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Geometry Seminar

Fall 2017

Date:September 1, 2017
Time:4:00pm
Location:BLOC 628
Speaker:E. Ventura, TAMU
Title:Catalecticants, antipolars, and ranks of forms
Abstract:The complex (or real) rank of a homogeneous polynomial is the smallest number of powers of complex (or real) linear forms such that the polynomial may be expressed as a linear combination of those. Such an expression is called a minimal Waring or symmetric decomposition. We will talk about real and complex ranks in the setting of bihomogeneous polynomials by the means of maps associated to them: catalecticants and antipolars. We will also explain how the latter ones are related to loci encoding information about minimal decompositions of these bihomogeneous polynomials.

Date:September 9, 2017
Time:4:00pm
Location:Denton
Title:AMS mtg

Date:September 10, 2017
Time:4:00pm
Location:Denton
Title:AMS mtg

Date:September 11, 2017
Time:3:00pm
Location:BLOC 628
Speaker:A. Conner, TAMU
Title:Geometry of symmetrized matrix multiplication

Date:September 15, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Souvik Goswami, TAMU
Title:Higher arithmetic Chow groups
Abstract:For a regular scheme, which is flat and quasi-projective over an arithmetic ring (typically the ring of integers of a number field), Gillet and Soul ́e defined an arithmetic version of the usual Chow groups, taking into account the complex embeddings of the scheme. Typically the complex embeddings add more complex analytic/ Hodge-theoretic informations to the usual Chow groups. On the other hand, higher Chow groups were defined by Spencer Bloch as a simple example of a motivic cohomology. In this talk, we will explore the possibility to obtain a good definition for higher arithmetic Chow groups. This is a joint work with Jos ́e Ignacio Burgos from ICMAT, Madrid.

Date:September 18, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Gregory Pearlstein, TAMU
Title:Mixed Hodge Metrics
Abstract:I will give an overview of recent work with P. Brosnan on the asymptotic behavior of archimedean heights, and with C. Peters on the differential geometry of mixed period domains.

Date:September 22, 2017
Time:4:00pm
Location:BLOC 628
Speaker:V. Makam, U. Michicgan
Title:Degree bounds for invariant rings of quivers
Abstract:The ring of polynomial invariants for a rational representation of a reductive group is finitely generated. Nevertheless, it remains a difficult task to find a minimal set of generators, or even a bound on their degrees. Combining ideas originating from Hochster, Roberts and Kempf with the study of various ranks associated to linear matrices, we prove "polynomial" bounds for various invariant rings associated to quivers. The polynomiality of our bounds have strong consequences in algebraic complexity. If time permits, we will discuss these as well as applications to lower bounds for border rank of tensors. This is joint work with Derksen.

Date:September 25, 2017
Time:3:00pm
Location:BLOC 628
Speaker:F. Gesmundo, Copenhagen
Title:On multiplicativity of various notions of rank
Abstract:Matrix rank has several different generalizations to the setting of tensors. Whereas for matrices it is easy to show that rank is multiplicative over tensor product (the matrix Kronecker product), multiplicativity is not straightforward (and in most cases false) in the setting of tensors. We discuss this problem for various notions of rank: tensor rank, partially symmetric tensor rank and tensor border rank in particular. The geometric framework allows for further generalizations, that we briefly present.

Date:September 29, 2017
Time:4:00pm
Location:BLOC 628
Speaker:JM Landsberg, TAMU
Title:Quantum max flow v. quantum min cut and the geometry of matrix product states

Date:October 2, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Boris Hanin, TAMU
Title:Pointwise Estimates in the Weyl Law on a Compact Manifold
Abstract:Let (M,g) be a compact smooth Riemannian manifold. This talk focuses on connecting the structure of geodesics on (M,g) to the behavior of eigenfunctions of the Laplacian at high frequencies. I will explain a physical heuristic for why such a connection should exist. I will then present some new estimates for the second term in the pointwise Weyl Law. These estimates imply that if the geodesics passing through a given point on M are dispersive (in a suitable sense), then the spectral projector of the Laplacian onto the frequency interval (lambda,lambda+1] has a universal scaling limit as lambda goes to infinity (depending only on the dimension of M). This is joint work with Y. Canzani.

