MenuGeometry Seminar
Spring 2019
| Date: | January 14, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | C. Farnsworth, Texas State University in San Marcos |
| Title: | Algebraic funtf Completion |
| Abstract: | 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. |
| Date: | January 17, 2019 |
| Time: | 1:00pm |
| Location: | BLOC 628 **note |
| Speaker: | Amy Hang Huang, University of Wisconsin Madison |
| Title: | Equations of Kalman Varieties |
| Abstract: | 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. |
| Date: | January 25, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Xiaoxian Tan, TAMU |
| Title: | Applying Algebraic Methods in Mathematical Biology |
| Abstract: | Many challenging problems in mathematical biology, for instance, in biochemical reaction networks and phylogenetics, are to solve non-linear 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. |
| Date: | February 4, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | C.J. Bott, TAMU |
| Title: | Mirror symmetry for K3 surfaces |
| Abstract: | Mirror symmetry is the phenomenon, originally discovered by physicists, that Calabi-Yau 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 Calabi-Yau 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 Landau-Ginzburg/Calabi-Yau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani-Boissière-Sarti (2011) originally showed that they agree when n=2, and Comparin-Lyon-Priddis-Suggs (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. |
| Date: | February 6, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 220 |
| Speaker: | Jose Burgos Gil, ICMAT, Madrid |
| Title: | Arithmetic of Toric Varieties, Lecture 1. |
| Abstract: | 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 algebro-geometric concepts to combinatorial concepts. With this dictionary many algebro-geometric 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). |
| Date: | February 11, 2019 |
| Time: | 1:00pm |
| Location: | BLOC 628 |
| Speaker: | Jose Burgos Gil, ICMAT, Madrid |
| Title: | Arithmetic of Toric Varieties, Lecture 2 |
| Abstract: | See lecture 1 for the series abstract. |
| Date: | February 13, 2019 |
| Time: | 10:30am |
| Location: | BLOC 628 |
| Speaker: | Jose Burgos Gil, ICMAT, Madrid |
| Title: | Arithmetic of Toric Varieties, Lecture 3 |
| Abstract: | See lecture 1 for series abstract. |
| Date: | February 25, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | A. Conner, TAMU |
| Title: | Kronecker powers of tensors and the exponent of matrix multilplication |
| Date: | March 1, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Visu Makam, IAS |
| Title: | Exponential degree lower bounds for invariant rings |
| Abstract: | 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 4-tuples of cubic forms. |
| Date: | March 4, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Tingran Gao, U. Chicago |
| Title: | Manifold Learning on Fibre Bundles |
| Abstract: | 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 non-scalar 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. |
| Date: | March 8, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Giuseppe Martone, University of Michigan |
| Title: | Hitchin representations and positive configurations of apartments |
| Abstract: | 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 Morgan-Shalen 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 Fock-Goncharov construction to describe the intersection patterns of apartments in invariant subsets of the building that arises in the boundary of the Hitchin component. |
| Date: | March 22, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Igor Zelenko, TAMU |
| Title: | Projective and affine equivalence of sub-Riemannian metrics: generic rigidity and separation of variables conjecture. |
| Abstract: | Two sub-Riemannian 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 Levi-Civita in 1898 and Eisenhart in 1923, respectively. In particular, a Riemannian metric admitting a nontrivial (i.e. non-constant 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 sub-Riemannian metrics. In particular, we will discuss genericity of metrics that do not admit non-constantly 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 non-constantly proportional affinely equivalent metrics. The talk is based on the collaboration with Frederic Jean (ENSTA, Paris) and Sofya Maslovskaya (INRIA, Sophya Antipolis). |
| Date: | March 25, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | E. Ventura, TAMU |
| Title: | Tensors and their symmetry groups |
| Abstract: | Tensors (multi-dimensional 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. |
| Date: | March 29, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Shilin Yu, TAMU |
| Title: | Deformation quantization of coadjoint orbits |
| Abstract: | The coadjoint orbit method/philosophy suggests that irreducible unitary representations of a Lie group can be constructed as quantization of coadjoint orbits of the group. I will propose a geometric way to understand orbit method using deformation quantization, in the case of noncompact real Lie groups. This approach combines recent studies on quantization of symplectic singularities and their Lagrangian subvarieties. This is joint work with Conan Leung. |
| Date: | April 8, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Bernd Siebert, University of Texas |
| Title: | Toric degenerations - a finite element method in algebraic geometry |
