### JANUARY 21 (Tuesday), 2020

Special Colloquium

4:00 p.m., 245 Altgeld Hall

Speaker: Jingyin Huang (Ohio State University)

Title: The Helly geometry of some Garside and Artin groups

Host: Nathan Dunfield

**Abstract:** Artin groups emerged from the study of braid groups and complex hyperplane arrangements, and they are connected to Coxeter groups, 3-manifold groups, buildings and many others. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required. This is joint work with D. Osajda.

### JANUARY 22 (Wednesday), 2020

Special Colloquium

4:00 p.m., 245 Altgeld Hall

Speaker: Di Qi (Courant Institute of Mathematical Sciences)

Title: Statistical reduced models and rigorous analysis for uncertainty quantification of turbulent dynamical systems

Hosts: Lee DeVille and Vera Hur

**Abstract: **The capability of using imperfect statistical reduced-order models to capture crucial statistics in turbulent flows is investigated. Much simpler and more tractable block-diagonal models are proposed to approximate the complex and high-dimensional turbulent flow equations. A rigorous statistical bound for the total statistical uncertainty is derived based on a statistical energy conservation principle. The systematic framework of correcting model errors is introduced using statistical response and empirical information theory, and optimal model parameters under this unbiased information measure are achieved in a training phase before the prediction. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the reduced-order model in various dynamical regimes with distinct statistical structures.

### JANUARY 23 (Thursday), 2020

Special Colloquium

4:00 p.m., 245 Altgeld Hall

Speaker: Gaku Liu (Max Planck Institute for Mathematics in the Sciences)

Title: Semistable reduction in characteristic 0

Host: Alex Yong

**Abstract:** Semistable reduction is a relative generalization of the classical problem of resolution of singularities of varieties; the goal is, given a surjective morphism f:X→B of varieties in characteristic 0, to change f so that it is "as nice as possible". The problem goes back to at least Kempf, Knudsen, Mumford, and Saint-Donat (1973), who proved a strongest possible version when B is a curve. The key ingredient in the proof is the following combinatorial result: Given any d-dimensional polytope P with vertices in Zd, there is a dilation of P which can be triangulated into simplices each with vertices in Zd and volume 1/d!. In 2000, Abramovich and Karu proved, for any base B, that f can be made into a weakly semistable morphism f′:X′→B′. They conjectured further that f′ can be made semistable, which amounts to making X′ smooth. They explained why this is the best resolution of f one might hope for. In this talk I will outline a proof of this conjecture. They key ingredient is a relative generalization of the above combinatorial result of KKMS. I will also discuss some other consequences in combinatorics of our constructions. This is joint work with Karim Adiprasito and Michael Temkin.

### JANUARY 24 (Friday), 2020

Special Colloquium

4:00 p.m., 245 Altgeld Hall

Speaker: Erik Walsburg (University of California Irvine)

Title: Logical and geometric tameness over the real line

Host: Philipp Hieronymi

**Abstract: **There are now a number of important and well-understood examples of logically tame first order structures over the real numbers such as the ordered field of real numbers and the ordered field of real numbers equipped with the exponential function. In these examples subsets of Euclidean space which are (first order) definable are geometrically very well behaved. Recent research had yielded general theorems in this direction. I will discuss one result in this subject: A first order structure on the real line which expands the ordered vector space of real numbers and defines a closed set X such that the topological dimension of X is strictly less then the Hausdorff dimension of X defines every bounded Borel set. Informally: An expansion of the ordered real vector space which defines a fractal is maximally wild from the viewpoint of logic. Joint with Fornasiero and Hieronymi.

### JANUARY 27 (Monday), 2020

Special Colloquium

4:00 p.m., 245 Altgeld Hall

Speaker: Jesse Thorner (University of Florida)

Title: A new approach to bounding L-functions

Host: Kevin Ford

**Abstract:** Analytic number theory began with studying the distribution of prime numbers, but it has evolved and grown into a rich subject lying at the intersection of analysis, algebra, combinatorics, and representation theory. Part of its allure lies in its abundance of problems which are tantalizingly easy to state which quickly lead to deep mathematics, much of which revolves around the study of L-functions. These extensions of the elusive Riemann zeta function ζ(s) are generating functions with multiplicative structure arising from either arithmetic-geometric objects (like number fields or elliptic curves) or representation-theoretic objects (automorphic forms). Many equidistribution problems in number theory rely on one's ability to accurately bound the size of L-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for ζ(s) and its extensions to other L-functions. I will discuss some motivating problems along with recent work (joint with Kannan Soundararajan) which produces new bounds for L-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis.

### JANUARY 30, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Plamen Stefanov (Purdue University)

Title: Local and global boundary rigidity

Host: Pierre Albin

**Abstract:** The boundary rigidity problem consist of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain an assumption of the existence of a strictly convex foliation. The problem can be reformulated as a recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local in nature. In particular, we can recover locally the compressional and the shear wave speeds for the elastic Earth model, given local information. The talk is based on a joint work with G.Uhlmann (UW) and A.Vasy (Stanford). We will also present results for a recovery of a Lorentzian metric from red shifts motivated by the problem of observing cosmic strings. The methods are based on Melrose’s scattering calculus in particular but we will try to make the exposition accessible to a wider audience without going deep into the technicalities.

