## Seminar Categories

## Current Series

Thu Sep 05 |
## Colloquium3:35pm - Vincent Hall 16Colloquium - Canceled Alina Chertock, NSCU Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction- diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems extremely delicate and challenging task. In this talk, I will present a family of high-order numerical methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to to multi-scale and coupled chemotaxisfluid system and will also be discussed. |

Thu Sep 19 |
## Colloquium3:35pm - Vincent Hall 16The Langlands Program: An Introduction and Recent Progress Solomon Friedberg,, Boston College The Langlands Program, connecting algebra, analysis and geometry in diverse ways, is foundational to modern number theory. I will introduce this program and indicate some recent progress. As we shall see, a great deal still remains to be done. |

Thu Oct 17 |
## Colloquium3:35pm - Vincent Hall 16Hopf monoids relative to a hyperplane arrangement Marcelo Aguiar, Cornell University The talk is based on recent and ongoing work with Swapneel |

Thu Oct 31 |
## Colloquium3:35pm - Vincent Hall 16Differential operators on invariant rings Anurag Singh, University of Utah Work of Levasseur and Stafford describes the rings of differential operators on various classical invariant rings of characteristic zero; in each of these cases, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of reductive groups over the complex numbers, Smith and Van den Bergh asked if reduction modulo p works for differential operators in this context. In joint work with Jack Jeffries, we establish that this is not the case for various classical groups. |

Thu Nov 07 |
## Colloquium2:30pm - Vincent Hall 16On various questions (and answers) in High-dimensional probability Galyna Livshyts, Georgia Tech In this talk, several topics from High-dimensional probability shall be discussed. This fascinating area is rich in beautiful problems, and several easy-to-state questions will be outlined. Further, some connections between them will be explained throughout the talk. I shall discuss several directions of my research. One direction is invertibility properties of inhomogeneous random matrices: I will present sharp estimates on the small ball behavior of the smallest singular value of a very general ensemble of random matrices, and will briefly explain the new tools I developed in order to obtain these estimates. Another direction is isoperimetric-type inequalities in high-dimensional probability. Such inequalities are intimately tied with concentration properties of probability measures. Among other results, I will present a refinement of the concavity properties of the standard gaussian measure in an n-dimensional euclidean space, under certain structural assumptions, such as symmetry. This result constitutes the best known to date estimate in the direction of the conjecture of Gardner and Zvavitch from 2007. The above topics will occupy most of the time of the presentation. In addition, I shall briefly mention other directions of my research, including noise-sensitivity estimates for convex sets, or, in other words, upper bounds on perimeters of convex sets with respect to various classes of probability distributions. If time permits, I will discuss my other results, such as small ball estimates for random vectors with independent coordinates, and partial progress towards Levi-Hadwiger illumination conjecture for convex sets in high dimensions. |

Thu Nov 07 |
## Colloquium3:35pm - Vincent Hall 16Eisenstein Series on Loop Groups and their Metaplectic Covers Manish Patnaik, University of Alberta Both the Langlands-Shahidi method of studying automorphic L-functions and approach via the theory of Weyl group multiple Dirichlet series to studying moments of L-functions now require new classes of groups with which to work. In this talk, I will explain our progress on extending these techniques to certain infinite-dimensional Kac-Moody groups, namely loop groups (and their metaplectic covers). Of note in our work is the presence of two quite different types of Eisenstein series that exist on the same group and which need to be considered in conjunction with one other. This is a report on joint work in progress with H. Garland, S.D. Miller, and A. Puskas. |

Tue Nov 12 |
## Colloquium2:30pm - Vincent Hall 16Unraveling Local Cohomology Emily Witt, Univ. of Kansas Local cohomology modules are fundamental tools in commutative algebra, due to the algebraic and geometric information they carry. For instance, they can help determine the number of equations necessary to define an affine variety. Unfortunately, however, the application of local cohomology is limited by the fact that these modules are typically very large (e.g., not finitely generated), and can be difficult to determine explicitly. In this talk, we discuss new techniques developed to understand the structure of local cohomology (e.g., coming from invariant theory). We also describe recently-discovered "connectedness properties" of spectra that local cohomology encodes. |