| Abstract: | Toric degenerations in the broad sense referred to here, are deformations with central fiber a union of toric varieties, intersecting pairwise along joint toric divisors. A typical example is a family of quartic hypersurfaces in projective 3-space with central fiber the union of four coordinate hyperplanes. The interesting thing about such degenerations is that there is a large class of examples that can be produced canonically out of discrete data, thus giving a vast generalization of toric geometry (joint work with Mark Gross). In the talk I will give an overview of such degenerations, the relation to tropical geometry and wall structures, the explanation of the mirror phenomenon in this framework and the appearance of special functions generalizing Riemannian theta functions. |
| Date: | April 12, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | David Ben-Zvi, UT Austin |
| Title: | Integrating quantum hamiltonians |
| Abstract: | Harish Chandra showed that symmetric spaces carry large commutative algebras of invariant differential operators, generalizing the Laplacian. The symbols of these differential operators provide the commuting Hamiltonians responsible for many classical integrable systems. I'll describe joint work with Sam Gunningham (arXiv/1712.01963) describing a universal ``integration" for these classical and quantum Hamiltonian systems, through a categorical analog of the Harish-Chandra isomorphism, constructed via the geometric Langlands correspondence. |
| Date: | April 15, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Jurij Volcic, TAMU |
| Title: | Free loci of noncommutative polynomials |
| Abstract: | Let f be a noncommutative polynomial. The free locus of f is the set of all tuples of matrices X such that f(X) is singular. That, is the free locus is an infinite family of determinantal hypersurfaces (one for each size of matrices). The adjective ``free'' relates to quickly emerging free analysis and free real algebraic geometry, which study noncommutative functions on matrices in a certain dimension-free setting. Free loci play are important in (noncommutative) control theory and convex optimization. However, in this talk we will connect them to factorization in free algebra: roughly speaking, components of the free locus of f correspond to distinct irreducible factors of f, and irreducible polynomials are determined by their free loci. We will focus on the role of invariant theory for the general (and special) linear group in the proof. The talk is based on joint work with Bill Helton and Igor Klep. |
| Date: | April 22, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | J. Maurice Rojas, Texas A&M University |
| Title: | Explicit Univariate Polynomials with Optimal Condition Number (after Beltran, et. al.) |
| Abstract: | The complexity of polynomial system solving depends not only on the input polynomials, but also on their distance from a suitable discriminant variety. One measure of this distance is the condition number, and there are now even theorems to estimate it with high probability. A consequence of these results is that "most" polynomial systems are "well-conditioned," meaning that "most" polynomial systems are "easy" to solve. Rigorously stated, this is the content of Lairez's recent solution to Smale's 17th Problem.
However, a vexing question left open was the construction of an explicit family (computable in polynomial-time) of univariate polynomials with low condition number. This is an instance of "finding hay in a haystack": an object occuring with high probability, lacking an explicit construction. We review a recent solution to this problem by Beltran, Etayo, Marzo, and Orega-Cerda, as well as its connection to Smale's 7th Problem on well-spaced points on the unit 2-sphere. |
| Date: | April 26, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Michael Di Pasquale, Colorado State University |
| Title: | The asymptotic containment problem for symbolic powers of ideals |
| Abstract: | The symbolic powers of an ideal I, denoted I^(s), are an important geometric analogue of taking regular powers. There is significant interest in the containment problem; that is studying which pairs (r,s) satisfy that I^(s) is contained in I^r. A celebrated result of Ein,Lazarsfeld, and Smith and Hochster and Huneke states that I^(hr) is contained in I^r where I is an ideal of big height h in a regular ring. In an effort to quantify these containment results more precisely, the notions of resurgence and asymptotic resurgence of an ideal were introduced by Bocci and Harbourne and Guardo, Harbourne, and Van Tuyl. We show that the asymptotic resurgence of an ideal can be computed using integral closures, which leads to a characterization of asymptotic resurgence as the maximum of finitely many Waldschmidt-like constants. For monomial ideals these constants can be computed by solving linear programs over the symbolic polyhedron introduced by Cooper, Embree, Ha, and Hoefel. This makes it reasonable to compute the asymptotic resurgence of many monomial ideals, leading to some interesting examples related to combinatorial optimization where asymptotic resurgence and resurgence are different. This is joint work with Chris Francisco, Jeff Mermin, and Jay Schweig. |
| Date: | April 29, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | A numerical toolkit for multiprojective varieties |
| Abstract: | A multiprojective variety is a subvariety of a product of projective spaces. They may be studied as projective varieties under the Segre embedding or locally as affine varieties in an affine patch. Both approaches have disadvantages, increasing complexity or ignoring structure. I will discuss methods from numerical algebraic geometry to study multiprojective varieties that take advantage of their structure and do not increase the complexity of the numerical representation. This is joint work with Hauenstein, Leykin, and Rodriguez. |
