
Date Time 
Location  Speaker 
Title – click for abstract 

01/14 3:00pm 
BLOC 628 
C. Farnsworth Texas State University in San Marcos 
Algebraic funtf Completion
In this talk we will define the finite unit norm tight frame (funtf) variety. The algebraic funtf completion problem is the determination the fiber of a projection of this funtf variety onto a set of coordinates. We will give a characterization of the bases of the algebraic matroid underlying the funtf variety in R^3 and give our partial results for higher dimension. 

01/17 1:00pm 
BLOC 628 **note 
Amy Hang Huang University of Wisconsin Madison 
Equations of Kalman Varieties
Given a subspace L of a vector space V, the Kalman variety
consists of all matrices of V that have a nonzero eigenvector in L. I will
discuss how to apply Kempf Vanishing technique with some more explicit
constructions to get a long exact sequence involving coordinate ring of
Kalman variety, its normalization and some other related varieties in
characteristic zero. Time permitting I will also discuss how to extract
more information from the long exact sequence including the minimal
defining equations for Kalman varieties. 

01/25 4:00pm 
BLOC 628 
Xiaoxian Tan TAMU 
Applying Algebraic Methods in Mathematical Biology
Many challenging problems in mathematical biology, for instance, in biochemical reaction networks and phylogenetics, are to solve nonlinear polynomial systems. Therefore, methods and tools in algebraic geometry and combinatorics are more applicable and powerful. One typical example is the multistationarity problem: whether a given biochemical reaction network has two or more positive steady states? In this talk, we introduce a simple criterion to determine multistationarity for networks arising from biology and to identify the parameter values for which the given network exhibits multistationarity. For linearly binomial networks, we prove our method is much less expensive than standard real quantifier elimination method in computational algebraic geometry. The two key ideas for improving the efficiency are: 1. whether a given network is linearly binomial can be read off easily from graphs associated to the network.
2. linearly binomial networks have nice algebraic and geometric structures. 

02/04 3:00pm 
BLOC 628 
C.J. Bott TAMU 
Mirror symmetry for K3 surfaces
Mirror symmetry is the phenomenon, originally discovered by physicists, that CalabiYau manifolds come in dual pairs, with each member of the pair producing the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. One important problem in mirror symmetry is that there may be several ways to construct a mirror dual for a CalabiYau manifold. Hence it is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?
We specifically consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry. BHK mirror symmetry was inspired by the LandauGinzburg/CalabiYau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely nonsymplectic automorphism of order n, we ask if these two constructions agree. Results of ArtebaniBoissièreSarti (2011) originally showed that they agree when n=2, and ComparinLyonPriddisSuggs (2012) showed that they agree when n is prime. However, the n being composite case required more sophisticated methods. Whenever n is not divisible by four (or n=16), this problem was solved by Comparin and Priddis (2017) by studying the associated lattice theory more carefully. We complete the remaining case of the problem when n is divisible by four by finding new isomorphisms and deformations of the K3 surfaces in question, develop new computational methods, and use these results to complete the investigation, thereby showing that the BHK and LPK3 mirror symmetry constructions also agree when n is composite. 

02/06 2:00pm 
BLOC 220 
Jose Burgos Gil ICMAT, Madrid 
Arithmetic of Toric Varieties, Lecture 1.
Abstract: Toric varieties form a very rich family of algebraic varieties that provide examples where explicit computations can be made. There is a toric dictionary that translates algebrogeometric concepts to combinatorial concepts. With this dictionary many algebrogeometric quantities can be computed. For example the degree of an ample line bundle on a toric variety is essentially given by the volume of an associated convex polytope.
In joint work with P. Philippon and M. Sombra we have extended the toric dictionary to relate arithmetic properties with convex analytical properties. For example the height of a toric variety with respect to a positive metrized line bundle can be computed as the integral of a convex function on the associated polytope.
The minicourse will consist of three lectures:
Lecture 1: Overview of the theory of toric varieties.
Lecture 2: The theory of heights and the analogy between geometry and arithmetic.
Lecture 3: Arithmetic properties of toric varieties.
Most of the material of the course is in the book:
Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín Arithmetic geometry of toric varieties. Metrics, measures and heights. Astérisque No. 360 (2014).


02/11 1:00pm 
BLOC 628 
Jose Burgos Gil ICMAT, Madrid 
Arithmetic of Toric Varieties, Lecture 2
See lecture 1 for the series abstract. 