Thu Nov 14 |
## Colloquium2:30pm - Vincent Hall 16Applications of Frobenius beyond prime characteristic. Daniel Hernández, Univ. of Kansas Abstract: Recall that the Frobenius morphism is simply the map sending an element in a ring of prime characteristic $p>0$ -- say, a polynomial with coefficients in a finite field -- to its $p$-th power. Though simple to define, Frobenius has proven to be a useful and effective tool in algebraic geometry, representation theory, number theory, and commutative algebra. Furthermore, and remarkably, some of the most interesting applications of Frobenius are to the study of objects defined over the complex numbers, and more generally, over a field of characteristic zero! In this talk, we will discuss some of these applications, with an eye towards classical singularity theory and birational algebraic geometry, both over the complex numbers. |

Thu Nov 14 |
## Colloquium3:35pm - Vincent Hall 16$p$-adic estimates for exponential sums on curves Joe Kramer-Miller, UC Irvine A central problem in number theory is that of finding rational or integer solutions to systems of polynomials in several variables. This leads one naturally to the slightly easier problems of finding solutions modulo a prime $p$. Using a discrete analogue of the Fourier transformation, this modulo $p$ problem can be reformulated in terms of exponential sums. We will discuss $p$-adic properties of such exponential sums in the case of higher genus curves as well as connections to complex differential equations. |

Tue Nov 19 |
## Colloquium2:30pm - Vincent Hall 16Random matrix theory and supersymmetry techniques Tatyana Shcherbyna, Princeton University Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a spacial structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY) . SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. |

Tue Nov 26 |
## Colloquium1:25pm - Vincent Hall 570K-stability and moduli spaces of Fano varieties Yuchen Liu, Yale University Fano varieties are positively curved algebraic varieties which form one of the three building blocks in the classification. Unlike the case of negatively curved varieties, moduli spaces of Fano varieties (even smooth ones) can fail to be Hausdorff. K-stability was originally invented as an algebro-geometric notion characterizing the existence of K\"ahler-Einstein metrics on Fano varieties. Recently, people have found strong evidence toward constructing compact Hausdorff moduli spaces of Fano varieties using K-stability. In this talk, I will discuss recent progress in this approach, including an algebraic proof of the existence of Fano K-moduli spaces, and describing these moduli spaces explicitly. This talk is partly based on joint works with H. Blum and C. Xu. |

Tue Nov 26 |
## Colloquium3:35pm - Vincent Hall 16Schrodinger solutions on sparse and spread-out sets Xiumin Du, University of Maryland If we want the solution to the Schrodinger equation to converge to its initial data pointwise, what's the minimal regularity condition for the initial data should be? I will present recent progress on this classic question of Carleson. This pointwise convergence problem is closely related to other problems in PDE and geometric measure theory, including spherical average Fourier decay rates of fractal measures, Falconer's distance set conjecture, etc. All these problems essentially ask how to control Schrodinger solutions on sparse and spread-out sets, which can be partially answered by several recent results derived from induction on scales and Bourgain-Demeter's decoupling theorem. |

Mon Dec 02 |
## Colloquium3:35pm - Vincent Hall 570Analysis and geometry of free boundaries: recent developments Mariana Smit Vega Garcia, Western Washington University In the applied sciences one is often confronted with free boundaries, which arise when the solution to a problem consists of a pair: a function u (often satisfying a partial differential equation (PDE)), and a set where this function has a specific behavior. Two central issues in the study of free boundary problems and related problems in calculus of variations and geometric measure theory are:(1) What is the optimal regularity of the solution u? In this talk, I will overview recent developments in obstacle type problems and almost minimizers of Bernoulli-type functionals, illustrating techniques that can be used to tackle questions (1) and (2) in various settings. The study of the classical obstacle problem - one of the most renowned free boundary problems - began in the 60s with the pioneering works of G. Stampacchia, H. Lewy and J. L. Lions. During the past five decades, it has led to beautiful and deep developments in the calculus of variations and geometric partial differential equations. Nowadays obstacle type problems continue to offer many challenges and their study is as active as ever. |

