10001708-1663071300-1663074600@colleges.claremont.edu
SUMMARY:Kriz's theorem via dynamics of linear operators (Yunied Puig de Dios\, CMC)
DESCRIPTION:The existence of a set $A\subset \N_0$ of positive upper Banach density such that $A-A:=\{m-n:m\, n\in A\, m>n\}$ does not contain a set of the form $S-S$ with $S$ a piecewise syndetic is in essence the content of a popular result due to K\v r\'{i}\v z in 1987. Since then at least four different proofs of this result have been given\, and all of them give basically the example originally exhibited by K\v r\'{i}\v z when viewed appropriately. We obtain a generalization of K\v r\'{i}\v z’s result. Our approach differs completely from the previous ones\, as this would be the first proof of K\v r\'{i}\v z’s Theorem which does not rely on Lov\'{a}sz’s Theorem for chromatic numbers of Kneser graphs. Furthermore\, it is done via operator theory\, namely using dynamics of bounded linear operators on infinite-dimensional complex separable Banach spaces. As a consequence\, our example is genuinely different from the one exhibited  originally by K\v r\'{i}\v z.
URL:https://colleges.claremont.edu/ccms/event/krizs-theorem-via-dynamics-of-linear-operators-yunied-puig-de-dios-cmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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