Department of Mathematics and Statistical Science Algebra Seminar Series

https://arxiv.org/abs/2303.01436.

There will be some individual guest speakers on other topics, and those will be updated here when known.  All other weeks will feature discussions of the following topic:

Schubert varieties are parameter spaces whose points correspond to configurations of subspaces (in a vector space) satisfying some specific incidence conditions. There are interesting questions about the geometry of Schubert varieties that frequently have answers in terms of the combinatorics of how these incidence conditions are given. The survey tackles an approach using combinatorics and commutative algebra to study these questions. I do expect to spend a few weeks just defining Schubert varieties and explaining how to work with them.

https://arxiv.org/abs/2303.01436.

https://arxiv.org/abs/2303.01436.

https://arxiv.org/abs/2303.01436.

Jiayu Yang, Mathematics Ph.D. Student, University of Idaho

Uniquely bi-embeddable Bipartite 2-regular graphs

Abstract: A bipartite 2-regular graph is a bipartite graph G such that G = C_{2n1} ∪ C_{2n2} ∪ · · · ∪ C_{2nk} with k ≥ 1, i.e., G is a vertex disjoint union of bipartite cycles. In this paper, we completely characterize the bipartite 2-regular graphs which can be uniquely bi-embedded into their bipartite complements. This work is an analogue of the result proved by Grzelec, Pilsniak and Wozniak ["A note on uniquely embeddable 2-factors," Applied Mathematics and Computation, 468, 2024].

https://arxiv.org/abs/2303.01436.

Ian Tan, Graduate Student, Auburn University

Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

Abstract: We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus's (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(C)xSL2(C)-->SO4(C), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under teh action of the SLOCC group.

https://arxiv.org/abs/2303.01436.

https://arxiv.org/abs/2303.01436.

https://arxiv.org/abs/2303.01436.

Jeyoung Song, Mathematics Ph.D. Student, University of Idaho

On Tensors and Ranks with Applications to Segre Varieties

Abstract:  We organize fundamental theories on tensor products and ranks in general settings.  We use these backgrounds to analyze tensor ranks and secant varieties over Segre varieties.

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 non-toric staged tree models in algebraic statistics.

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 p-1 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 Weil-Deligne 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 Fang-Ting 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.

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.

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.

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 CM-field of low degree can have. A full description of the possibilities for a quartic CM-field's Galois group will be given, and the key ideas of how to determine possible Galois groups for CM-fields of higher degree is described, but the details are much more complicated, but we will engage in some exploration of the possibilities for sextic CM-fields.

Dr. Faqruddin Azam (Lewis-Clark State College)

Divisor labeling of staircase diagrams and fiber bundle structures on Schubert varieties

Abstract: Let Gr (r,n) denote the Grassmannian of r-dimensional subspaces of C^n. Each Gr (r,n) contains a unique codimension-1 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.

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].

Dr. Alexander Woo (University of Idaho)

An affine paving for Delta-Springer fibers

Abstract: In some joint work with Sean Griffin and Jake Levinson, we needed an affine paving for Delta-Springer 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 Delta-Springer fibers with Schubert cells gives such an affine paving. (After untangling the definitions, this uses only elementary linear algebra.)

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.

End of Year Picnic -Brink Hall Faculty-Staff Lounge - 12:00 to 2:00 p.m. - Pizza and drinks provided.  Feel free to bring something else to share!

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.

Dr. Jennifer Johnson Leung (University of Idaho)

Continuation of discussion from last week's seminar.

 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.

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.

John Pawlina (University of Idaho)

Lower Bounds for the Minimum Distances of Evaluation Codes using Algebraic Properties of Corresponding 0-dimensional 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^{k-1} (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 alpha-invariant 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.

Jake Sapozhnikov (University of Idaho)

Introverts and potty stars: Sufficient conditions for 2-packings of bipartite and tripartite graphs

Abstract: In 2019, Hong Wang demonstrated the existence of a fixed-point-free 2-packing 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 2-packing for all tripartitions of trees, and generalize the result to sufficient conditions for the existence of a 2-packing of a tripartite graph of order n and size n-1.

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 so-called 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 algebro-geometric approach to the problem of finding the generic rank of partially symmetric tensors, that is, the rank of a generic partially symmetric tensor.

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 R-modules 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.

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.

Dr. Hirotachi Abo (University 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.

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.

Shiliang Gao (University of Illinois at Urbana-Champaign)

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
weight of minimal paths between any pair of permutations. Building upon that, we obtain a simple criterion for the tilted Bruhat order studied by Brenti-Fomin-Postnikov. These results motivate the definition of tilted Richardson varieties, which provides geometrical interpretations of tilted Bruhat orders. Tilted Richardson varieties are indexed by pairs of permutations and generalize Richardson varieties in the flag variety. This is a joint work with Jiyang Gao and Yibo Gao.

 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.