### FEBRUARY 6, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Xiang Tang (Washington University St. Louis)

Title: Analytic Grothendieck Riemann Roch Theorem

**Abstract**: In this talk, we will introduce an interesting index problem naturally associated to the Arveson-Douglas conjecture in functional analysis. This index problem is a generalization of the classical Toeplitz index theorem, and connects to many different branches of Mathematics. In particular, it can be viewed as an analytic version of the Grothendieck Riemann Roch theorem. This is joint work with R. Douglas，M. Jabbari, and G. Yu.

Host: Rui Fernandes

### FEBRUARY 20, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Niall Mangan (Northwestern University)

Title: Data-driven methods for model identification and parameter estimation of dynamical systems

Host: Jared Bronski

**Abstract: **Inferring the structure and dynamical interactions of complex systems is critical to understanding and controlling their behavior. I am interested in discovering models from the time-series in order to understand biological systems, material behavior, and other dynamical systems. One can frame the problem as selecting which interactions, or model terms, are most likely responsible for the observed dynamics from a library of possible terms. Several challenges make model selection and parameter estimation difficult including nonlinearities, varying parameters or equations, and unmeasured state variables. I will discuss methods for reframing these problems so that sparse model selection is possible including implicit formulation and data clustering. I will also discuss preliminary results for parameter estimation and model selection for deterministic and chaotic systems with hidden or unmeasured variables. We use a variational annealing strategy that allows us to estimate both the unknown parameters and the unmeasured state variables.

### FEBRUARY 24-28, 2020

Arthur B. Coble Memorial Lectures

4:00 p.m., 245 Altgeld Hall

Speaker: Frank Calegari (University of Chicago)

Title: To be announced

Host: Nathan Dunfield and Patrick Allen

### MARCH 5, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: George Shabat (Russian State University for the Humanities and Independent University of Moscow)

Title: Plane Trees and Algebraic Numbers

Host: Alexander Tumanov

**Abstract:** The main part of the talk will be devoted to an elementary version of the deep relations between the combinatorial topology and the arithmetic geometry. Namely, an object defined over the field of algebraic numbers, a polynomial with algebraic coefficients and only two finite critical values, will be associated to an arbitrary plane tree. Some applications of this construction will be presented, including polynomial Pell equations and quasi-elliptic integrals (going back to N.-H. Abel). The relations with finite groups and Galois theory will be outlined. At the end of the talk the possible generalizations will be discussed, including the *dessins d'enfants* theory initiated by Grothendieck.

### MARCH 12, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Kevin Zumbrun (Indiana University Bloomington)

Title: Stability of roll wave solutions in inclined shallow-water flow

Host: Vera Hur

**Abstract:** We review recent developments in stability of periodic roll-wave solutions of the Saint Venant equations for inclined shallow-water flow. Such waves are well-known instances of hydrodynamic instability, playing an important role in hydraulic engineering, for example, flow in a channel or dam spillway. Until recently, the analysis of their stability has been mainly by formal analysis in the weakly unstable or ``near-onset'' regime. However, hydraulic engineering applications are mainly in the strongly unstable regime far from onset. We discuss here a unified framework developed together with Blake Barker, Mat Johnson, Pascal Noble, Miguel Rodrigues, and Zhao Yang for the study of roll wave stability across all parameter regimes, by a combination of rigorous analysis and numerical computation. The culmination of our analysis is a complete stability diagram, of which the low-frequency stability boundary is, remarkably, given explicitly as the solution of a a cubic equation in the parameters of the solution space.

### MARCH 26, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Jared Wunsch (Northwestern University)

Title: Trapping, resonances, and the decay of waves

Host: Pierre Albin

**Abstract:** I will discuss some results, new and old, involving the influence of the geometry on the decay of waves. The quantum correspondence principle dictates that at high frequency, the dynamics of particle trajectories should be related to the rate at which the energy of a solution to the wave or Schrödinger equation decays. This relationship is mediated by the existence of resonances, which correspond to states with a finite (but possibly long) lifetime that ultimately decay owing to tunneling effects. I will discuss what we know about the existence and nonexistence of resonances, and focus on some recent results about resonances associated to the subtle effects of diffraction in classical and quantum problems that have singular structures in a metric or potential.

### APRIL 2, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Kevin Purbhoo (University of Waterloo)

Title: To be announced

Host: Alex Yong

**Abstract:** To come

### APRIL 9, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Mikhail Ostrovskii (St. John's University)

Title: To be announced

Hosts: Denka Kutzarova and Marius Junge

**Abstract:** To come

APRIL 30, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Elizabeth Meckes (Case Western Reserve University)

Title: To be announced

Hosts: Pierre Albin

**Abstract:** To come

### MAY 7, 2020

4:00 p.m., 245 Altgeld Hall

Speaker: Sami Assaf (University of Southern California)

Title: To be announced

Host: Alex Yong and Reuven Hodges

**Abstract:** To come