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# Geometry Seminar

## Fall 2019

 Date: August 26, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Guangbo Xu, Texas A&M University Title: Introduction to Floer homology Abstract: In this talk, after reviewing the classical finite dimensional Morse theory, I will give a short introduction to various versions of Floer homology theories, for example, Lagrangian intersection Floer homology in symplectic geometry and instanton Floer homology in gauge theory. I will also explain the conceptual picture of the celebrated Atiyah-Floer conjecture. Non-experts, especially graduate students are welcome. Date: September 2, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Mounir Nisse, Xiamen University Malaysia Title: Phase tropical varieties are topological manifolds Abstract: After defining some tropical tools and giving an overview of the subject, we prove first that phase tropical curves, that are limits of algebraic smooth complete intersection curves, are topological manifolds. We generalize this fact to phase tropical k-planes, and then to phase tropical varieties that are limits of algebraic smooth complete intersection varieties (work is in progress). Date: September 20, 2019 Time: 4:00pm Location: BLOC 628 Speaker: A. Conner, TAMU Title: Border Apolarity of tensors and matrix multiplication Abstract: I will present a new method for border rank lower bounds of tensors which exploits continuous symmetry. I will discuss new results from this method's application to the matrix multiplication tensor, whose border rank is fundamentally related to the complexity of matrix multiplication. Date: September 27, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Souvik Goswami, Texas A&M University Title: Height pairing on Bloch's higher cycles and mixed Hodge structures. Abstract: In a previous work with José Ignacio Burgos, we have studied the higher arithmetic Chow groups. As a by product, an Archimedean height pairing between higher cycles has been defined. Classically, Hain has shown that the Archimedean component of the height pairing between ordinary cycles can be interpreted as the class of a biextension in the category of mixed Hodge structures. In the current work we study the mixed Hodge structure defined by a pair of higher cycles intersecting properly and show that, in a special case, the Archimedean height pairing is one of the periods attached to such mixed Hodge structure. This is joint work in progress with Greg Pearlstein and José Ignacio Burgos. Date: October 4, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Ursula Whitcher, University of Michigan Title: Zeta functions of alternate mirror Calabi-Yau families Abstract: Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. For example, one can use mirror symmetry to explore the structure of the zeta function, which encapsulates information about the number of points on a variety over a finite field. We prove that if two Calabi-Yau invertible pencils in projective space have the same dual weights, then they share a common polynomial factor in their zeta functions related to a hypergeometric Picard-Fuchs differential equation. The polynomial factor is defined over the rational numbers and has degree greater than or equal to the order of the Picard-Fuchs equation. This talk describes joint work with Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, and John Voight. Date: October 5, 2019 Time: 10:00am Location: BLOC 628 Title: Fall TAGS program. Abstract: The Texas Algebraic Geometry Symposium is a joint seminar of Rice University, Texas A&M University, and the University of Texas at Austin. This conference aims to bring to a regional audience the latest developments in Algebraic Geometry. This Fall there will be a weekend program to complement the main conference series. These events will be held at Texas A&M campus on October 5 and October 6. The speakers will be: Jennifer Balakrishnan, Boston University. Daniel Erman, University of Wisconsin-Madison Sarah Frei, Rice University. Jessica Sidman, Mt. Holyoke College Emanuele Ventura, Texas A&M University. Ursula Whitcher, University of Michigan Date: October 7, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Matthew Ballard, University of South Carolina Title: Can the derived category detect rationality? Abstract: Rationality questions and the structure of derived categories of coherent sheaves have been shown to be intimately tied together over the past 30 years - both via evidence and tantalizing conjectures. Perhaps the most basic question one can ask is: are there natural conditions on D(X) which would imply the rationality of X? One of the simplest structural conditions on D(X) is that it can be broken into pieces that are derived categories of (smooth) points. In joint work with Duncan, Lamarche, and McFaddin, we show that this is insufficient to guarantee the existence of a k-point in general much less rationality. However if one assumes all the points are just Spec k then we verify this guarantees rationality for toric varieties. Date: October 11, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Taylor Brysiewicz, Texas A&M University Title: The degree of Stiefel manifolds