Mathematics Colloquia, Fall 2020

December 3, 2020

Time: 4 p.m. CST (Informal chat with speaker begins at 3:30).
Speaker: Mikhail Ostrovskii (St. John's University)
Title:  Geometry of transportation cost (also known as Earth Mover or Wasserstein distance)
Abstract:  We consider (finitely supported) transportation problems on a metric space M. They form a vector space TP(M). The optimal transportation cost for such transportation problems is a norm on this space. Such normed spaces were introduced and studied by Kantorovich and his students in 1940-1950s years. This development lead to terms: Kantorovich distance, Kantorovich-Rubinstein distance, and transportation cost. Simultaneously, another group of researchers, starting with Markov (1941) started the study of algebraically “free” topological (or metric) structures which contain a given topological (or metric) structure as a substructure. On these lines Arens and Eells (1956) constructed the space which coincides with the space of transportation problems with the norm equal to the optimal transportation cost. In this connection you can meet such terms as Arens-Eells space and Lipschitz-free space. I am going to talk about geometry of such normed spaces. My results presented in this talk, mentioned in it, or related to it, can be found in joint papers with Stephen Dilworth, Seychelle Khan, Denka Kutzarova, Mutasim Mim, and Sofiya Ostrovska (available on arXiv).
Zoom Info: please email currid@illinois.edu

 

November 5, 2020

Time: 4 p.m. CST
Speaker: Greta Panova (University of Southern California)
Title: Computational complexity meets algebraic combinatorics
Host: Alexander Yong

Abstract:  How hard is a problem? How nice is a solution? Such questions can actually be formalized using the theory of Computational Complexity. Yet, distinguishing the different computational complexity classes, like P vs NP, are major problems. Algebraic Combinatorics studies discrete structures originating in Algebra/Representation Theory via combinatorial methods and vice versa. Some of the main longstanding open problems concern the “combinatorial interpretation” of structure constants and multiplicities originally defined via representation theory like Kronecker and plethysm coefficients. In this talk we will discuss the two-way interaction between the fields via such structure constants. First, how Kronecker coefficients appear in the distinction of algebraic complexity classes via the Geometric Complexity Theory. Second, how computational complexity explains why the problem of finding a combinatorial interpretation is hard. Meeting Info: please email ayong@illinois.edu 

 

October 22, 2020

NOTE SPECIAL TIME: 12 noon, CDT, via Zoom
Speaker: Ralf Hiptmair (ETZ Zurich)
Title: The Topology of Circuit-Field Coupling
Host: Anil Hirani

Colloquium Poster
Slides
Video

Abstract:  Imagine an electromagnetic wave impinging on a small electric device. The wave is governed by Maxwell's equations and propagates in a "field domain", whereas the device occupies a "circuit domain" and is described by a circuit model with well-defined ports located on the common interface. I am going to discuss how to incorporate ports into (finite-element discretized) variational formulations of Maxwell's equations. It turns out that two types of ports have to be distinguished, electric and magnetic. Each of them is linked to two port quantities, one "potential" and one "current". A key insight concerns the close relationship of the port quantities with generating fundamental cycles of the relative cohomology of the common interface minus the port areas. These cycles become instrumental for linking the circuit to Maxwell field models also in the context of finite-element discretization. Therefore I am going to elaborate the construction of fundamental cycles on triangulated surfaces by means of spanning-tree techniques. Another key insight is that in case of non-trivial topology of circuit domain ports alone are not sufficient to connect fields and circuits. "Linked fluxes" associated with handles of circuit domain also have to be taken into account and this entails knowledge about the topological properties of the circuit beyond the scope of customary descriptions. For demonstration I am going to give a striking numerical example. (Joint work with J. Ostrowski). > NOTE: This Colloquium talk is at 12:00 PM Central time, NOT the usual Colloquium time due to the time difference with speaker's time zone. The talk will be on Zoom for which the link will be sent out to the department and other mailing lists. If you don't receive such an email, please email hirani@illinois.edu from your Illinois email address to receive the Zoom link.

 

Previous Seminars This Semester

September 3, 2020

4:00 p.m., 245 Altgeld Hall
Speaker: Marius Junge, University of Illinois at Urbana-Champaign
Title: TBA
Abstract: To come.

 

September 10, 2020

4:00 p.m., via Zoom (please contact Jozsef Balogh)
Speaker: Noga Alon (Princeton University and Tel Aviv)
Title: Fair representation
Hosts: Jozsef Balogh and Alexandr Kostochka
Abstract: A substantial number of results and conjectures deal with the existence of a set of prescribed type which contains a fair share from each member of a finite collection of objects in a space. Examples include the Ham-Sandwich Theorem in Measure Theory, the Hobby-Rice Theorem in Approximation Theory, the Necklace Theorem and the Ryser Conjecture in Discrete Mathematics, and more. The techniques in the study of these results combine combinatorial, topological, geometric and algebraic tools. I will describe the topic, focusing on several recent results.

 

Coming Soon!

November 5, 2020

4:00 p.m., via Zoom (please contact Alex Yong for information)
Speaker: Greta Panova (University of Southern California)
Title: TBA
Host: Alex Yong
Abstract: To come 
 

December 3, 2020

4:00 p.m., via Zoom 
Speaker: Mikhail Ostrovskii
Title: TBA
Hosts: Marius Junge, Denka Kutzarova
Abstract: To come 

 

Date TBA

4:00 p.m., 245 Altgeld Hall
Speaker: Andrew Granville (University of Montreal)
Title: TBA
Host: Kevin Ford
Abstract: To come
 

Date TBA

4:00 p.m., 245 Altgeld Hall
Speaker: Bjorn Sandstede (Brown University)
Title: TBA
Host: Jared Bronski
Abstract: To come
 

Date TBA

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.
 

Date TBA

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.
 

Date TBA

4:00 p.m., 245 Altgeld Hall
Speaker: Kevin Purbhoo (University of Waterloo)
Title: To be announced
Host: Alex Yong
Abstract: To come
 

Date TBA

4:00 p.m., 245 Altgeld Hall
Speaker: Elizabeth Meckes (Case Western Reserve University)
Title: To be announced
Hosts: Pierre Albin
Abstract: To come
 

Date TBA

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