Department of Mathematics and Statistical Science Algebra Seminar Series
About the Seminar
The Department of Mathematics and Statistical Science Algebra Seminar is a graduate student led seminar series that provides graduate students and advanced undergraduate students in mathematics the opportunity to give professional mathematical talks. These talks are open to any who would like to attend, and their content reflects a variety of mathematical disciplines.
Times and Locations
For Spring 2023, the Algebra Seminar meets on Tuesdays, from 1:00 to 1:50 p.m., on Zoom. It will also meet in person in TLC 029.
- We will use the same Zoom Link each week. Passcode: 818108
- Check the schedule below for more information.
Date |
Speaker / Title / Abstract |
|---|---|
| Current Semester: | |
| February 28, 2023 at 1:00 pm |
Dr. Alex Woo (Univ of Idaho) The cohomology of Hessenberg varieties Abstract: I will describe two perspectives on the cohomology of Hessenberg varieties. The first is GKM (Goresky-Kottwicz-Macpherson) theory, which gives a combinatorial description of (equivariant) cohomology as the space of labellings of the vertices of a graph by polynomials satisfying certain conditions. The second is a geometric basis given by the closures of the intersections of Hessenberg varieties with Schubert cells. Hopefully I will get to some ongoing joint work with Erik Insko (Florida Gulf Coast University) and Martha Precup (Washington University in St. Louis) to understand this geometric basis in certain cases and also outline an approach to understanding the second perspective in terms of the first. |
| March 7, 2023 at 1:00 pm | Dr. Brooks Roberts (Univ of Idaho) Dirichlet characters and Dirichlet's theorem Abstract: In this expository talk we will recall the L-series associated to a Dirichlet character, and we will describe how these L-series can be used to prove Dirichlet's famous theorem on primes in an arithmetic progression. We will also indicate how this theorem implies a basic property of a Dirichlet character when viewed as an adelic automorphic form. |
| March 21, 2023 at 1:00 pm |
Dr. Hirotachi Abo (Univ of Idaho) Algebro-geometric approaches to mixed Nash equilibria Abstract: Mixed Nash equilibrium is a concept in game theory that determines an optimal solution of a non-cooperative finite game. Using so-called payoff tensors (that is, tensors which express the possible choices for the players as well as the outcomes of such choices), one can interpret mixed Nash equilibria as points in the tensor space (called Nash points). In this talk, we discuss an algebro-geometric interpretation of the tight upper bound for the number of Nash points for a “generic” game (that is, the game with generic payoff tensors) obtained by R. McKelvey and A. McLennan. The formula for the upper bound indicates when a generic game has no Nash points. If a generic game has no Nash points, then the payoff tensors, with which the game has Nash points, form a subvariety. If time permits, I will talk about the geometry of such a subvariety. This is a preliminary report of the (still ongoing) project with Luca Sodomaco and Irem Portakal. |
| March 28, 2023 at 1:00 pm |
Dalton Bidleman (Auburn University) Restricted Secants of Grassmannians Abstract: Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmanians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay 2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory. |
| April 4, 2023 at 1:00 pm | Shiliang Gao (Univ of Illinois at Urbana-Champaign) |
| May 2, 2023 |
Social Event - Details TBA |
Past Semesters: |
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| January 18, 2022 at 11:00 am Virtual meeting via Zoom |
Dr. Stefan Tohaneanu (Univ Idaho) Ideals generated by fold products of linear forms Abstract: On method for computing the minimum distance of a linear code is to check if certain projective schemes are empty or not. The defining ideals of these schemes are ideals generated by fold products of the linear forms that are dual to the generating matrix of the linear code. Geometrically, the projective varieties defined by these ideals are union of subspaces of various dimensions, and they generalize the more classical objects known as star configurations. In this talk I will review some (most) of the algebraic properties of these ideals, taking the chance to introduce the audience to some of the classical concepts in commutative algebra. |
| Please note: Special time and location Friday, February 11 at 4:00 pm In TLC 022 and via Zoom |
Dr. Alex Woo (Univ Idaho) From order statistics to knot theory Abstract: Suppose one wishes to do statistics purely with rank data, inferring the order certain items come in where the only ‘experimental’ data available consists of the order these items come in under various ‘measurements’. Then one needs some notion of ‘distance’ between orderings, or, equivalent, between permutations. Three such notions of distance are known to statisticians as Spearman’s footrule, Kendall’s tau, and transposition distance, and to combinatorialists respectively as total displacement, inversion number, and reflection length. Diaconis and Graham showed in the 1970s that Spearman’s footrule is at least the sum of Kendall’s tau and transposition distance, and asked for a characterization of equality. Recently, Cornwell and McNew gave a way of associating a knot or link to a permutation. Using the results of Cornwell and McNew as well as a recursive answer, due to Hadjicostas and Monico, to the question of Diaconis and Graham, we show that the permutations satisfying equality are precisely the ones associated to an unlinked collection of unknotted loops. |