and spaces of Parseval frames. Abstract: The (k, n)-th Stiefel manifold is the space of k×n matrices M with the property that M*M^T=Id. Equivalently, this is the space of Parseval n-frames for k-dimensional space. The polynomial equations characterizing the Stiefel manifold define an embedded affine algebraic variety. We will sketch our proof of a formula for its degree using aspects of representation theory, Gelfand-Tsetlin polytopes, and the combinatorics of non-intersecting lattice paths. [joint work with Fulvio Gesmundo] Date: October 18, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Paulo Lima-Filho, Texas A&M University Title: Transforms of geometric currents under correspondences and regulators for Higher Chow groups. Abstract: In this talk we show how equidimensional algebraic correspondences between complex algebraic varieties can be used to construct pull-backs and transforms on a class of currents representable by integration. As a main application we exhibit explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth quasi-projective complex algebraic varieties to Deligne-Beilinson cohomology, utilizing the original simplicial description of Higher Chow groups with integral coefficients. The main ingredients come from Suslin's equidimensionality results, which show that suitable complexes of equidimensional correspondences are quasi-isomorphic to Bloch's original complex. We indicate how this can be applied to Voevodsky's motivic complexes and realizations of mixed motives. The GMT constructions may be extended to more general metric spaces, such as rigid analytic spaces. This is joint work with Pedro dos Santos and Robert Hardt. Date: October 21, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Donghao Wang, MIT Title: Finite Energy Monopoles on $\C \times \Sigma$ Abstract: The Seiberg-Witten (monopole) equations and the monopole invariants introduced by Witten have greatly influenced the study of smooth 4-manifolds since 1994. By studying its dimensional reduction in dimension 3, Kronheimer-Mrowka defined the monopole Floer homology for any closed 3-manifolds. In this talk, we continue this reduction process and consider the moduli space of solutions on $X=\mathbb{C}\times\Sigma$, where $\Sigma$ is a compact Riemann surface. We will classify solutions to the Seiberg-Witten equations on $X$ with finite analytic energy and estimate their decay rates at infinity according to the algebraic input. The motivation is to extend the construction of Kronheimer-Mrowka for compact 3-manifolds with boundary, and this work is the first step towards this goal. Date: October 25, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Jordyn Harriger, Indiana University Title: Planar Algebras Related to the Symmetric Groups. Abstract: What makes the symmetric groups special ? Well, one interesting thing about S_n is that it has a subgroup of index n and that the permutation representation of S_n comes from inducing the trivial representation of that subgroup. How could this generalize if n was not an integer? Using planar algebras we can describe Rep(S_n) graphically. Then using this graphical description we can construct a planar algebra for Rep(S_t), where t is not an integer, via inter- polation between the Rep(S_n)'s. Additionally, I will describe how this planar algebra can from a special biadjunction between tensor categories, which gen- eralizes the induction and restriction relations between S_n and S_{n-1}. I will also discuss the relationship between these planar algebras and usual partition algebra description of Rep(S_t). Date: November 1, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Margaret Regan, Notre Dame Title: Applications of Parameterized Polynomial Systems. Abstract: Many problems in computer vision and engineering can be formulated using a parameterized system of polynomials which must be solved for given instances of the parameters. Due to the nature of these applications, solutions and behaviour over the real numbers are those that provide meaningful information for the system. This talk will describe using homotopy continuation within numerical algebraic geometry to solve these parameterized polynomial systems. It will also discuss applications regarding 2D image reconstruction in computer vision and 3RPR mechanisms in kinematics. Date: November 4, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Nida Obatake, Texas A&M University Title: Polyhedral methods for chemical reaction networks Abstract: Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. A major problem asks about the stability of steady states of these networks. Rubinstein et al. (2016) showed that the ERK network exhibits multiple steady states, bistability, and undergoes periodic oscillations for some choice of rate constants and total species concentrations. The ERK network reduces to the processive dual-site phosphorylation network when certain reactions are omitted, and this network is known to have a unique, stable steady state (Conradi and Shiu, 2015). To investigate how the dynamics change as reactions are removed from the ERK network, we analyze subnetworks of the ERK network. In particular, we prove that oscillations persist even after we greatly