Algebra Seminars
Seminars are held in 358 Parker Hall on Tuesdays at 2:30pm.
Upcoming Algebra Seminars
Past Algebra Seminars
DMS Graduate Student Seminar
Feb 26, 2025 03:00 PM
010 ACLC
DMS Algebra Seminar
Dec 03, 2024 02:30 PM
ZOOM
\(A\hat{\sigma}_f B = f(A^{-1}\sharp B)Af(A^{-1}\sharp B),\)
DMS Algebra Seminar
Nov 19, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Oct 22, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Oct 15, 2024 02:00 PM
358 Parker Hall
DMS Algebra Seminar
Oct 08, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 24, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 17, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 10, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 03, 2024 02:30 PM
358 Parker Hall
Feb 26, 2025 03:00 PM
010 ACLC
Speaker: Dr. Michael Brown (Auburn University)
Title: Introduction to Free Resolutions
Abstract: Given a ring R and an R-module M, a free resolution of M is an approximation of M in terms of the simplest kind of module: namely, free modules. Free resolutions play an important role in commutative algebra, algebraic geometry, and many other neighboring fields. This talk will be an introduction to free resolutions: how to construct them, some key results about them, and some open questions.
DMS Algebra Seminar
Dec 03, 2024 02:30 PM
ZOOM
Speaker: Jose Franco (University of North Florida)
Title: A theory of alternative means of matrices
Abstract: In this talk, we introduce a theory of alternative means of positive definite matrices. We define the alternative mean associated to a normalized operator monotone function by
for positive definite matrices \(A\) and \(B\). We study the adjoint, transpose, and duals of these means and obtain a characterization of the spectral geometric means in terms of alternative means and their duals. Additionally, we show the deep connection between the alternative means and the near order of matrices. In particular, we obtain in-betweenness results of the type \(A\preceq A\hat{\sigma}_f B \preceq B.\)
DMS Algebra Seminar
Nov 19, 2024 02:30 PM
358 Parker Hall
Speaker: John Cobb (Auburn)
For graduate students: John will also give a “pre-talk” aimed at grads starting at 1:50, in the same room, to help supply background for his talk.
Title: Multigraded Syzygies
Abstract: I’ll describe two results that fit a recent theme of research in commutative algebra and algebraic geometry: pushing known results from the graded to the multigraded setting. One involves using logic to get a bound on projective dimension (Stillman’s Conjecture), and the other using homological techniques to get a bound on the regularity of curves. In the pretalk, I’ll introduce the main objects and theirs motivation more thoroughly.
DMS Algebra Seminar
Oct 22, 2024 02:30 PM
358 Parker Hall
Speaker: Ian Tan (Auburn)
Title: Four-qubit critical states
Abstract: Let H be the Hilbert space of unnormalized four-qubit state vectors. By a result of Verstraete, Dehaene, and De Moor, ||f||^(1/m) is an entanglement monotone for any complex homogeneous polynomial \(f\) on \(H\) of degree \(m > 0\) invariant under the action of the SLOCC group. We observe that many highly entangled or useful four-qubit states that appear in prior literature are stationary points of \(||f||\) for natural choices of \(f\). This motivates the search for more stationary points. Using the notion of critical points (in the sense of the Kempf-Ness theorem) together with results from Vinberg’s theory of theta groups, we reduce the complexity of the problem significantly. After reduction, we solve systems of polynomial equations with techniques from numerical algebraic geometry, recovering the stationary points known to us at the beginning and more.
DMS Algebra Seminar
Oct 15, 2024 02:00 PM
358 Parker Hall
Time: Tuesday, October 15 from 2:00 - 2:50. (Note the unusual time!)
Speaker: Justin Lyle (Auburn)
Title: Homological Aspects of Commutative Rings
Abstract: Homological methods have been integral to our understanding of both commutative and noncommutative algebras since the so-called "Homological Invasion" of the 1950s, and have led to solutions of numerous problems not intrinsically homological in nature. Despite this, there are several fundamental aspects that remain mysterious, with several problems from the 1950s-1970s remaining stubbornly open, including the Finitistic Dimension Conjecture, the Nakayama Conjecture and its generalized variant, the Tachikawa Conjectures, and the Auslander-Reiten Conjecture. In this talk, we give an overview of the historical and broader contexts of these problems, and discuss recent progress of the speaker on some related questions, as a consequence leading to new broad cases of the Auslander-Reiten conjecture.
DMS Algebra Seminar
Oct 08, 2024 02:30 PM
358 Parker Hall
Speaker: Hal Schenck (Auburn)
Title: Syzygies of permanental ideals
Abstract: We describe the minimal free resolution of the ideal of 2×2 subpermanents of a 2×n generic matrix M. In contrast to the case of 2×2 determinants, the 2×2 permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of 2×2 permanents of a generic matrix with our previous work connecting the initial ideal of 2×2 permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence.
Joint work with F. Gesmundo, H. Huang, J. Weyman.
DMS Algebra Seminar
Sep 24, 2024 02:30 PM
358 Parker Hall
Speaker: Michael Brown (Auburn)
Title: Noncommutative sheaf cohomology
Abstract: I will give a brief introduction to noncommutative projective geometry, with a view toward a pair of projects concerning noncommutative sheaf cohomology (one joint with Daniel Erman and Greg Smith, and another joint with Prashanth Sridhar).
DMS Algebra Seminar
Sep 17, 2024 02:30 PM
358 Parker Hall
Speaker: Colin Crowley (University of Oregon)
Title: Orlik-Terao algebras and internal zonotopal algebras
Abstract: In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement.
Joint with Nick Proudfoot.
DMS Algebra Seminar
Sep 10, 2024 02:30 PM
358 Parker Hall
Speaker: Sean Grate (Auburn University)
Title: Betti numbers of connected sums of graded Artinian Gorenstein algebras
Abstract: Considered as an algebraic analog for the connected sum construction from topology, the connected sum construction introduced by Ananthnarayan, Avramov, and Moore is a method to produce Gorenstein rings. Joint with Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu, and Junzo Watanabe, we determine the graded Betti numbers for connected sums and fiber products of Artinian Gorenstein algebras, where the fiber product in the local setting was obtained by Geller. We also show that the connected sum of doublings is the doubling of a fiber product ring. I will discuss these results through some examples and Macaulay2 code.
DMS Algebra Seminar
Sep 03, 2024 02:30 PM
358 Parker Hall
Speaker: Ayah Almousa (University of South Carolina)
Title: Stirling numbers and Koszul algebras with symmetry
Abstract: Stirling numbers \(c(n,k)\) and \(S(n,k)\) of the first and second kind are the answers to two counting problems: how many permutations of n letters have \(k\) cycles, and how many set partitions of \([n]\) have \(k\) blocks? The \(c(n,k)\) also give the Hilbert function for certain well-studied Koszul algebras with symmetry: the cohomology of configurations of \(n\) distinct labeled points in \(d\)-space, also known as the Orlik-Solomon algebras and graded Varchenko-Gelfand algebras for type A reflection hyperplane arrangements. We discuss how the \(S(n,k)\) give the Hilbert series for their less-studied Koszul dual algebras. This includes relating the symmetric group action on the original algebras and on their Koszul duals, representation stability in the sense of Church and Farb, and branching rules that lift Stirling number recursions.
This is joint work with Vic Reiner and Sheila Sundaram.