# Calendar

## Yesterday

## Breuil-Kisin-Fargues modules

In the next two talks, we will study properties of Breuil-Kisin-Fargues modules and have a sketch of the proof for two theorems of Fargues on BKF modules used in BMS' paper.

### Department Colloquium, Prof. Guillaume Bal, University of Chicago, MATH 175

Tuesday, Feb 20 3:30 pm - 4:20 pm

Hybrid imaging and over-determined systems of PDEs

Abstract: Several recent coupled-physics (a.k.a. hybrid) medical imaging modalities aim to combine a high-contrast, low-resolution, modality with a high-resolution, low-contrast, modality in order to achieve both high contrast and high resolution. These modalities involve the reconstruction of constitutive parameters in partial differential equations (PDE) from knowledge of internal functionals of the parameters and the PDE solutions. The resulting inverse problems may mathematically be recast as over-determined systems of nonlinear PDEs, with the number of constraints (equations) depending on the number of available measurements. This talk will present recent analyses of such problems, focusing in particular on uniqueness and non-uniqueness results, stability estimates, and explicit reconstructions whenever available.

## Today

### Probability Seminar, Prof. Takashi Owada, Purdue University, REC 113

Wednesday, Feb 21 1:30 pm - 2:45 pm

Sub-tree counts on hyperbolic random geometric graphs

We shall consider a geometric graph model on the "hyperbolic" space, which is characterized by a negative Gaussian curvature. Among several equivalent models representing the hyperbolic space, we treat the most commonly used d-dimensional Poincare ball model. One of the main characteristics of geometric graphs on the hyperbolic space is tree-like hierarchical structure. From this viewpoint, we discuss the asymptotic behavior of sub-tree counts. The spatial distribution of sub-trees is crucially determined by an underlying curvature of the space. For example, if the space gets flatter and closer to the Euclidean space, sub-trees are asymptotically scattered near the center of the Poincare ball. On the contrary, if the space becomes "more hyperbolic" (i.e., more negatively curved), the distribution of trees is asymptotically determined by those concentrated near the boundary of the Poincare ball. We investigate the asymptotics of the expectation and variance of sub-tree counts. Moreover, we prove the corresponding central limit theorem as well.

This is joint work with Yogeshwaran D. at Indian Statistical Institute.

### Spectral and Scattering Theory Seminar, Peter Hintz, Clay Math Institute and UC Berkeley, MATH 431

Wednesday, Feb 21 1:30 pm - 2:30 pm

## Stability of Minkowski space and asymptotics of the metric

Abstract: I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of $\mathbb{R}^4$ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.### Commutative Algebra Seminar, Alessandra Costantini, Purdue University, UNIV 101

Wednesday, Feb 21 1:30 pm - 2:20 pm

## Cohen-Macaulayness of Rees algebras of modules

Frobenius descent for convergent isocrystals and a conjecture of Berthelot

Let k be a field of characteristic p > 0 and W the ring of Witt vectors of k. In this talk, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over k relative to W. This proof allows us to deduce an analogue of the de Rham complexes comparaison theorem of Berthelot without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot's conjecture on the preservation of convergent F-isocrystals under the higher direct image by a smooth proper morphism of k-varieties.

### Special Colloquium, Jing Wang, Doob research assistant professor, UIUC, BRNG 2290

Wednesday, Feb 21 3:30 pm - 4:30 pm

## Degenerate diffusions and their limiting behaviors

Abstract: We discuss degenerate hypoelliptic diffusion processes and their limiting behaviors in both large time and small time. For a diffusion process on a sub-Riemannian manifold, questions related to large time behavior such as stochastic completeness and convergence to equilibrium are closely related to global geometric bounds including Ricci curvature lower bound. Small time behavior of its transition density falls into the regime of large deviation estimate, which connects to the sub-Riemannian distance.

We are particularly interested in the study of small time behavior of a general strict-degenerate diffusion process (weak Hörmander's type), which has been a longstanding open problem. In this talk we will present a recent progress in this problem by developing a graded large deviation principle for diffusions on a nilpotent Lie group.

Parts of this talk are based on joint work with Fabrice Baudoin and Gerard Ben Arous.

## An Introduction to O-Minimal Structures

In the bulk of this talk, we will define and introduce basic properties of o-minimal structures. An o-minimal structure is similar to a topology in that it is a set of subsets of a total space, in our case a real closed field. O-minimal structures generalize defining properties of semi-algebraic sets, and with some work it can be shown that a number of other interesting subsets share these properties. After exploring different properties of o-minimal structures, we will use what we have learned to prove an interesting result in real algebraic geometry.