| February 22, 2022 | Dr. Jennifer Johnson-Leung (Univ Idaho) Details TBA Hecke characters, L-functions, and the ideles Abstract: We will give a motivation of the ideles by considering the generalizations of Dirichlet L-functions to characters of ideals in number fields. As time allows, we will explain how these Hecke characters also can arise in the study certain complex surfaces. This is an expository talk and all are welcome. |
| March 1, 2022 | Dr. Ben Breen (Clemson Univ) Computing Hilbert modular forms using a trace formula. Abstract: We present an explicit method for computing spaces of Hilbert modular forms using a trace formula. We describe the main algorithmic challenges and discuss the advantages and shortcomings of this method in comparison to other methods for computing spaces of Hilbert modular forms. We conclude with computations. |
| March 22, 2022 | We will watch the Clay lectures given by Fields medalist Akshay Venkatesh at the Arizona Winter School earlier this month. |
| March 29, 2022 | We will continue to watch the Clay lectures given by Fields medalist Akshay Venkatesh at the Arizona Winter School earlier this month. |
| April 5, 2022 | Dr. Brooks Roberts (Univ Idaho) The group law on binary quadratic forms: Bhargava's new perspective Abstract: The study of integral binary quadratic forms has contributed to the development of many parts of mathematics including algebraic number theory, abelian class field theory, and complex multiplication. As part of this development, Gauss and Dirichlet showed that the set of equivalence classes of primitive integral binary quadratic forms of fixed discriminant has a group law making this set into a finite abelian group. Beginning with his 2001 PhD thesis, Manjul Bhargava presented a new way to describe this group law, and found thirteen other higher composition laws. Bhargava was awarded the Fields Medal in 2014. In this talk we will review Bhargava's remarkable account of the group law for integral primitive binary quadratic forms. |
| April 19, 2022 | Jordan Hardy (Univ Idaho) Chains of Large Prime Gaps and Sieve Methods Abstract: Within Mathematics, number theory is a discipline that has become infamous for having problems that are easy to state in a form understandable to most members of the general public, but require incredibly sophisticated mathematical techniques to solve. In this talk, we examine a few versions of a simple question: "How big can the gap between consecutive prime numbers be and how many 'big' gaps can occur in a row?." The easiest versions of this problem, we will see are an accessible exercise for students taking their first course in number theory while more advanced versions have only been tackled using advances in Sieve methods made during the late 20th and early 21st century. |
| April 26, 2022 | Jordan Hardy (Univ Idaho) Some Conjectures About Nonprincipally Polarized Abelian Surfaces With Complex Multiplication Abstract: The theory of abelian varieties with complex multiplication was developed for abelian varieties of arbitrary genus in a book called Complex Multiplication of Abelian Varieties and Its Applications to Number Theory in 1961. This is a powerful theory which identifies certain special abelian varieties whose endomorphism rings are as big as possible. These abelian varieties turn out to be computationally easier to do with, and they correspond to special points in the siegel modular variety. When you evaluate a siegel modular function at these points you get algebraic numbers which can be used to generate class fields over the reflex field of CM fields. Almost all of the theory that has been developed in this area concerns abelian varieties which carry a principal polarization. What happens when we shift our focus to abelian varieties with nonprincipal polarizations? Earlier, I developed the theory to identify when it is possible to give an abelian surface with complex multiplication by a given CM field, but the criterion developed is clunky and difficult to quickly verify. In this talk we will discuss some conjectures about simpler necessary and sufficient conditions for there to be an abelian variety with complex multiplication by a given CM field which carries a polarization of a desired type. |
| May 3, 2022 | Jake Sapozhnikov (Univ Idaho) Bipackings of Bipartite Graphs Abstract: Let G = G(X_1,X_2) be a bipartite graph. Consider the partite-preserving functions from G to a complete bipartite graph B_{n+1} with the same vertex set and one extra vertex added to one of the partites. We say such a function f is a bipacking of G into B_{n+1} if no edges of G and f(G) coincide. In 2019, Dr. Hong Wang proved that there exists a bipacking for every bipartite graph of girth at least 12 and conjectured that this property holds for all bipartite graphs of girth at least 8. We will discuss the background of the problem, the techniques from Wang(2019), and partial progress toward a proof of the tighter bounds using these techniques. |