Date:October 13, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Emre Sen, Northeastern
Title:Singularities of dual varieties associated to exterior representations
Abstract:For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Pl\"{u}cker embeddings. We give a complete classification of the irreducible components of the singular locus of several representation classes. Basically, they admit two types of singularities: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point respectively. In particular, our results include a description of  singularities of dual Grassmannian varieties.

Date:October 16, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Tian Yang, TAMU
Title:Rigidity of hyperbolic cone metrics on triangulated 3-manifolds
Abstract:In this joint work with Feng Luo, we prove that a hyperbolic cone metric on an ideally triangulated compact 3-manifold with boundary consisting of surfaces of negative Euler characteristic is determined by its combinatorial curvature. The proof uses a convex extension of the Legendre transformation of the volume function.

Date:October 20, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Michael Di Pasquale, Oklahoma State University
Title:Homological Obstructions to Freeness of Multi-arrangements
Abstract:

If the module of vector fields tangent to a multi-arrangement is free over the underlying polynomial ring, we say that the multi-arrangement is free. It is of particular interest in the theory of hyperplane arrangements to investigate the relation of freeness to the combinatorics of the intersection lattice - the holy grail here is Terao's conjecture that freeness of arrangements is detectable from the intersection lattice. It is known that corresponding statements for multi-arrangements fail.

Given a multi-arrangement, we present a co-chain complex derived from work of Brandt and Terao on k-formality whose exactness encodes freeness of the multi-arrangement. The cohomology groups of this co-chain complex thus present obstructions to freeness of multi-arrangements. Using this criterion we give an example showing that the property of being totally non-free is not detectable from the intersection lattice. This builds on previous work with Francisco, Schweig, Mermin, and Wakefield.


Date:October 23, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Shilin Yu, TAMU
Title:Families of representations of Lie groups
Abstract:Beilinson and Bernstein generalized the Borel-Weil-Bott theorem and showed that representations of a (noncompact) reductive Lie group G can be realized as D-modules on flag variety. In this talk, I will show that such D-modules live naturally in families, which explains a mysterious analogy between representation theory of the group G and a related semidirect product group due to Mackey, Higson and Afgoustidis. Connection with Kirillov's coadjoint orbit method will be discussed. The talk is based partially on joint projects with Qijun Tan, Yijun Yao and Conan Leung.

Date:October 27, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Xiaoxian Tang, TAMU
Title:Investigating multistationarity in structured reaction networks
Abstract:Many dynamical systems arising in applications exhibit multistationarity (two or more positive steady states), but it is often difficult to determine whether a given system is multistationary, and if so to identify a witness to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. We prove two new sufficient conditions for multistationarity: (1) when there are no boundary steady states and a certain critical function changes sign, and (2) when the steady-state equations can be replaced by equivalent triangular-form equations. We also investigate the mathematical structure of this critical function, and give conditions that guarantee that triangular-form equations exist.

Date:November 3, 2017
Time:4:00pm
Location:BLOC 628
Speaker:John Calabrese, Rice University
Title:Comparison Formulae for Donaldson-Thomas Invariants
Abstract:Donaldson-Thomas numbers are virtual enumerative invariants attached to Calabi-Yau threefolds. I will begin the talk to provide some intuition for these, in the context of curve-counting. I will then discuss a few comparison formulae.

Date:November 6, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Bo Lin, UT Austin
Title:The tropical Fermat-Weber points
Abstract:In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well-known that the Fermat-Weber point of a sample is not necessarily unique in the metric space. We investigate the computation of Fermat-Weber points under the tropical metric on the quotient space $\mathbb{R}^{n} \!/ \mathbb{R} {\bf 1}$ with a fixed $n \in \mathbb{N}$, motivated by its application to the space of equidistant phylogenetic trees with $N$ leaves (in this case $n=\binom{N}{2}$) realized as the tropical linear space of all ultrametrics. We show that the set of all tropical Fermat-Weber points of a finite sample is always a classical convex polytope, and we present a combinatorial formula for the minimal sum of distances from an arbitrary point to the points in the sample (which is attained at this set). We identify conditions under which this set is a singleton. We apply numerical experiments to analyze the set of the tropical Fermat-Weber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical Fermat-Weber points and the $k$-ellipses that are generalizations of the Fermat-Weber points. This is a joint work with Ruriko Yoshida.

