Auburn Algebra Seminar
Spring 2025
Schedules from past semestersSeminars will be held in Parker 354 on Tuesdays from 2:30 to 3:20, unless otherwise noted.
Schedule:
Janauary 14:Title:
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January 21:
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January 28:
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February 4:
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February 11:
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February 18
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February 25: Michael Brown (Auburn)
Title: Ranks of matrix factorizations and sheaf cohomology
Abstract: Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings. The goal of this talk is to explain how the graded version of Buchweitz-Greuel-Schreyer’s conjecture implies a novel conjecture concerning the cohomology of sheaves over non-Fano projective hypersurfaces. This is joint work with Mark Walker.
March 4: Hal Schenck (Auburn)
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.
March 17 (Monday): 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.
March 18: 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.
March 25: Ian Tan (Auburn)
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.
April 1:
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April 8: Jose Franco (University of North Florida). This talk is on Zoom: https://auburn.zoom.us/j/4402370368.
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 $0\le \alpha \le 1$.
April 15:
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April 22: Jayan Mukherjee (Oklahoma State University)
Title: Syzygies of ribbons on higher genus curves
Abstract: In this talk, we will discuss the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus $p_a$ on a general curve of genus $g$ with $p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}$. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for $C$ and it implies Green's conjecture for all double covers $C'$ of $C$ with $g(C') \geq \textrm{max}\{3g+2, 6g-4\}$. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to $C$, which may be of independent interest.
April 29: Rankeya Datta (University of Missouri). This talk is on Zoom: https://auburn.zoom.us/j/4402370368.
Title: Enhancements of flatness in commutative algebra
Abstract: The notion of flatness plays a fundamental role in algebraic geometry and commutative algebra. A basic property of a flat module is that expansion of ideals to the module commutes with intersection for a finite family of ideals. In this talk I will introduce several enhancements of flat modules that arise from examining the natural question of when ideal expansion to a flat module commutes with intersection for an arbitrary (i.e. possibly infinite) family of ideals. These enhancements of flatness were first explored by Raynaud-Gruson in their work on the faithfully flat descent of projectivity and by Ohm-Rush in their work on content functions and trace ideals. Later, Hochster-Huneke encountered these enhancements in their development of tight closure theory in prime characteristic. Most of my talk will focus on defining and exploring these flatness enhancements and how they relate to each other. A particularly pleasing picture emerges in the local setting. If time permits, I will mention consequences for some outstanding questions in prime characteristic singularity theory.