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 faculty, 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 2024, the Algebra Seminar meets on Thursdays, from 2:10 to 3:00 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:  
January 25, 2024 at 2:10 pm 
Dr. Arpan Pal (University of Idaho, Institute for Interdisciplinary Data Sciences) Symmetry Lie Algebras of Varieties with Applications to Algebraic Statistics Abstract: The motivation for this project is to detect when an irreducible projective variety V is not toric. This is achieved by analyzing the Lie group and Lie algebra associated with V. If the dimension of V is strictly less than the dimension of the aforementioned objects, then V is not a toric variety. I'll briefly discuss an algorithm to compute the Lie algebra of an irreducible variety and use it to present examples of nontoric staged tree models in algebraic statistics. 
February 15, 2024 at 2:10 pm 
Dr. Michael Allen (Louisiana State University) Hypergeometric Functions and Explicit Results in Modularity Abstract: Under certain conditions, it is often the case that the value of a given hypergeometric series truncated at p1 agrees modulo p with the pth Fourier coefficient of a prescribed modular form for all but the first few primes p. In rarer instances, these congruences hold modulo higher powers of p. As an immediate application, these supercongruences  along with the WeilDeligne bounds  yield exceptionally efficient computations for Fourier coefficients of modular forms. We give a broad overview of results and techniques in this area of hypergeometric supercongruences. We then discuss ongoing work in progress with Brian Grove, Ling Long, and FangTing Tu utilizing supercongruences, Ramanujan's theory of alternative bases, and commutative formal group laws to produce explicit modularity results for certain hypergeometric Galois representations before finishing with a few explicit applications. 
February 22, 2024 at 2:10 pm 
Dr. Brooks Roberts An introduction to modular forms Abstract: In this talk we will motivate the definition of modular forms by considering an interesting example from number theory. Besides presenting the definition, we will also describe the connection between modular forms and the representation theory of GL(2). As a final enticement, we will mention the astounding connection, via string theory, between certain modular functions and the representation theory of sporadic finite simple groups. 
February 29, 2024 at 2:10 pm 
Arthur Huey (University of Idaho) Computing the minimal complementary dual order ideals of principal order ideals in Coxeter systems of type D Abstract: This presentation is designed to be accessible to anyone with basic knowledge of the symmetric group. An introduction to Coxeter systems and the Bruhat order is included. GIven any sigma in S_n the problem of finding the minimal noncomparable elements is well understood. This problem is now considered in a more complicated subgroup of the signed permutations. 
March 21, 2024 at 2:10 pm 
Dr. Jordan Hardy (University of Idaho) Galois Groups of CM Fields of Low Degree Abstract: The aim of this talk is to explore the possible Galois groups that a CMfield of low degree can have. A full description of the possibilities for a quartic CMfield's Galois group will be given, and the key ideas of how to determine possible Galois groups for CMfields of higher degree is described, but the details are much more complicated, but we will engage in some exploration of the possibilities for sextic CMfields. 
March 28, 2024 at 2:10 pm 
Dr. Faqruddin Azam (LewisClark State College) Divisor labeling of staircase diagrams and fiber bundle structures on Schubert varieties Abstract: Let Gr (r,n) denote the Grassmannian of rdimensional subspaces of C^n. Each Gr (r,n) contains a unique codimension1 Schubert subvariety called the Schubert divisor of the Grassmannian. In this talk, we will discuss the correspondence between the set of permutations avoiding the patterns 3412, 52341, 52431 and 53241, and the set of Schubert varieties in the complete flag variety which are iterated fiber bundles of Grassmannians or Grassmannian Schubert divisors. Using this geometrical structure, we calculate the generating function that enumerates the permutations avoiding these patterns. 
April 4, 2024 at 2:10 pm 
Troy Rice (University of Idaho) Groebner Basis and their Properties Abstract: In this talk we will consider the Ideal Membership Problem in k[x_1, ..., x_n] where k is a field. We will introduce the solution to this problem in the single variate case using the division algorithm for single variate polynomials, and seek to abstract this idea to the multivariate case. We will introduce the division algorithm in k[x_1 ..., x_n] which will lead us to the definition of a Groebner Basis for an ideal in k[x_1 ... x_n]. We will discuss properties of Groebner Bases including their relationship with the division algorithm, and the Ideal Membership Problem in k[x_1, ..., x_n]. 