Tue Dec 03 |
## Colloquium3:30pm - Vincent Hall 16Mirror symmetry and canonical bases for quantum cluster algebras Travis Mandel, Univ. of Edinburgh Mirror symmetry is a phenomenon which relates the symplectic geometry of one space X to the algebraic geometry of another space Y. One consequence is that a canonical basis of regular functions on Y can be defined in terms of certain counts of holomorphic curves in X. I'll discuss the application of this to (quantum) cluster algebras --- certain combinatorially defined algebras whose definition was motivated by the appearance of canonical bases in representation theory and Teichmüller theory. |

Thu Dec 05 |
## Colloquium3:35pm - Vincent Hall 16Modular forms on exceptional groups Aaron Pollack, Duke When G is a reductive (non-compact) Lie group, one can consider automorphic forms for G. These are functions on the locally symmetric space X_G associated to G that satisfy some sort of nice differential equation. When X_G has the structure of a complex manifold, the _modular forms_ for the group G are those automorphic forms that correspond to holomorphic functions on X_G. They possess close ties to arithmetic and algebraic geometry. For certain exceptional Lie groups G, the locally symmetric space X_G is not a complex manifold, yet nevertheless possesses a very special class of automorphic functions that behave similarly to the holomorphic modular forms above. Building upon work of Gan, Gross, Savin, and Wallach, I will define these modular forms and explain what is known about them. |

Tue Jan 21 |
## Colloquium3:30pm - Vincent Hall 16Optimal Transport as a Tool in Analytic Number Theory and PDEs Stefan Steinerberger, Yale University Optimal Transport is concerned with the question of how to best move one measure to another (this could be sand on a beach or products from a warehouse to consumers). I will explain the basic definition of Wasserstein distance and then describe how it can be used as a tool to say interesting things in other fields. (1) How to get new regularity statements for classical objects in number theory almost for free (irrational rotations on the torus, quadratic residues in finite fields). (2) How to best distribute coffee shops over downtown Minneapolis. (3) Finally, how to obtain higher dimensional analogues of classical Sturm-Liouville theory: simply put, Sturm-Liouville theory says that eigenfunctions of the operator Ly = -y''(x) +p(x)y(x) (think of sin(kx) and cos(kx)) cannot have an arbitrary number of roots; we present a generalization to higher dimensions that is based on a simple (geometric) inequality. |

Thu Jan 23 |
## Colloquium3:35pm - Vincent Hall 16Modularity and the Hodge/Tate conjectures for some self-products Laure Flapan, MIT If X is a smooth projective variety over a number field, the Hodge and Tate conjectures describe how information about the subvarieties of X is encoded in the cohomology of X. We explore the role that certain automorphic representations, called algebraic Hecke characters, can play in understanding which cohomology classes of X arise from subvarieties. We use this to deduce the Hodge and Tate conjectures for certain self-products of varieties, including some self-products of K3 surfaces. This is joint work with J. Lang. |

Thu Mar 05 |
## Colloquium3:35pm - Vincent Hall 16Colloquium Richard McLaughlin, UNC, Chapel HIll |

Thu Mar 19 |
## Colloquium3:35pm - Vincent Hall 16Colloquium Mauro Maggioni, Ordway Visitor, Johns Hopkins University |

Thu Apr 16 |
## Colloquium3:35pm - Vincent Hall 16Colloquium Andrea Montanari, Stanford University |

Thu Apr 23 |
## Colloquium3:35pm - Vincent Hall 16Colloquium Dionisios Margetis - Ordway Visitor, University of Maryland, College Park |