Mathematics Colloquia, Spring 2020

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. DouglasM. Jabbari, and G. Yu.  
Host: Rui Fernandes

FEBRUARY 20, 2020

4:00 p.m., 245 Altgeld Hall
Speaker: Niall Mangan (Northwestern University)
Title: To be announced
Host: Jared Bronski

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

Abstract: To come

MARCH 5, 2020

4:00 p.m., 245 Altgeld Hall
Speaker: George Shabat (Russian State University for the Humanities)
Title: To be announced
Host: Alexander Tumanov

Abstract: To come

MARCH 12, 2020

4:00 p.m., 245 Altgeld Hall
Speaker: Kevin Zumbrun (Indiana University Bloomington)
Title: To be announced
Host: Vera Hur

Abstract: To come

MARCH 26, 2020

4:00 p.m., 245 Altgeld Hall
Speaker: Jared Wunsch (Northwestern University)
Title: To be announced
Host: Pierre Albin

Abstract: To come

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