# Calendar

## Wednesday

## Next Week

Refreshments served in the Library Lounge

### CCAM Seminar, Dr. Xiu Yang , Pacific Northwest National Lab, UNIV 103

Monday, Nov 27 4:30 pm - 5:20 pm

## Alternating direction method for enhancing sparsity of the representation of uncertainty

Abstract: Compressive sensing has become a powerful tool for uncertainty quantification when only limited data is available. We provide a general framework using alternating direction method to enhance the sparsity of the representation of uncertainty in the form of generalized polynomial chaos expansion. This method identifies new sets of random variables through iterative rotations such that the new representation of the uncertainty using these variables is sparser. Consequently, we increase both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. We demonstrate the effectiveness of this method with applications in analyzing uncertainties in high-dimensional complex systems.## Dirichlet's principle (the first one)

Abstract: Also known as Pigeonhole principle, the principle says that if more than n objects are placed in n pigeonholes, then one pigeonhole will contain more than one object. I will discuss how this idea evolved to a powerful tool in the study of equations, for example partial differential equationsRefreshments served in the Library Lounge prior to Colloquium.

## Quantifying congruences between Eisenstein series and cusp forms

Consider the following two problems in algebraic number theory:

- For which prime numbers p can we easily show that the Fermat equation x^p + y^p =z^p has no non-trivial integer solutions?
- Given an elliptic curve E over the rational numbers, what can be said about the group of rational points of finite order on E?

Research Area: Algebraic number theory and modular forms

### Spectral and Scattering Theory Seminar, Oran Gannot, Northwestern University, MATH 731

Wednesday, Nov 29 1:30 pm - 2:20 pm

## Logarithmic decay of waves on spacetimes bounded by event horizons.

I will describe the decay of linear waves on a class of stationary spacetimes bounded by non-degenerate Killing horizons. Without any assumptions on the trapped set, solutions of the wave equation exhibit logarithmic energy decay. This is analogous to well known results in other geometric settings. The proof follows from new high frequency bounds on the resolvent.### Topology Seminar, Akhil Mathew, University of Chicago, BRNG B242

Wednesday, Nov 29 2:30 pm - 3:20 pm

## Kaledin's noncommutative degeneration theorem and topological Hochschild homology

Abstract: For a smooth proper variety over a field of characteristic zero, the Hodge-to-de Rham spectral sequence (relating the cohomology of differential forms to de Rham cohomology) is well-known to degenerate, via Hodge theory. A "noncommutative" version of this theorem has been proved by Kaledin for smooth proper dg categories over a field of characteristic zero, based on the technique of reduction mod p. I will describe a short proof of this theorem using the theory of topological Hochschild homology, which provides a canonical one-parameter deformation of Hochschild homology in characteristic p.Refreshments served in the Library Lounge

### Algebraic Geometry Seminar, Tatsunari Watanabe, Purdue University, MATH 731

Wednesday, Nov 29 3:30 pm - 4:20 pm

## The section conjecture for the universal hyperelliptic curves

In this talk, I will briefly introduce the Grothendieck's anabelian philosophy and anabelian conjectures, including the section conjecture. The section conjecture predicts that if the fundamental group of a hyperbolic curve over the rational numbers is "far from being abelian", then the structure of the group is rich enough to determine the existence of a rational solution to the polynomial equations defining the curve.

The problem of the section conjecture lies in the intersection of diophantine geometry, algebraic geometry, topology and group theory. For example, Hain has proved that the section conjecture holds for the universal curves over the rational numbers. Hain's proof involves the study of the (relative) completions of the mapping class groups and Torelli groups and their relations with the obstruction for the existence of a rational point of the universal curves. In this talk, I will introduce the universal hyperelliptic curves and sketch a proof of the section conjecture for the universal hyperelliptic curves. The key computation uses the hyperelliptic Johnson homomorphism and a certain pair of commuting symmetric Dehn twists to produce an obstruction that we want.