April 11, 2024 at 2:10 pm 
Dr. Alexander Woo (University of Idaho) An affine paving for DeltaSpringer fibers Abstract: In some joint work with Sean Griffin and Jake Levinson, we needed an affine paving for DeltaSpringer fibers to compute the dimension of their cohomology rings. I will try to give some idea of what all the words in the previous sentence mean, and then show some of the details of the proof that the intersection of DeltaSpringer fibers with Schubert cells gives such an affine paving. (After untangling the definitions, this uses only elementary linear algebra.) 
April 25, 2024 at 2:10 pm 
Jonathan Webb (University of Idaho) Constructing binary Parseval frames Abstract: A binary frame is a spanning set for a binary vector space. Computing the coefficients used to represent a given vector in terms of a frame may be difficult, but it is easy to do if the frame is Parseval. In this talk, we describe a class of Parseval frames consisting of vectors of equal weight. We then give conditions on the weights of vectors in a general Parseval frame. We show how the problem of extending a given set to a Parseval frame may be reduced to solving a linear system. We conclude with a connection to graph theory and a conjecture regarding the minimum number of vectors one must add in the extension problem. 
May 2, 2024 
End of Year Picnic Brink Hall FacultyStaff Lounge  12:00 to 2:00 p.m.  Pizza and drinks provided. Feel free to bring something else to share! 
Past Semesters: 

September 19, 2023 at 12:30 pm 
Dr. Jennifer Johnson Leung (University of Idaho) A Quaternionic Maass Space Abstract: In the 1970's Maass studied a subspace of Siegel modular forms of degree two for which the Fourier coefficients satisfy certain relations. Later, it was shown that this space is the image of a theta lift. It turns out that there is a theta lifting between any dual pair of reductive groups which has led to much rich mathematics, but not all of these groups admit modular forms with Fourier developments. In this talk, I will review some of the history of this subject and discuss new results giving Maass relations which characterize the image of the theta lift from Sp(4) to SO(8). This latter group does not admit holomorphic modular forms, so we work with the Fourier expansions of quaternionic modular forms. This is joint work with Finn McGlad, Isabella Negrini, Aaron Pollack, and Manami Roy. 
September 26, 2023 at 12:30 pm 
Dr. Jennifer Johnson Leung (University of Idaho) Continuation of discussion from last week's seminar. 
October 3, 2023 at 12:30 pm 
Jonathan Webb (University of Idaho) Lattices and their associated theta series for linear codes over GH(8) Abstract: Let K be the number field given by adjoining to the rationals a root of some irreducible cubic polynomial f. We give conditions on f under which 2 is inert in K and show that these conditions are satisfied when K is monogenic and Galois. Let K have rind of integers R. Because 2 is inert, the quotient ring R/2R is isomorphic to GF(8) so a linear code over this field may be identified with an element of the quotient ring. The preimage of the code under the surjection from R to the quotient ring is a lattice. We show that this lattice is integral with respect to the trace form. The lattice and trace form together generate a theta series. We compute examples of this theta series with the lattice being R for various monogenic and Galois K. 
October 10, 2023 at 12:30 pm 
Dr. Brooks Roberts (University of Idaho) The structure of the paramodular Hecke algebra and some applications Abstract: Hecke operators act on vector spaces of modular forms and are an essential tool for extracting information. Viewed abstractly, Hecke operators are the images of elements of certain rings called Hecke algebras. In this talk, we will describe the structure and some applications of the Hecke algebra associated tot he paramodular congruence group of level p for p a prime. Interestingly, this graded algebra is neither commutative nor of Iwahori type. This talk will be understandable to first year graduate students and is joint work with Jennifer Johnson Leung and Joshua Parker. 
October 24, 2023 at 12:30 pm 
John Pawlina (University of Idaho) Lower Bounds for the Minimum Distances of Evaluation Codes using Algebraic Properties of Corresponding 0dimensional Projective Varieties Abstract: Let K be any field. Let n and k be integers with n at least k. Let X be a set of n distinct points in P^{k1} (over K) not contained in a hyperplane. Let a be a positive integer and let d(X)_a be the minimum distance of the evaluation code of order a associated to X. In this talk I will share lower bounds for d(X)_a. The first bound, true for any X as described, is found using the alphainvariant of the defining ideal of X. The second bound applies to the case when X is in general (linear) position and is found using the socle degree of X. Both results improve or generalize previously established lower bounds. 
October 31, 2023 at 12:30 pm 
Jake Sapozhnikov (University of Idaho) Introverts and potty stars: Sufficient conditions for 2packings of bipartite and tripartite graphs Abstract: In 2019, Hong Wang demonstrated the existence of a fixedpointfree 2packing for all bipartite graphs of girth at elast 12, and conjectured that the result holds for graphs of girth at least 8. We extend this result to bipartite graphs of girth exactly 10. We also demonstrate the existence of a 2packing for all tripartitions of trees, and generalize the result to sufficient conditions for the existence of a 2packing of a tripartite graph of order n and size n1. 
