Fall 2025

This seminar met on Mondays 3:00-3:50 PM and Fridays 4:00-4:50 PM in Blocker 302. The organizers were Frank Sottile and others.

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Date	Speaker	Affiliation	Title	Other
August 25, 2025	Justin Lacini	Texas A&M	Logarithmic bounds on Fujita’s conjecture	View Abstract
August 29, 2025			No seminar	(promotion talks)
September 5, 2025			No seminar	(promotion talks)
September 12, 2025			No seminar	(promotion talks)
September 22, 2025	Matthew Faust	Michigan State	Algebraic Geometry in the Study of the Eigenvalues of Discrete Periodic Operators	View Abstract
September 25, 2025	Jordy Lopez	Notre Dame	Toric compactifications of periodic graph operators	View Abstract Special time, Thursday 11-12 Bloc 605AX
October 6, 2025	Debaditya Raychadhury	University of Arizona	Singularities of Secant varieties	View Abstract
October 17, 2025				Algebra and Combinatorics Seminar
October 24, 2025	Nicklas Day	Texas A&M	Local Geometry of Distributions: Symplectification, Cartan Prolongation, and Maximality of Class	View Abstract Special time: 3-4 PM
November 3, 2025	Thomas Tony	University of Münster	Scalar Curvature Comparison Geometry and Product Rigidity	View Abstract
November 17, 2025	Algebra	Group	Courses	Discussion
November 21, 2025	No seminar
December 5, 2025	No seminar

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A longstanding conjecture of T. Fujita asserts that if $X$ is a smooth complex projective variety of dimension $n$ and if $L$ is an ample line bundle, then $K_X+mL$ is basepoint free for $m\geq n+1$. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if $m$ is larger than a quadratic function in $n$. We show that for $n\geq 2$ the conjecture holds for $m$ larger than $n(\log\log(n)+3)$. This is joint work with L. Ghidelli.

Given a periodic graph $G$, we consider a class of discrete periodic operators given by the sum of a weighted adjacency operator and a potential. We wish to study the spectrum of this operator acting on the Hilbert space of square-summable functions on the vertices of $G$. By Floquet theory, the spectrum can be realized as the projection of a finite number of band functions. If one of these band functions is constant, the operator is said to have a flat band, corresponding to eigenvalues of infinite multiplicity. In this talk, we will introduce the relevant background and discuss recent results that demonstrate that, generically, discrete periodic operators do not exhibit flat bands. This is based on joint work with Wencai Liu.

A periodic graph operator is a weighted Laplacian plus potential acting on functions on the vertices of a periodic graph. It is well-known that the spectrum of a periodic graph operator is the projection of an affine algebraic variety known as the Bloch variety. Motivated by Bättig, we compactify the Bloch variety of a periodic graph operator inside the normal toric variety associated to its Newton polytope. For a family of periodic graphs, we extend this operator to such toric variety by expressing the compactification as the support of a kernel sheaf. We outline a few spectral-theoretic consequences of this compactification. This is joint work with Matthew Faust (MSU), Stephen Shipman (LSU), and Frank Sottile.

Secant varieties are classical objects in algebraic geometry. Given a smooth projective variety inside a projective space, its secant variety is by definition the closure of the union of secant lines. It is almost always singular and sits inside the same projective space by its construction. In this talk, we will discuss the singularities of secant varieties when the embedding is sufficiently positive. In particular, we will study the Du Bois complex of secant varieties and will also discuss about its local cohomology modules. The results are obtained in various collaborations with Q. Chen, B. Dirks, S. Olano and L. Song.

In 1970, N. Tanaka gave a method for assigning a canonical frame to distribution with constant Tanaka symbol. In 2009, B. Doubrov and I. Zelenko utilized a symplectification procedure to obtain a canonical frame for distributions independent of their Tanaka symbol, but an additional condition called maximality of class was required of the distribution. In a work with I. Zelenko, we recently proved that all rank 2 distributions with 5-dimensional cube are of maximal class at a generic point; this allowed us to assign a canonical frame at a generic point to every rank 2 distribution which is not Goursat. In the rank 2 case, I will give an interpretation of the symplectification procedure in terms of a classical construction called Cartan prolongation. Time permitting, I will also sketch what is known about a collection of fundamental invariants called harmonic invariants in the case of rank 2 distributions in dimensions 6 and 7 along with differential relations called syzygies satisfied by these invariants.
In contrast to the rank 2 case, we found examples of rank 3 distributions with 6-dimensional square which are not of maximal class. In particular, I will present a (3, 8) distribution of nonmaximal class with 29-dimensional symmetry containing a semi-direct sum of the exceptional Lie group G_2 with a copy of its adjoint module.

In the late 1990s, Llarull proved that the round n-sphere is area-extremal in the sense that one cannot simultaneously increase its scalar curvature and its metric, confirming a rigidity phenomenon anticipated by Gromov. Goette and Semmelmann generalized Llarull’s rigidity result to certain area non-increasing spin maps $f: M \to N$ of nonzero $A$-degree, showing that under an appropriate lower bound on the scalar curvature, the map $f$ must be a Riemannian submersion. In this talk, I will give a brief introduction to scalar curvature comparison geometry and discuss several generalizations of the Goette–Semmelmann rigidity theorem. In particular, I will present a generalization in which the topological condition on the Â-degree is replaced by a weaker assumption involving the so-called higher mapping degree. I will also describe an ongoing joint project with Oskar Rieder, in which we strengthen the rigidity conclusion: the map $f$ is not only a Riemannian submersion, but the manifold M even splits locally as a Riemannian product of $N$ with a Ricci-flat manifold $F$.

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Abstract