Algebra Seminars

COSAM Departments Mathematics & Statistics Research Seminars Algebra



Seminars are held in 358 Parker Hall on Tuesdays at 2:30pm.

Upcoming Algebra Seminars
DMS Algebra Seminar
Apr 08, 2025 02:30 PM
ZOOM


josefranco

Speaker: Jose Franco (University of North Florida)

Title: On the Chaotic Thompson Metric of Matrices

 
Abstract: In this talk, we introduce and study properties of the Thompson metric induced by the chaotic order on the cone of positive definite matrices. Furthermore, we introduce an approach to characterize geodesic curves associated to Thompson metrics and study in-betweenness type results for the metric induced by the chaotic order. Among other results, we obtain that the map \(A\mapsto d_\ll(A,Q_{\alpha,z}(B,A)); \quad z>0,\) satisfies Audenaert's in-betweenness for \(0\le \alpha \le 1\).

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Past Algebra Seminars
DMS Algebra Seminar
Mar 25, 2025 02:30 PM
354 Parker Hall


Speaker: Ian Tan (Auburn University)

Title: Orbits and Invariants in Quantum Information Theory

Abstract: In the field of quantum information theory, one studies—among other things—the information processing tasks that can be achieved by taking advantage of a quantum phenomenon known as entanglement. There is a mathematical formalism that captures the notion of entanglement by elements of a vector space called state vectors. The local unitary and SLOCC (stochastic local operations with classical communication) groups act on this space, producing natural equivalence classes of state vectors. In this work, we consider these group actions and their invariants which are used to classify and distinguish state vectors.

DMS Algebra Seminar
Mar 18, 2025 02:30 PM
354 Parker Hall


pandey

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) 

kim

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!

 

hal

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


brown

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


josefranco
 
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
\(A\hat{\sigma}_f B = f(A^{-1}\sharp B)Af(A^{-1}\sharp B),\)
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


cobb
 
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!)
 
justinLyle
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


hal
 
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.

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