simplify the model by making all reactions irreversible and removing intermediates. To prove this, we introduce the Newton-polytope Method: an algorithmic procedure that uses techniques from polyhedral geometry to construct a positive point where a pair of polynomials achieve certain desired sign conditions. We then use our algorithm to apply an algebraic criterion for Hopf bifurcations that relies on analyzing polynomials (Yang, 2002). Additionally, we investigate the maximum number of steady states of a system by defining a notion of a mixed volume of a chemical reaction network. In general, the mixed volume is an upper bound on the number of complex-number steady states, but we show that this bound is tight for ERK networks. Joint work with Anne Shiu, Xiaoxian Tang and Angelica Torres. Date: November 11, 2019 Time: 3:00pm Location: BLOC 628 Speaker: M. Michalek, MPI Leipzig Title: Singularities of secant and tangential varieties of Segre-Veronese varieties Abstract: We will show applications of ideas from statistics to study classical objects in algebraic geometry. A change of coordinates, inspired by computation of cumulants, reveals a toric structure on secant variety of any Segre-Veronese variety. We will show how to exploit this structure to study the singularities. Date: November 14, 2019 Time: 4:00pm Location: BLOC 628 Speaker: M. Michalek, MPI Leipzig Title: C^*-actions, ML-degree and various Grassmannians Abstract: Central questions in many branches of mathematics are related to understanding the cohomology class of a graph of a Cremona transformation. One case is statistics, where the Maximum Likelihood degree of Gaussian models can be expressed exactly in this terms. I will report on work in progress with Wisniewski, where we upgrade the relevant Cremona transformation to a C^* action on a blow up of a homogeneous variety. This leads to a new computational approach towards the ML degree. Date: November 18, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Elise Walker, Texas A&M University Title: Toric degenerations and optimal homotopies from finite Khovanskii bases Abstract: Homotopies are useful numerical methods for solving systems of polynomial equations. I will present such a homotopy method using Khovanskii bases. Finite Khovanskii bases provide a flat degeneration to a toric variety, which consequentially gives a homotopy. The polyhedral homotopy, which is implemented in PHCPack, can be used to solve for points on a general linear section of this toric variety. These points can then be traced via the Khovanskii homotopy to points on a general linear slice of the original variety. This is joint work with Michael Burr and Frank Sottile. Date: November 22, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Guillem Cazassus, Indiana University Title: Equivariant Lagrangian Floer homology and extended Field theory Abstract: Given a Hamiltonian G-manifold endowed with a pair of G-Lagrangians, we provide a construction for their equivariant Floer homology. Such groups have been defined previously by Hendricks, Lipshitz and Sarkar, and also by Daemi and Fukaya. A similar construction appeared independently in the work of Kim, Lau and Zheng. We will discuss an attempt to use such groups to construct topological Field theories: these should be seen as 3-morphism spaces in the Hamiltonian 3-category, which should serve as a target for a Field theory corresponding to Donaldson polynomials. Date: December 2, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Charles Doran, University of Alberta, Canada Title: Calabi-Yau Geometry of the Multiloop Sunset Feynman Integrals Abstract: We will explore the "geometric gems" that emerge naturally when computing the simplest infinite family of Feynman integrals. These include Hessians of cubic surfaces, complete intersections in permutohedral varieties, and Landau-Ginzburg mirrors of weak Fano varieties. An iterative fibration structure on Calabi-Yau varieties, and a consequent iterative description of their periods, is ultimately crucial to understanding these Feynman integrals. We derive from this a conjectural "motivic mirror" principle that recasts Feynman integrals in terms of Landau-Ginzburg models fibered by motivic Calabi-Yau varieties. Date: December 13, 2019 Time: 4:00pm Location: BLOC 628 Speaker: Tim Seynnaeve, MPI Leipzig Title: Uniform Matrix Product States from an Algebraic Geometer’s point of view Abstract: Uniform matrix product states are certain tensors that describe physically meaningful states in quantum information theory. We apply methods from algebraic geometry to study the set of uniform matrix product states. In particular, we provide many instances in which the set of uniform matrix product states is not closed, answering a question posed by Hackbusch. We also confirm a conjecture of Critch and Morton asserting that, under some assumptions, matrix product states are identifiable''. Roughly speaking, this means that the parametrizing map is as injective as it could possibly be. Finally, we managed to compute defining equations for the variety of uniform matrix product states for small parameter values. This talk is based on joint work with Adam Czaplinski and Mateusz Michalek.