Confirmed Plenary Speakers


PDF Schedule



Poster Session


Click here for the poster presenters and titles.


All TAGS activities will take place in the Blocker Building of the Texas A&M campus. For more details, specific rooms, a map and parking directions, see the Local Information.

Titles and Abstracts

Lectures

  • David Ben-Zvi, UT Austin.
    Title: Geometric Theory of Harish Chandra Characters
    Abstract:
    I'll discuss recent work with David Nadler (Northwestern) exploring the theory of characters for Lie group actions on categories. The general theme is the surprisingly tight analogy between the categorified theory and the classical representation theory of finite groups. As an application I'll explain a purely geometric approach to Harish Chandra's theory of distributional characters for (infinite dimensional) unitary representations of Lie groups.
  • Tommaso de Fernex, University of Utah.
    Title: Rationality in families of threefolds
    Abstract:
    How does the property of being rational behaves in families of varieties? The question whether the locus of rational fibers in a smooth family of complex projective varieties is the union of at most countably many closed subfamilies has been around for some time. In joint work with Davide Fusi, we prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this gives a positive answer to the previous question in dimension three.
  • Harm Derksen, University of Michigan.
    Title: Quiver Representations in Schubert Calculus
    Abstract:
    I will give several examples of how the theory of quiver representations can be used to attack problems in Schubert Calculus.
  • Sean Keel, UT Austin (Special Lecture for Graduate Students).
    Title: Mirror Symmetry made easy
    Abstract:
    I will describe my conjecture (and theorem in dimension two), joint with Hacking and Gross, which gives an elementary synthetic construction of the mirror to an affine Calabi Yau, as the spectrum of an explicit ring: A vector space with canonical basis parametrized by a natural mori theoretic generalisation of the thurston boundary to teichmuller space, and with multiplication rule given by counts of rational curves. No knowledge of mirror symmetry is required - If you know what is meant by the order of zero or pole of a rational function along a divisor, you know enough to understand the construction, as well as some surprising implications, e.g. that an affine CY variety with maximal boundary (for example an affine cubic surface, or the complement to a nodal plane cubic, or the character variety of a riemann surface) has "theta functions" - a canonical basis of global functions.

  • Abhinav Kumar, MIT.
    Title: Rational Elliptic Surfaces with High Mordell-Weil Rank and Multiplicative Bad Fibers
    Abstract:
    Elliptic surfaces are a natural generalization of elliptic curves, by replacing the Weierstrass coefficients by polynomials in one variable. One of the most basic questions for an elliptic curve is: what is its Mordell-Weil group?
    In the case of elliptic surfaces, this is (isomorphic to) the group of sections of the elliptic fibration. In the simplest possible case, the elliptic surface is birational to P^2. Here, the possible Mordell-Weil groups that could arise were classified by Oguiso and Shioda.
    The study of specific Mordell-Weil lattices has connections to invariant theory and inverse Galois theory (for instance, of some Weyl groups of root lattices), and Shioda has used these techniques to construct "excellent" families of rational elliptic surfaces with additive reduction and Mordell-Weil lattice E8, E7, E6, D4 etc, and also for multiplicative reduction and Mordell-Weil lattice E6.
    I will describe joint work with Shioda which deals with the multiplicative reduction case with Mordell-Weil lattice E8 or E7. The parameters of the "excellent" families are related to the fundamental multiplicative invariants of the corresponding Weyl groups. We use our results to produce examples of elliptic surfaces for which the splitting field has large Galois group, and also examples for which it has trivial Galois group (all sections defined over Q).
  • Brian Lehmann, Rice University (Special Lecture for Graduate Students).
    Title: An introduction to the minimal model program
    Abstract:
    The minimal model program relates the geometry of a variety to properties of its cotangent bundle. I will explain the goals of the program and give an informal introduction to recent developments in the field.
  • Rekha Thomas, University of Washington.
    Title: The Triangulation Problem in Computer Vision
    Abstract: A fundamental problem in computer vision is to reconstruct the 3D coordinates of a scene from noisy images of it in a set of cameras. This problem can be modeled as a polynomial optimization problem with determinantal constraints. These determinants define the multi-view ideal which is endowed with rich combinatorics. In joint work with Chris Aholt and Bernd Sturmfels we determine a universal Groebner basis for this ideal. We show that the Hilbert scheme of this ideal is connected and that it has a distinguished component that carries all multi-view ideals when there are at least three cameras. In joint work with Sameer Agarwal and Chris Aholt, we then solve the polynomial optimization problem using convex relaxation methods that rely on semidefinite programming and apply the methods to reconstructing well known point sets in the computer vision literature.
  • Josephine Yu, Georgia Tech.
    Title: Computing Tropical Resultants
    Abstract:
    We fix the supports A=(A_1,...,A_k) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We show that TR(A) equals the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time. The tropical resultant inherits a fan structure from the secondary fan of the Cayley configuration of A and we present algorithms for the traversal of TR(A) in this structure. We also present a new algorithm for recovering a Newton polytope from the support of its tropical hypersurface. We use this to compute the Newton polytope of the sparse resultant polynomial in the case where TR(A) is of codimension 1. Finally we consider the more general setting of specialized tropical resultants and report on experiments with our implementations. This is joint work with Anders Jensen.