02/13 10:30am 
BLOC 628 
Jose Burgos Gil ICMAT, Madrid 
Arithmetic of Toric Varieties, Lecture 3
See lecture 1 for series abstract. 

02/25 3:00pm 
BLOC 628 
A. Conner TAMU 
Kronecker powers of tensors and the exponent of matrix multilplication 

03/01 4:00pm 
BLOC 628 
Visu Makam IAS 
Exponential degree lower bounds for invariant rings
The ring of invariants for a rational representation of a
reductive group is finitely generated and graded. We give a general
technique that can be used to show that an invariant ring is not generated
by invariants of small degree. The main ingredients are Grosshans principle
and the moment map, which I will explain. As an example, we apply this
technique to show "exponential" lower bounds for the action of SL(n) on
4tuples of cubic forms. 

03/04 3:00pm 
BLOC 628 
Tingran Gao U. Chicago 
Manifold Learning on Fibre Bundles
Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a nonscalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diff‚usion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology. 

03/08 4:00pm 
BLOC 628 
Giuseppe Martone University of Michigan 
Hitchin representations and positive configurations of apartments
Hitchin singled out a preferred component in the character variety of representations from the fundamental group of a surface to PSL(d,R). When d=2, this Hitchin component coincides with the Teichmuller space consisting of all hyperbolic metrics on the surface. Later Labourie showed that Hitchin representations share many important differential geometric and dynamical properties.
Parreau extended previous work of Thurston and MorganShalen to a compactification of the Hitchin component whose boundary points are described by actions of the fundamental group of the surface on a building.
In this talk, we offer a new point of view for the Parreau compactification, which is based on certain positivity properties discovered by Fock and Goncharov. Specifically, we use the FockGoncharov construction to describe the intersection patterns of apartments in invariant subsets of the building that arises in the boundary of the Hitchin component. 

03/22 4:00pm 
BLOC 628 
Igor Zelenko TAMU 
Projective and affine equivalence of subRiemannian metrics: generic rigidity and separation of variables conjecture.
Two subRiemannian metrics are called projectively equivalent if they have the same geodesics up to a reparameterization and
affinely equivalent if they have the same geodesics up to affine reparameterization. In the Riemannian case both equivalence
problems are classical: local classifications of projectively and affinely equivalent Riemannian metrics were established by
LeviCivita in 1898 and Eisenhart in 1923, respectively. In particular, a Riemannian metric admitting a nontrivial (i.e.
nonconstant proportional) affinely equivalent metric must be a product of two Riemannian metrics i.e. certain separation of
variable occur, while for the analogous property in the projectively equivalent case a more involved (``twisted") product
structure is necessary. The latter is also related to the existence of sufficiently many commuting nontrivial integrals
quadratic with respect to velocities for the corresponding geodesic flow. We will describe the recent progress toward the
generalization of these classical results to subRiemannian metrics. In particular, we will discuss genericity of metrics
that do not admit nonconstantly proportional affinely/projectively equivalent metrics and the separation of variables on
the level of linearization of geodesic flows (i.e. on the level of Jacobi curves) for metrics that admit nonconstantly
proportional affinely equivalent metrics. The talk is based on the collaboration with Frederic Jean (ENSTA, Paris) and Sofya
Maslovskaya (INRIA, Sophya Antipolis). 

03/25 3:00pm 
BLOC 628 
E. Ventura TAMU 
Tensors and their symmetry groups
Tensors (multidimensional matrices) appear
in many areas of pure and applied mathematics. I will discuss their
use in algebraic complexity theory.
Matrix multiplication is a tensor and its complexity is encoded in its
tensor rank. To analyze the complexity of the matrix multiplication tensor,
Strassen introduced a class of tensors that vastly generalize it, the tight tensors.
These tensors have continuous symmetries. Pushing Strassen’s ideas
forward, with A. Conner, F. Gesmundo, and J.M. Landsberg,
we investigate tensors with large symmetry groups and
classify them under a natural genericity assumption. Our study provides new paths
towards upper bounds on the complexity of matrix multiplication. 

03/29 4:00pm 
BLOC 628 
Shinlin Yu TAMU 
TBA 

04/08 3:00pm 
BLOC 628 
Bernd Siebert University of Texas 
TBA 

04/12 4:00pm 
BLOC 628 
David BenZvi UT Austin 
TBA 

04/15 3:00pm 
BLOC 628 
Jurij Volcic TAMU 
TBA 

04/26 4:00pm 
BLOC 628 
Michael Di Pasquale Colorado State University 
TBA 