| Date: | February 16, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Julia Lindberg, University of Texas |
| Title: | On the typical and atypical solution to the Kuramoto equations |
| Abstract: | The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has been much work to effectively bound the number of equilibria to the Kuramoto model for a given network. By formulating the Kuramoto equations as a system of algebraic equations, we first relate the complex root count of the Kuramoto equations to the combinatorics of the underlying network by showing that the complex root count is generically equal to the normalized volume of the corresponding adjacency polytope of the network. We then give explicit algebraic conditions under which this bound is strict and show that there are conditions where the Kuramoto equations have infinitely many equilibria. |
|
| Date: | February 19, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | Welschinger signs and the Wronski map |
| Abstract: | A general real rational plane curve C of degree d has 3(d-2) flexes and (d-1)(d-2)/2 complex double points. Those double points lying in RP^2 are either nodes or solitary points. The Welschinger sign of C is (-1)^s, where s is the number of solitary points. When all flexes of C are real, its parameterization comes from a point on the Grassmannian under the Wronskii map, and every parameterized curve with those flexes is real (this is the Mukhin-Tarasov-Varchenko Theorem). Thus to C we may associate the local degree of the Wronskii map, which is also 1 or -1. My talk will discuss work with Brazelton and McKean towards a possible conjecture that that these two signs associated to C agree, and the challenges to gathering evidence for this. |
|
| Date: | February 26, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Stefano Marini, University of Parma |
| Title: | On finitely Levi nondegenerate closed homogeneous CR manifolds |
| Abstract: | A complex flag manifold F= G /Q decomposes into finitely many real orbits under the action of a real form of G. Their embedding into F define on them CR manifold structures. We give a complete classification of all closed simple homogeneous CR manifolds which have finitely nondegenerate Levi forms. |
|
| Date: | March 8, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | T. Mandziuk, TAMU |
| Title: | Border varieties of sums of powers |
| Abstract: | The variety of sums of r powers (VSP(F,r)) of a homogeneous degree d polynomial F is the closure in the Hilbert scheme of the set of all those r-tuples of points of the d-th Veronese variety that contain F in their linear span. As a main ingredient in their border apolarity theory, Buczyńska and Buczyński introduced the notion of a border variety of sums of powers. During the talk I will compare VSP(F,r) (and a similarly defined subset of the Hilbert scheme) with the border variety of sums of powers. The talk is based on a joint work with Emanuele Ventura. |
|
| Date: | March 22, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Paulina Hoyos Restrepo, UT Austin |
| Title: | Manifold Learning in the Presence of Symmetries |
| Abstract: | Graph Laplacian-based algorithms for data lying on a manifold have proven effective for tasks such as dimensionality reduction, clustering, and denoising. Consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. In this talk, I will show how to construct a G-invariant graph Laplacian (G-GL) by incorporating the distances between all the pairs of points generated by the action of G on the data set. The G-GL converges to the Laplace-Beltrami operator on the data manifold, with a significantly improved convergence rate compared to the standard graph Laplacian, which uses only the distances between the points in the given data set. |
|
| Date: | April 5, 2024 |
| Time: | 6:00pm |
| Location: | TBA |
| Title: | Texas Algebraic Geometry Symposium (TAGS) |
| Abstract: | see https://franksottile.github.io/conferences/TAGS24/index.html |
|
| Date: | April 6, 2024 |
| Time: | 10:00am |
| Location: | TBA |
| Title: | Texas Algebraic Geometry Symposium (TAGS) |
| Abstract: | see https://franksottile.github.io/conferences/TAGS24/index.html |
|
| Date: | April 12, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Brandon Ashley, Southern Oregon University |
| Title: | A Group-Theoretic Approach to Darboux Integrable $f$-Gordon Equations |
| Abstract: | Classically, a partial differential equation is said to be Darboux integrable if its general solution can be found by only solving a system of ordinary differential equations. In this talk, we describe a transformation group-theoretic approach to the study of Darboux integrable equations and highlight how this approach can be used to solve the equivalence problem for so-called $f$-Gordon equations of the form $u_{xy} = f(x,y,u,u_x,u_y)$. |
|
| Date: | April 15, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | R. Ramkumar, Cornell |
| Title: | Cartwright-Sturmfels ideals and their moduli |
| Abstract: | Cartwright-Sturmfels ideals, CS-ideals for short, are multigraded ideals whose generic initial ideals are radical. First studied by Cartwright and Sturmfels, some examples include the ideals of maximal minors of a matrix of linear forms, binomial edge ideals, closure of linear spaces, and multiview ideals. In this talk, I will discuss the geometry of CS-ideals inside the multigraded Hilbert scheme, with a particular focus on bigraded CS-ideals. This is joint work with Alessio Sammartano. |
|
| Date: | April 19, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Alex Cohen, MIT |
| Title: | An optimal inverse theorem for the rank of tensors |
| Abstract: | A polynomial f(x_1, … x_n) over a finite field has a large bias if its output distribution is far from uniform. It has rank `r' if we can write `f' as a function of polynomials g_1, …, g_r that each have smaller degree. Bias measures the amount of randomness, and rank measures the amount of structure. It is known that if `f' has small rank, it must have large bias. Green and Tao proved an inverse theorem stating that if `f' is significantly biased, its rank is bounded. Their bound was qualitative, however, and several authors gave quantitative improvements. We prove an optimal inverse theorem: the rank and the log of the bias are equivalent up to linear factors (over large enough fields). Our techniques are very different from the usual methods in this area, we rely on algebraic geometry rather than additive combinatorics. This is joint work with Guy Moshkovitz. |
|
| Date: | April 29, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Akash Sengupta, U. Waterloo |
| Title: | Uniform bounds on Sylvester-Gallai type configurations of polynomials |
| Abstract: | The classical Sylvester-Gallai theorem says that if a finite set of points in the Euclidean plane has the property that the line joining any two points contains a third point from the set, then all the points must be collinear. More generally, a Sylvester-Gallai type configuration is a finite set of geometric objects with certain local dependencies. A remarkable phenomenon is that the local constraints give rise to global dimension bounds for linear SG-type configurations, and such results have found far reaching applications to complexity theory and coding theory.
In this talk we will discuss non-linear generalizations of SG-type configurations which consist of polynomials. We will discuss how the commutative-algebraic principle of Stillman uniformity can shed light on low dimensionality of SG-configurations. I’ll talk about results showing that these non-linear SG-type configurations are indeed low-dimensional as conjectured by Gupta. This is based on joint works with A. Garg, R. Oliveira and S. Peleg.
|
|
| Date: | May 3, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Ronan Conlon, UT Dallas |
| Title: | A family of Kahler flying wing steady Ricci solitons |
| Abstract: | Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension n>2. This is joint work with Pak-Yeung Chan and Yi Lai. |
|
| Date: | May 6, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | F. Gesmundo, U. Toulouse |
| Title: | Collineation varieties of tensors |
| Abstract: | A linear space of matrices gives rise naturally to a family of algebraic
varieties: the k-th collineation variety arises from the parametrization
induced by the minors of size k. We propose the study of collineation
varieties to obtain `equations' for interesting varieties in the space
of tensors. In recent work with Hanieh Keneshlou, we classify these
varieties in the case of pencils of matrices and nets of matrices of
small dimension. In this seminar, I will introduce this construction and
show to what extent the collineation varieties, and their geometric
invariants (e.g. their degree), separate orbit-closures and allow us to
recover varieties of interest in the study of border rank and border
subrank of tensors. |
|
Menu