Posters

  • Christopher Brooks, Texas A&M University.
    Title: Galois groups of Schubert problems of lines
    Abstract:
    Schubert problems are a class of geometric counting problems, and the goal of our work is to study the structure of these problems through their Galois groups. Experimentally, we found that the Galois groups of Schubert problems involving lines always contain the alternating group, which implies they have no special geometric structure. Building on work of Vakil and Sottile, we proved this fact using a novel approach which reduces the problem to an inequality of integrals. This is joint work with Abraham Martín del Campo and Frank Sottile.
  • Taylor Dupuy, University of New Mexico.
    Title: Arithmetic Deformation Theory of Curves
    Abstract:
    We study a construction of and Deligne- Illusie which associates to a lift of a scheme defined over a field in characteristic p a cohomology class in the Frobenius tangent bundle. This class can be thought of as an Arithmetic Kodaira-Spencer class.
  • Zac Griffin,Texas A&M University.
    Title: Real Solutions and Parameter Continuation
    Abstract:
    We propose a homotopy-based method which attempts to compute the real solution of a polynomial system of minimum distance from the given point. This method is modeled on the optimization method of gradient descent. We apply this method to compute points on the discriminant locus as well as move in a parameter space to change the number of real solutions for a parameterized family of polynomial systems.
  • Nickolas Hein, Texas A&M University.
    Title: A mod four congruence in the symmetric real Schubert calculus
    Abstract:
    A Schubert problem is an intersection of Schubert varieties containing a finite positive number of points. In 1994 Boris and Michael Shapiro conjectured that if the Schubert varieties are all osculating the same rational normal curve at distinct real points then all points in the intersection are real. A special case was proven by Eremenko and Gabrielov in 2002, and the conjecture was proven for all Grassmannian Schubert problems by Mukhin-Tarasov-Varchenko in 2009. Also in 2002, Eremenko-Gabrielov studied a related problem for which the reality condition on osculation points is somewhat relaxed, but the intersected Schubert varieties are hypersurfaces. In this case, not all solutions must be real, but they proved nontrivial lower bounds on the numbers of real solutions to such systems. While investigating these lower bounds we observed that for problems with some symmetry the number of real solutions is invariant mod 4. We give some insight on this phenomenon and furnish some theorems.
  • Jen-Chieh Hsiao, Purdue University.
    Title: Cartier modules on toric varieties
    Abstract:
    In this joint work with K. Schwede and W. Zhang, we describe combinatorially the ideals on a toric variety that are fixed by a given Cartier algebra. We also show that geometrically such ideals can be view as certain generalization of the defining ideals of unions of log canonical centers in the minimal model program.
  • Harlan Kadish, Texas A&M University.
    Title: A better ring for algebraic equivalence relations
    Abstract:
    When an algebraic group G acts on a variety V, the functions in k[V] usually fail to separate the orbits. So extend k[V] with a new "quasi-inverse" operation that computes the reciprocal of a polynomial where defined. Then there exists an efficient algorithm that computes a finite set of functions in this "quasi-inverse ring" that are G-invariant and separate orbits. It turns out that functions in the quasi-inverse ring can separate the classes of some other equivalence relations on a variety: those arising from algebraic dynamical systems, from formulas in first order logic, or from ideals of polynomials on V x V. In the latter case, in fact finitely many functions suffice.
  • Robert Krone, Georgia Institute of Technology.
    Title: Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
    Abstract:
    An ideal of a local polynomial ring can be described by calculating a standard basis, but in contexts where only approximate numerical computations are allowed this process is not numerically stable. Ideals can be described numerically by the dual functionals that annihilate it. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for zero-dimensional ideals. I present a stopping criterion for the positive-dimensional case based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.
  • Kuei-Nuan Lin, University of California, Riverside.
    Title: The Rees Algebras and Higher Secant Varieties of Determinantal varieties
    Abstract:
    Given a determinantal variety, we consider the diagonal ideal, kernel of the multiplication map of the homogeneous coordinate ring. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the first secant variety of the variety. We can construct the s-th secant variety via (s-1)-th secant variety and first secant variety. We consider the Rees algebra of the diagonal ideal that special fiber is corresponding to s-th secant variety. By finding the defining ideals of the Rees algebras, we are able to understand the special fiber for free.
  • Tina Mai, Texas A&M University.
    Title: The convex hulls of two circles
    Abstract:
    We describe the boundaries of convex hulls of perhaps the simplest compact curves in three-dimensional space, pairs of circles. The edge surface of such a convex hull is in general an irrational ruled surface whose rulings form a (2, 2)-curve in the product of circles. We are showing that every smooth real (2, 2)-curve arises from the edge surface of convex hull of some two circles.
  • Brian Miller, Texas Tech University.
    Title: On the integration algebraic functions: Computing the logarithmic part
    Abstract:
    We provide an algorithm to compute the logarithmic part of an integral of an algebraic function. The algorithm finds closed-form solutions that have been difficult to compute by other means. We are able to fully justify the algorithm using techniques of Gröbner bases, differential algebra, and algebraic geometry.
  • Jose Rodriguez, UC Berkeley.
    Title: Homotopies for Maximum Likelihood Estimation
    Abstract:
    The MLE-problem is to optimize a likelihood function on a discrete statistical model for given data. To solve this problem, we compute the critical points of a function on the Zariski closure of a statistical model for given data. These critical points are candidates for solutions to the MLE problem. By doing the expensive computation of computing critical points once for generic data, we then use numerical algebraic algebraic geometry to construct a homotopy that will solve the MLE-problem for arbitrary data quickly.
  • Letao Zhang, Rice University.
    Title: Modular Forms and Special Cubic Fourfolds
    Abstract:
    We study the degree of the special cubic fourfolds in the Hilbert scheme of cubic fourfolds via a computation of the generating series of Heegner divisors of even lattice of signature (2, 20). (Joint work with Z. Li)