Auburn Algebra Seminar
Spring 2024
Schedules from past semestersSeminars will be held in Parker 358 on Tuesdays from 2:30 to 3:20.
Schedule:
Janauary 16: Michael Brown (local)Title: A short proof of the Hanlon-Hicks-Lazarev Theorem.
Abstract: We give a short, new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. This is joint work with Daniel Erman.
January 23: Matthew Speck (local)
Title: Recent Progress on the Marcus-de Oliveira Conjecture
Abstract:.Marcus and de Oliveira conjectured bounds on the determinant of a sum of normal n-by-n matrices given their eigenvalues. We will discuss representation theoretic and combinatorial tools which have yielded a novel proof of the conjectured bounds for n=3, and we will outline how these tools may be used for higher values of n. This is joint work with Luke Oeding.
January 30: Sara Marziali (University of Siena)
Title: Product of Tensors and Description of Networks
Abstract: Introduction to the Bhattacharya-Mesner Product for tensors of order n. This product is particularly suitable for the study of neural networks described by acyclic directed graphs in which each node decides its own activation through tensors of choice. The final state of the network can be determined from the product of node tensors.
February 6: Trung Hoa Dinh (Troy University)
Title: The α-z-Bures Wasserstein divergence and α-z-quantum fidelity.
Abstract: In this talk, we introduce the alpha-z Bures Wasserstein divergence and study the least squares problem with respect to the new quantum divergence. We also introduce the quantum alpha-z-fidelity, obtain some variational formulas for it, and discuss the quantum alpha-z-fidelity between orbits.
February 13: Sean Grate (local)
Title: Betti tables and Lefschetz properties
Abstract: For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti diagram which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Join with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
February 20:
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February 27: Joachim Jelisiejew (University of Warsaw) (virtual: https://auburn.zoom.us/j/87035894762)
Title: New invariants of minimal border rank tensors
Abstract: The border rank of a tensor is a basic measure of its complexity. Yet, very little is known even for small sized tensors. In the talk I will explain the main open problems and recent developments connected to newly introduced 111-algebras which allow to apply commutative algebraic and deformation theoretic results to this problem. For most part of the talk, this is joint with J.M. Landsberg and A. Pal.
March 12: Ian Tan (local)
Title: Tensor factorizations and orbit classification
Abstract: We introduce a general framework for obtaining tensor factorizations with respect to various group actions. Our motivation comes from orbit classification problems that arise in quantum information. For example, the singular value decomposition gives real diagonal normal forms under left and right multiplication by unitary matrices. Similarly, the higher order singular value decomposition of De Lathauwer et al. was used by B. Kraus to find normal forms for generic n-qubit tensors under the action of the local unitary group. Our perspective explains these tensor factorizations and gives new ones for the "local" special orthogonal group and for the group of Stochastic Local Operations with Classical Communication (SLOCC). Consequently, we obtain SLOCC normal forms for generic n-qubit tensors. This is joint work with Luke Oeding.
March 19: Hal Schenck (local)
Title: Free curves, eigenschemes, and pencils of curves (arXiv:2306.09443)
Abstract: Let R=K[x,y,z]. A reduced plane curve C = V(f) in P^2 is free if its associated module of tangent derivations Der(f) is a free R-module, or equivalently if the corresponding sheaf T_{P^2}(−log C) of vector fields tangent to C splits as a direct sum of line bundles on P^2. In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in P^2. The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves. This is joint work with Roberta Di Gennaro, Giovanna Ilardi, Rosa Maria Mirò-Roig Hal Schenck, Jean Vallès.
March 26: Sean Grate (local). This talk will be in Parker 250.
Title: Problems in computational algebraic geometry: Lefschetz properties and toric varieties
Abstract: Starting with Lefschetz properties, moving on to toric varieties and Castelnuovo-Mumford regularity, and finishing with other miscellaneous projects, I will give an overview of the research I have conducted while at Auburn University. A common theme among all these projects is the strong presence (and necessity) of computation. As such, there will be many examples written in Macaulay2 and Python to help understand where these projects came from and how they were completed.
April 2:
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April 9:
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April 16:
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April 23: Matthew Speck (local). This talk will be at an ususual time: 1:00 pm. It will be in person, but there is also a Zoom link: https://auburn.zoom.us/j/3418391635?omn=83942308355.
Title: Determinants of Sums of Normal Matrices
Abstract: Recent efforts in matrix theory have been concerned with describing invariants of matrices with "nice" properties. In this talk, we address a conjecture on the determinant of the sum of a pair of normal matrices. Reducing this conjecture to the problem of providing non-negative solutions for a system of linear equations without full rank, we use tools from representation theory to describe the modifications required to provide such solutions. This is my doctoral defense and joint work with my committee chair, Luke Oeding.