## Tomorrow

### Automorphic Forms and Representation Theory Seminar, Prof. Manish Patnaik, University of Alberta, UNIV 317

Thursday, Feb 22 1:30 pm - 2:20 pm

## Metaplectic covers of Kac-Moody groups and Whittaker functions

We will describe how to construct metaplectic covers of Kac-Moody groups, generalizing a classical construction of Matsumoto. In the case of non-archimedean local fields, we will then explain how to formulate Whittaker functions on such groups, and compute them via a Casselman-Shalika type formula. Joint work with with A. Puskas.### PDE Seminar, Prof. Alex Misiats, Virginia Commonwealth University, REC 121

Thursday, Feb 22 3:30 pm - 4:20 pm

## Convex Duality in Nonconvex Variational Problems.

We consider the minimization problem related to modeling materials with memory, e.g. shape memory alloys. I will start my presentation with a visual illustration of shape memory effect. Physical experiments suggest that if two distinct phases of such material are present at opposite sides of a rectangular sample, a zig-zag pattern is formed. Our goal is to understand if this pattern is energy minimizing. The mathematical model of this phenomenon involves the minimization of singularly perturbed elastic energy. I my talk I will review classic minimization approach via convex duality. Despite the fact that the problem is highly nonconvex, in my talk, I will describe a relaxation method which allows to use the convex duality technique for the purpose of obtaining a sharp lower bound.

### Mathematics Society, Prof. Steve Bell, Purdue University, REC 108

Thursday, Feb 22 6:00 pm - 7:00 pm

## Fourier Series are Music to My Ears

Most people think that Fourier series were invented by J. B. Fourier to study heat flow in a wire. They were really invented years earlier by Johann Bernoulli, in competition with his brother, Jacob, to study the motion of a vibrating string. I will explain how understanding Fourier series leads to a deeper understanding of music.

I will also reveal how this subject lured me away from Acoustical Engineering as an undergraduate at University of Michigan to pure math as a PhD student at MIT, and how it still reverberates in my present research.

## Next Week

### Geometric Analysis Seminar, Robert Hardt, Rice University, MATH 731

Monday, Feb 26 3:30 pm - 4:30 pm

## A Linear Isoperimetric Inequality for Algebraic and Analytic Varieties

Abstract: In R^n, the classical n dimensional isoperimetric inequality bounds the volume of any smooth region U by c_n [H^{n-1}(BdryU)]^{n/(n-1)}, where c_n is the constant giving equality with U being an n ball. In higher codimension, any integral k-1 dimensional cycle B in R^n with k < n , is the boundary of some k chain T with mass(T) \leq c_k mass(B)^ {k/(k-1)}. The proof by Federer and Fleming in 1960 generalized to an ambient space X like a compact Riemannian manifold, a polyhedron, or more generally a Euclidean Lipschitz neighborhood retract, with the constant $C_X$ depending on X. However, this relation may fail for various singular spaces X such as algebraic varieties. With T. DePauw (Paris VII), we prove a {\it linear} isoperimetric inequality valid for cycles of any dimension in a compact set X defined by polynomial or real analytic equalities or inequalities. This generalizes earlier work on codimension one boundaries by L.Bos -P.Milman and B.Hua-FH.Lin.</p>### CCAM Seminar, Haizhao Yang, National University of Singapore, REC 308

Monday, Feb 26 4:30 pm - 5:30 pm

## A Unified Framework for Oscillatory Integral Transform: When to use NUFFT or Butterfly Factorization?

This talk introduces a nearly optimal fast algorithm for the matvec $g=Kf$ for $K\in \mathbb{C}^{N\times N}$, which is the discretization of the oscillatory integral transform $g(x) = \int K(x,\xi) f(\xi)d\xi$ with a kernel function $K(x,\xi)=\alpha(x,\xi)e^{2\pi i\Phi(x,\xi)}$, where $\alpha(x,\xi)$ is a smooth amplitude function , and $\Phi(x,\xi)$ is a piecewise smooth phase function with $O(1)$ discontinuous points in $x$ and $\xi$. A unified framework is proposed to compute $Kf$ with $O(N\log N)$ time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an $O(N)$ fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicite formulas for the amplitude and phase functions are known; 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an $O(N\log N)$ algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm; 2) a new stable and nearly optimal BF with amplitude and phase functions in form of a low-rank factorization (IBF-MAT) is proposed to evaluate the matvec $Kf$. Numerical results are provided to demonstrate the effectiveness of the proposed framework.

### Algebraic Geometry seminar, Christine Berkesch, University of Minnesota, MATH 731

Wednesday, Feb 28 3:25 pm - 4:25 pm

## Parametric behavior of A-hypergeometric solutions.

Abstract: A-hypergeometric systems are the D-module counterparts of toric ideals, and theirbehavior is linked closely to the combinatorics oftoric varieties. I will discuss recent work that aims to explain the behavior of the solutions of these systems as their parameters vary. In particular, we stratify the parameter space so that solutions are locally analytic within each (connected component of a) stratum. This is joint work with Jens Forsgård and Laura Matusevich.### Automorphic Forms and Representation Theory Seminar, Dr. Qing Zhang, Sun Yat-sen University, UNIV 317

Thursday, Mar 1 1:30 pm - 2:20 pm

## Local converse theorem for unitary groups

Local converse theorem says that an irreducible generic representation of a classical group over a p-adic field should be determined by its various local gamma factors twisting with GL_n. In this talk, I will describe a sketch of a proof of local converse theorems for quasi-split unitary groups. A main ingredient of the proof is certain properties of partial Bessel functions developed by Cogdell-Shahidi-Tsai.