November 7, 2023 at 12:30 pm 
Dr. Hirotachi Abo (University of Idaho) On the rank of a partially symmetric tensor Abstract: Every partially symmetric tensor can be expressed as a linear combination of a finite number of socalled decomposable partially symmetric tensors. The rank of a partially symmetric tensor is defined as the smallest positive number r such that the partially symmetric tensor can be written as a linear combination of r decomposable partially symmetric tensors. In this talk, we discuss an algebrogeometric approach to the problem of finding the generic rank of partially symmetric tensors, that is, the rank of a generic partially symmetric tensor. 
November 14, 2023 at 12:30 pm 
Dr. Stefan Tohaneanu (University of Idaho) Brief intro to free resolutions Abstract: Given M a finitely generated module over a commutative ring R, a free resolution is an exact complex of free Rmodules that measures how closed or far M itself is from being free. I will present a mild introduction to these very important tools in commutative algebra, emphasizing more on the case when M is a graded module over the graded ring of polynomials with coefficients in a field, with standard grading given by the degree. If time permitting, I will present some instances where free resolutions show up in my own research. 
November 28, 2023 at 12:30 pm 
Alex Barrios (University of St. Thomas) Lower bounds for the modified Szpiro ratio Abstract: Let a, b, and c be relatively prime positive integers such that a + b = c. How does c compare to rad(abc), where rad(n) denotes the product of distinct prime factors of n? According to the explicit abc conjecture, it is always the case that c is less than the square of rad(abc). This simple statement is incredibly powerful, and as a consequence, one gets a (marginal) proof of Fermat's Last Theorem for exponent n greater than 5. In this talk, we introduce Masser and Oesterle's abc conjecture and discuss some of its consequences, as well as some of the numerical evidence for the conjecture. We will then introduce elliptical curves and see that the abc conjecture has an equivalent formulation in this setting, namely, the modified Szpiro conjecture. We conclude the talk by discussing a recent result that establishes the existence of sharp lower bounds for the modified Szpiro ratio of an elliptic curve that depends only on its torsion structure. 
December 5, 2023 TBD 
Social Event 
February 28, 2023 at 1:00 pm 
Dr. Alex Woo (University of Idaho) The cohomology of Hessenberg varieties Abstract: I will describe two perspectives on the cohomology of Hessenberg varieties. The first is GKM (GoreskyKottwiczMacpherson) 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 (University of Idaho) Dirichlet characters and Dirichlet's theorem Abstract: In this expository talk we will recall the Lseries associated to a Dirichlet character, and we will describe how these Lseries 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 (University of Idaho) Algebrogeometric approaches to mixed Nash equilibria Abstract: Mixed Nash equilibrium is a concept in game theory that determines an optimal solution of a noncooperative finite game. Using socalled 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 algebrogeometric 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 kplanes 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 nondefectivity 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 (University of Illinois at UrbanaChampaign) Quantum Bruhat graph and tilted Richardson varieties Abstract: The quantum Bruhat graph is introduced by Postnikov to study structure constants of the quantum cohomology ring of the flag variety, with very rich combinatorial structures. We provide an explicit formula for the

April 25, 2023 at 1:00 pm 
Jonathan Webb (University of Idaho) Extending a set of vectors to a Parseval frame in a binary vector space Abstract: A frame for a binary vector space is a spanning set. Although any vector may be represented in terms of a frame, the coefficients used in this representation may be difficult to compute. A Parseval frame is a special case for which this computation is very easy. In this talk, we address the problem of adding vectors to a given set such that the result is a Parseval frame. We first discuss how this problem may be reduced to solving a linear system and then work towards minimizing the number of vectors which must be added.

May 5, 2023 
End of Year Picnic  Ghormley Park  3:00 to 5:00 p.m.  Pizza, drinks, and dessert provided. Feel free to bring something else to share! 
January 18, 2022 at 11:00 am Virtual meeting via Zoom 
Dr. Stefan Tohaneanu (University of 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 (University of 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 JohnsonLeung (University of Idaho) Details TBA Hecke characters, Lfunctions, and the ideles Abstract: We will give a motivation of the ideles by considering the generalizations of Dirichlet Lfunctions 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 University) 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 (University of 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 (University of 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 (University of 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 (University of Idaho) Bipackings of Bipartite Graphs Abstract: Let G = G(X_1,X_2) be a bipartite graph. Consider the partitepreserving 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. 