## Singular Loci of Sextic Curves

This talk will entertain the following question: Let C be a rational plane curve of degree 6. Let (m,n) be the degrees of the generators of the syzygy module. A general curve with (m,n) =(3,3) will have ten double points as singularities. The same is true for a general curve with (m,n) =(2,4). How can we distinguish these two sets of points geometrically? I will discuss some other algebraic distinctions between the two sets of curves and, time permitting, I'll touch on some of the potential methods used to provide a geometric distinction. Despite the technical language in the description of the problem, I hope to provide enough detail to show how down-to-earth the proof methods are for these kinds of results."## An algorithm for overcoming the curse of dimensionality in Hamilton-Jacobi Equations, and super-resolution for imaging high contrast targets in inverse medium scattering problem

In this talk we discuss two topics.

We first discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems. A major contribution of our works is to consider an optimization problem over a single vector of the same dimension as the dimension of the HJ PDE instead. To do so, we consider a Hopf-type formula. The sub-problems are now independent and they can be implemented in an embarrassingly parallel fashion. That is ideal for perfect scaling in parallel computing.

Next we consider a topics on inverse problem. In particular we discuss super-resolution in imaging high contrast targets in inverse medium scattering problem. Super-resolution refers to breaking the diffraction barrier from far-field measurement, which places a fundamental limit on the minimal distance at which we can resolve. We try to mathematically analyze the experimentally-observed phenomenon of super-resolution in imaging the target shape using the concept of scattering coefficients.

Research Area: Numerical solutions to PDE, optimal control, and algorithms for optimization.### Automorphic Forms and Representation Theory Seminar, Prof. Aaron Pollack, Duke University, BRNG 1260

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

## Title: TBA

Refreshments served in the Library Lounge

### CCAM Lunch Seminar, Prof. David Gleich, Purdue University, LWSN B134

Friday, Dec 1 11:30 am - 12:20 pm

## TBD

## Two Weeks

### CCAM Seminar, Prof. Padmanabhan Seshaiyer , National Science Foundation, UNIV 103

Monday, Dec 4 4:30 pm - 5:30 pm

## Computational modeling, analysis and simulation of multi-physics applications in biological, bio-inspired and engineering systems

Abstract: In this talk, we will present modeling, analysis and simulation for nonlinear interaction of multi-physics applications described via coupled differential equation models that arise from examples such as flow-structure interactions to understand rupture of aneurysms and dynamics of micro-air vehicles as well as modeling infectious diseases to understand spread of Zika. Some theoretical and numerical results that validate the reliability and robustness of the computational methodology employed will also be presented.### Special Colloquium, Prof. Wai-Tong (Louis) Fan, University of Wisconsin-Madison, REC 121

Monday, Dec 4 4:30 pm - 5:30 pm

## Stochastic and deterministic spatial models for complex systems

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws in complex systems. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.

In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in mathematical modeling. I will also present novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.

Research Area: Probability, stochastic analysis, biological modeling

Refreshments served in the Library Lounge

### Mathematical Physics Seminar, Emil Prodan, Yeshiva University, NY, UNIV 319

Thursday, Dec 7 1:30 pm - 2:30 pm

## The K-theoretic Bulk-Boundary Principle for Patterned Resonators

This talk will be about point-patterns and equivariant systems of discrete Hamiltonians over such patterns. I will first present a kaleidoscope of numerical examples which display spectral properties akin to two- and higher-dimensional Integer Quantum Hall Effects. In the second part, I will demonstrate how K-theory can be used to understand and predict the bulk and the edge spectra of such systems. In particular, a simple K-theoretic version of the bulk-boundary principle will be presented, which enables one to see from bulk data when topological edge spectrum is to be expected. This last part will be supported again with a kaleidoscope of numerical examples.Refreshments served in the Library Lounge

## Three Weeks

### Automorphic Forms and Representation Theory Seminar, Prof. Ila Varma, Columbia University, BRNG 1260

Thursday, Dec 14 1:30 pm - 2:20 pm