DMS Algebra Seminar
Mar 18, 2025 02:30 PM
354 Parker Hall
Speaker: Vaibhav Pandey (Purdue University)
Title: Symbolic powers of maximal minors under general linkage
Abstract: The symbolic powers of the minors of a generic matrix are well understood as determinantal rings are Algebras with Straightening Laws (ASLs). In particular, the ASL structure readily yields the fact that the symbolic powers of maximal minors agree with the ordinary powers. We prove that, quite surprisingly, the equality of the symbolic and ordinary powers holds for `the most general link' of maximal minors as well. Better still, the blowup algebras of the general link have precisely the same homological properties as those of the maximal minors. The key point is that these facts do not follow from standard techniques in liaison theory; we develop the novel tool of Gröbner degeneration of links in order to attack the problem. This is joint work with Matteo Varbaro.
DMS Algebra Seminar
Mar 17, 2025 02:30 PM
354 Parker Hall
March 17 (Monday)
Speaker: Sejong Kim (Chungbuk National University)
Title: Quasi-Wasserstein mean of positive definite matrices
Abstract: The typical examples of Kubo-Ando's operator means are the weighted arithmetic, geometric, and harmonic means, which are monotonically interpolated by the power mean (introduced by Lim and Palfia). There are other important means of non Kubo-Ando's operator means such as the weighted spectral geometric and Wasserstein mean. We define quasi-Wasserstein means, which interpolate the weighted spectral geometric and Wasserstein mean. We study their properties including monotonicity for near-order, trace and norm inequalities.
DMS Algebra Seminar
Mar 04, 2025 02:30 PM
354 Parker Hall
Note the new room this semester!
Speaker: Hal Schenck
Title: From approximation theory to homological algebra--splines
Abstract: I will discuss the use of homological algebra in attacking a problem in approximation theory: determining the space of piecewise polynomial functions on a polyhedral subdivision of a region in Euclidean space. I will also touch on an interesting connection to toric geometry.
DMS Graduate Student Seminar
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
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
\(A\hat{\sigma}_f B = f(A^{-1}\sharp B)Af(A^{-1}\sharp B),\)
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).
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