# Calendar

## Yesterday

### Mathematical Physics Seminar, Annegret Burtscher, Rutgers University, UNIV 319

Thursday, Nov 16 1:30 pm - 2:30 pm

## A mathematical approach to black hole formation

Abstract: Singularities are a generic feature of Einstein's general theory of relativity. Like the Big Bang, singularities may be present at the initial state of our universe, or develop over time when stars collapse, like black holes. Mathematically such situations occur when a solution to the Einstein equations is geodesically inextendible. I this talk I will review the mechanisms that lead to the dynamical formation of singularities in the case of perfect fluid matter and discuss some of the open problems associated to singularity formation in general.### Automorphic Forms and Representation Theory Seminar, Dr. Ozlem Ejder, Colorado State University, BRNG 1260

Thursday, Nov 16 1:30 pm - 2:20 pm

## Sporadic Points on $X_1(n)$

Abstract: Roughly speaking, the points on the modular curve $X_1(n)$ classifies the pairs $(E,P)$ up to isomorphism where $E$ is an elliptic curve and $P$ is a point on $E$ of order $n$. We call a closed point $x$ on $X_1(n)$ sporadic if there are only finitely many closed points of degree at most $\deg(x)$. Hence the problem of determining sporadic points on $X_1(n)$ is closely related to classifying the torsion subgroups of elliptic curves over a degree $d$ field. When $d=1$ or 2, Mazur and Kamienny's work show that there are no sporadic points of degree $d$ on $X_1(n)$. In this talk, we show that for a fixed field $k$, the set of $k$-rational $j$-invariants of sporadic points is finite. We show in particular that there are no sporadic points with a rational $j$-invariant on $X_1(n)$ when $n$ is a prime. This project has started in Women in Numbers Workshop in Banff and it is joint with A. Bourdon, Y. Liu, F. Odumudu and B. Viray.Refreshments served in the Library Lounge

## On the Optimal Shape for the Heat Insulation Energy Problem

The heat insulation energy problem is to minimize \begin{eqnarray} \label{min} J(u,\Omega):=\frac{1}{2}\int_{\Omega} |\nabla u|^2dx+\frac{1}{2m}\left(\int_{\partial \Omega} |u| d\mathcal{H}^{n-1}\right)^2-\int_{\Omega} fu dx \end{eqnarray} over all $u \in H^1(\Omega)$ and open set $\Omega$ with $|\Omega|$ prescribed, where $m>0$ is fixed. Two open questions are asked by Bucur-Buttazzo-Nitsch (i) Can the infimum of the energy functional \eqref{min} be attained by an optimal shape? \\ (ii) It's proved that ball of any radius $R$ is stationary. Is such ball an optimal shape?\\ In this talk, I will present some progress towards these questions. We prove that the infimum can be attained over convex domains, and we discuss the stability of ball by calculating the second variation of both function and domain. With stability results, we are able to show ball is not optimal for any $m>0$ or any radius $R>0$. This is a joint work with Hengrong Du and Prof. Changyou Wang.## Projections in Banach Spaces and Harmonic Analysis

In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.

Research Area: Harmonic Analysis and PDE

## Today

## From Atomistic Model to Peierls-Nabarro Model

The Peierls-Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this talk, we present the connection between a fully atomistic model and a PN model with $\gamma$-surface for the dislocation in a bilayer system (e.g. bilayer graphene). Under some stability condition, we prove that the displacement field of the atomistic model is asymptotically close to that of the dislocation solution of the PN model. Our work can be considered as a generalization of the analysis of the convergence from atomistic model to Cauchy-Born rule for crystals without defects in the literature. This is a joint work with Prof. Pingbing Ming and Prof. Yang Xiang.## Small Cancellation theory.

Note for special time and location. Refreshments will be served as usual.Refreshments served in the Library Lounge

## Reconstructing inclusions from Electrostatic Data

Abstract: In this talk, we will discuss the use of a Sampling Method to reconstruct impenetrable inclusions from Electrostatic Cauchy data. Sampling Methods allow one to recover unknown obstacles with little to no a prior information. These methods are computationally simple to implement and analytically rigorous. We consider the case of an Impedance (Robin) inclusion where we show that the Dirichlet-to-Neumann mapping can be used to reconstruct such impenetrable sub-regions. We also propose a non-iterative method based on Boundary Integral Equations to reconstruct the impedance parameter from the knowledge of multiple Cauchy pairs. Some numerical reconstructions will be presented in two dimensions. We will also briefly discuss the extension to other Inverse Boundary Value Problems. This is joint work with W. Rundell.

Research Area: Direct and inverse problems for PDE, acoustic and electromagnetic scattering

## Next Week

Refreshments served in the Library Lounge

## Transport for Seismic Inversion

Abstract: Optimal transport has become a well-developed topic in the analysis since it was first proposed by Monge in 1781. Due to their ability to incorporate differences in both intensity and spatial information, the related Wasserstein metrics have been adopted in a variety of applications, including seismic inversion. The quadratic Wasserstein metric ($W_2$) has ideal properties like convexity and insensitivity to noise, while conventional $L^2$ norm is known to suffer from local minima. We propose two ways of using W2 in seismic inversion, a trace-by-trace comparison solved by 1D exact formula, and the global comparison which requires the numerical solution of Monge-Ampère equation. The 1D approach has been successfully applied to field data in collaboration with PGS, Inc. I will discuss the connection between the Wasserstein metric $W_p$ and the Sobolev space $W^{-1,p}$. This is a joint work with Dr. Björn Engquist, Brittany Froese, Junzhe Sun and Lingyun Qiu.## Two Weeks

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.Refreshments 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

Refreshments served in the Library Lounge

## 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."## TBA

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

## Three 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

## TBA

Research Area: Probability, stochastic analysis, biological modelingRefreshments served in the Library Lounge prior to Colloquium.

Refreshments served in the Library Lounge

Refreshments served in the Library Lounge

## December

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

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