Date:November 10, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Alicia Harper, Brown University
Title:Factorization of maps of Deligne-Mumford stacks
Abstract:The weak factorization theorem provides a tool for explicitly relating birational varieties by a sequence of smooth blowups and blowdowns. In the setting of Deligne-Mumford stacks, one can hope to do something similar, but first one has to grapple with the fact that stacks also carry a local group structure, and thus one needs to use root stacks, a geometric operation that only exists in the stacky world, in addition to blowups and blowdowns. In this talk I will give a - not too technical - introduction to the above concepts, then discuss how to actually go about proving a stacky weak factorization theorem for non-representable morphisms.

Date:November 13, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Abraham Martín del Campo, CIMAT
Title:Semi-algebraic regions for phylogenetic algorithms
Abstract:

In Biology, phylogenetic trees encode evolutionary relations among observed species, sometimes computed using distance based methods. These methods are iterative processes that take a distance matrix for input, and decide about the next evolutionary relation from partial information in the matrix.

For some algorithms, the decision criteria are polynomial inequalities in the entries of the matrix, decomposing the space of all possible input matrices into semi-algebraic cones.

In this talk, which is suitable for graduate students, I will present some partial results from a joint work with Ruth Davidson (UIUC) where we study these regions in the Neighbor-Joining algorithm.


Date:November 17, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Taylor Brysiewicz, TAMU
Title:Counting polynomially parametrized interpolants via Necklaces
Abstract:We consider the problem of locally approximating an analytic curve in the complex plane plane by a polynomial parametrization t -> (x_1(t),x_2(t)) of bidegree (d_1,d_2). Contrary to Taylor approximations, these parametrizations can achieve a higher order of contact at the cost of losing uniqueness and possibly the reality of the solution. We study the extent to which uniqueness fails by counting the number of such curves as the number of aperiodic combinatorial necklaces on d_1 white beads and d_2 black beads. We analyze when this count is odd as an initial step in studying when real solutions exist.

Date:November 20, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Zheng Zhang, TAMU
Title:On the moduli space of pairs consisting of a cubic threefold and a hyperplane
Abstract:The period map is a powerful tool for studying moduli spaces, and has been applied successfully to abelian varieties, K3 surfaces, cubic threefolds/fourfolds, and hyper-Kahler manifolds. However, for some interesting moduli problems (e.g. moduli spaces for pairs of varieties) there might be no obvious way to construct periods. Joint with R. Laza and G. Pearlstein, we construct a period map for cubic pairs consisting of a cubic threefold and a hyperplane using a variation of the construction by Allcock, Carlson and Toledo (which allows us to encode a cubic pair as a “lattice polarized” cubic fourfold). The main result is that the period map induces an isomorphism between a GIT model of the moduli of cubic pairs and the Baily-Borel compactification of some locally symmetric domain.

Date:November 27, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Souvik Goswami, TAMU
Title:Higher Arithmetic Chow Groups - Part 2
Abstract:We continue in our quest to define an analogue of arithmetic Chow groups, for higher Chow groups. In this talk, I will introduce the notion of higher Chow groups a la Bloch, and then define an arithmetic version of it. We will also propose an intersection theory, which generalizes the one given by Gillet and Soul ́e. This is a joint work (in progress) with Jos ́e Ignacio Burgos Gil from ICMAT, Madrid.

Date:November 30, 2017
Time:3:00pm
Location:BLOC 220
Speaker:Patricio Gallardo , Washington University, St. Louis
Title:Point and line to plane
Abstract: We will discuss higher dimensional generalizations of the moduli of n labeled points in the sphere. In particular, we will compare the standard generalizations, constructed using the minimal model program, with new constructions based on configuration spaces. Most of the results are joint work with E. Routis.

Date:December 1, 2017
Time:4:00pm
Location:BLOC 628
Speaker:Jose Rodriguez, University of Chicago
Title:Numerical computation of Galois groups and braid groups
Abstract:Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In the first part of this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We give examples from formation shape control and algebraic statistics. In the second part, we discuss the generalization to braid groups. Braid groups were first introduced by Emil Artin in 1925 as a generalization of the symmetric group and have more refined information than the Galois group. We develop algorithms to compute a set of generators for these groups using homotopy continuation. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile and with Botong Wang.