## Splendor in the Graphs

## Two Weeks

### Geometric Analysis Seminar, Chika Mese, Johns Hopkins University, MATH 731

Monday, Mar 5 3:30 pm - 4:30 pm

Title: Harmonic maps into CAT(1) spaces

The pioneering works of Gromov-Schoen and Korevaar-Schoen established the theory of harmonic maps into NPC spaces, i.e. complete metric space of non-positive curvature. In this talk, we will discuss harmonic map theory when we relax the curvature assumption and consider metric spaces with curvature bounded from above.

### Math is Key Lecture, Dr. Eugenia Cheng, School of the Art Institute of Chicago

Tuesday, Mar 6 3:30 pm - 4:20 pm

## How To Bake Pi: making abstract mathematics palatable

Why does mathematics inspire love in some people and fear in others? Why do some people think mathematics is important for everyone while others think it is a collection of gibberish touching little of the world beyond the brains of some rare geniuses? Why do some think it is a creative art akin to poetry and music, while others think it is a boring tool for producing answers? In this talk I will present mathematics as a way of thinking, and not just about numbers. I will use a variety of unexpectedly connected examples including music, juggling and baking, as in the title of my recent book. My aim is to show that math can be made fun, intriguing and relevant for people of all ages, by means of hand-on activities, examples that everyone can relate to, and peculiar anecdotes. I will present surprisingly high level mathematics including some advanced abstract algebra usually only seen by maths undergraduates or PhD students, yet show how to make it accessible even to children. My message is relevant to those who wish to spread their love of math, as well as those who wish to overcome their fear of it. There will be a distinct emphasis on edible examples.## March

### CCAM Seminar, Prof. Jie Liang, University of Illinois, Chicago, REC 308

Monday, Mar 19 4:30 pm - 5:30 pm

## TBA

## A partial Laplacian as an infinitesimal generator on the Wasserstein space

We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space.

### Automorphic Forms and Representation Theory Seminar, Prof. Shiang Tang, University of Utah, UNIV 317

Thursday, Mar 22 1:30 pm - 2:30 pm

### Department Colloquium, Prof. Cristina Villalobos, University of Texas, Rio Grande Valley, MATH 175

Tuesday, Mar 27 3:30 pm - 4:20 pm

## TBD

### Automorphic Forms and Representation Theory Seminar, Dr. Fan Gao, Purdue University, UNIV 317

Thursday, Mar 29 1:30 pm - 2:20 pm

## TBA

### Mathematics Society, Prof. Donatella Danielli, Purdue University, REC 108

Thursday, Mar 29 6:00 pm - 7:00 pm

## April

### Department Colloquium, Prof. Dinakar Ramakrishnan, California Institute of Technology, MATH 175

Tuesday, Apr 3 3:30 pm - 4:20 pm

## TBD

### Automorphic Forms and Representation Theory Seminar, Prof. Mark Van Hoeij, Florida State University, UNIV 317

Thursday, Apr 5 1:30 pm - 2:30 pm

### Automorphic Forms and Representation Theory Seminar, Prof. Melissa Emory, University of Missouri, REC 113 (Note Special Day/Time/Location)

Friday, Apr 6 1:30 pm - 2:30 pm

### Department Colloquium, Prof. Frank Thorne, University of South Carolina, MATH 175

Tuesday, Apr 10 3:30 pm - 4:20 pm

## TBD

### Automorphic Forms and Representation Theory Seminar, Prof. Ravi Ramakrishna, Cornell University, UNIV 317

Thursday, Apr 12 1:30 pm - 2:20 pm

## TBA

### Automorphic Forms and Representation Theory Seminar, Prof. Adriana Salerno, Bates College, REC 113 (Note Special Day/Time/Location)

Friday, Apr 13 1:30 pm - 2:30 pm

### Department Colloquium, Prof. Matt Gursky, University of Notre Dame, MATH 175

Tuesday, Apr 17 3:30 pm - 4:20 pm

## TBD

### Automorphic Forms and Representation Theory Seminar, Dr. Cris Negron, MIT, UNIV 317

Thursday, Apr 19 1:30 pm - 2:20 pm

## TBD

### CCAM Lunch Seminar, Prof. Georgios Karagiannis, Durham University, REC 303

Friday, Apr 20 11:30 am - 12:30 pm

## TBA

### CCAM Seminar, Dr. Andrew Hill, Centers for Disease Control and Prevention, REC 308

Monday, Apr 23 4:30 pm - 5:30 pm