Auburn Algebra Seminar
Fall 2026
Schedules from past semestersSeminars will be held in STEM Ag 2111 on Tuesdays from 2:30 to 3:20, unless otherwise noted.
Schedule:
August 25: Michael Brown (Auburn)Title: A noncommutative Horrocks splitting criterion
Abstract: I will discuss a version of the Horrocks splitting criterion for vector bundles on noncommutative schemes. This is joint work in progress with Greg Smith.
September 1:
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September 8:
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September 15:
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September 22:
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September 29:
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October 6: Peter Wei (University of Arkansas)
Title: Weighted syzygies through kernel bundles
Abstract: A celebrated theorem of Mark Green relates the positivity of a line bundle on a smooth projective curve to the linearity of its syzygies, expressed by property (N_p). Brown and Erman established a weighted analogue of Green’s theorem for curves embedded by log complete series. In this talk, we introduce weighted kernel bundles and develop a geometric approach in the spirit of Green and Lazarsfeld. We use this method to recover and strengthen Brown–Erman’s results. We then discuss other applications, including a result of Banks, Cobb, and Sayrafi on weighted syzygies of pointed curves. If time permits, we will discuss extensions to stacky curves.
October 13: Trung Dung Vuong (Vietnam National University)
Title: Alternative-mean trace divergences: geometry, data processing, and barycenters
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Let \(f : (0,\infty) \to (0,\infty)\) be a nontrivial normalized operator monotone function and set \(s = f'(1)\). We introduce the alternative-mean trace functional
\[ \widehat{\Phi}_f(A,B) := \operatorname{Tr}(A \nabla_s B) - \operatorname{Tr}\!\big( f(A^{-1} \sharp B)\, A\, f(A^{-1} \sharp B) \big) \]on the positive definite cone. We prove that \(\widehat{\Phi}_f\) is a quantum divergence in the sense of Bhatia–Gaubert–Jain whose diagonal Hessian induces a positive multiple of the Bures–Wasserstein Riemannian metric. We also establish the sharp comparison
\[ s(1-s)\, d_{BW}(A,B)^2 \;\le\; \widehat{\Phi}_f(A,B) \;\le\; (1 - s + s^2)\, d_{BW}(A,B)^2 . \]We further establish several equivalent conditions characterizing those \(f\) for which \(\widehat{\Phi}_f\) satisfies data processing under all positive trace-preserving maps; in particular, diagonal pinching on faithful qubit states is a complete test. For the associated barycenter problem, we prove existence of closed-cone minimizers, derive a first-order equation for interior critical points, and give criteria ensuring uniqueness and positive definiteness. For the power family \(f(x) = x^t\), we determine exactly when every finite data set in the positive definite cone admits a unique closed-cone minimizer, which is then necessarily positive definite.
October 20: Richard Haburcak (Ohio State University)
Title: K3 surfaces and Brill-Noether theory
Abstract: Brill--Noether theory studies linear systems on an algebraic curve. K3 surfaces have been central in Brill--Noether theory since Lazarfeld's proof of the Brill--Noether theorem, which describes the linear series on a general curve of genus $g$. In this talk, we'll survey some recent results concerning the Brill--Noether theory of curves on K3 surfaces. Specifically, we'll discuss a refined Brill--Noether theory, which aims to understand the linear series on curves with a given Brill--Noether special linear series, and related unstable vector bundles on K3 surfaces with applications to the dimensions of components of Giesker--Petri loci in $\mathcal{M}_g$, parameterizing curves admitting a linear series with non-injective Petri map.
October 27: Muritala Ibrahim (Auburn)
Title: Beyond Classical Matrix Means: Near Geometric Curves, Near Spectral Geometric Curves, and Order Relations.
Abstract: This talk explores the order and spectral properties of the near geometric and near spectral geometric curves. Unlike conventional matrix means, these curves do not generally interpolate between their two inputs. Nevertheless, their spectral properties provide mean-like relationships between the spectra of the inputs, while their geometric structure gives geodesic-like descriptions of paths from a fixed point on the manifold of positive definite matrices. For the near geometric curve, we establish a normalized eigenvalue comparison with the arithmetic mean, give a sufficient condition for a stronger comparison in the Löwner order, and demonstrate why additional hypotheses are needed for certain log-majorization inequalities. For the near spectral geometric curve, we describe its connection with the classical spectral geometric mean and investigate its endpoint, monotonicity, and commutativity properties. We also discuss midpoint isospectrality, constant-speed and interpolation identities, and a Lie–Trotter limit connecting these constructions to averages of matrix logarithms. This is joint work with Dr. Huajun Huang.
November 3: Justin Lyle (Auburn)
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November 10: Caitlin Davis (Auburn)
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November 17: John Cobb (Auburn)
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December 1:
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