• Speaker: Brendon Rhoades, University of California, San Diego
• Title: The combinatorics, algebra, and geometry of ordered set partitions
• Date: 09/20/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

An {\em ordered set partition} of size $n$ is a set partition of $\{1, 2, \dots, n \}$ with a specified order on its blocks. When the number of blocks equals the number of letters $n$, an ordered set partition is just a permutation in the symmetric group $S_n$. We will discuss some combinatorial, algebraic, and geometric aspects of permutations (due to MacMahon, Carlitz, Chevalley, Steinberg, Artin, Lusztig-Stanley, Ehresmann, Borel, and Lascoux-Sch\"utzenberger). We will then describe how these results generalize to ordered set partitions and discuss a connection with the Haglund-Remmel-Wilson {\em Delta Conjecture} in the field of Macdonald polynomials. Joint with Jim Haglund, Brendan Pawlowski, and Mark Shimozono.

• Speaker: Yang Yang, Michigan State University
• Title: Some inverse source and coefficient problems for the wave operators (special colloquium)
• Date: 10/04/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Inverse problems seek to infer causal factor from the resulting observation, and waves are among the most prevalent and significant observations in nature. In this talk, we will discuss two inverse problems for the acoustic wave equation and its generalizations. The first is an inverse source problem where one attempts to determine an instantaneous source from the boundary Dirichlet data. We give sharp conditions on unique and stable determination, and derive an explicit reconstruction formula for the source. The second is an inverse coefficient problem on a cylinder-like Lorentzian manifold (M,g) for the Lorentzian wave operator perturbed by a vector field A and a function q. We show that local knowledge of the Dirichlet-to-Neumann map (DN-map) stably determines the jets of (g,A,q) up to gauge transformations, and global knowledge of the DN-map stably determines the lens relation of g as well as the light ray transforms of A and q. This is based on joint work with P. Stefanov.

• Speaker: Frank Morgan, Williams College
• Title: Double Soap Bubbles and Densities
• Date: 10/18/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The familiar double soap bubble is the least-area way to enclose and separate two given volumes in Euclidean space. What if you give space a density, such as r^2 or e^r^2 or e^-r^2? The talk will include recent results and open questions. Students welcome.

• Speaker: Rustum Choksi, McGill University
• Title: Nonlocal Geometric Variational Problems: Isotropic and Anisotropic Extensions of Gamow's Liquid Drop Problem and Beyond
• Date: 11/01/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The liquid drop (LD) model, an old problem of Gamow for the shape of atomic nuclei, has recently resurfaced within the framework of the modern calculus of variations. The problem takes the form of a nonlocal isoperimetric problem on all 3-space with nonlocal interactions of Coulombic type. In this talk, we first state and motivate the LD problem, and then summarize the state of the art for global minimizers. We then address certain recent anisotropic variants of the LD problem in the small mass regime, with a particular focus on the minimality of the Wulff shape. In the second half of the talk, we address a related nonlocal geometric problem based solely on competing interaction potentials of algebraic type. This problem is directly related to a wide class of self-assembly/aggregation models for interacting particle systems (eg. swarming). This talk includes joint work with Almut Burchard (Toronto), Robin Neumayer (IAS and Northwestern), and Ihsan Topaloglu (Virginia Commonwealth).

• Speaker: Jared Speck
• Title: Singularity Formation in General Relativity
• Date: 11/08/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The celebrated Hawking–Penrose theorems are breakdown results for solutions to the Einstein equations of general relativity, which are a system of highly nonlinear wave-like PDEs. These theorems show that, under appropriate assumptions on the matter model, a large, open set of initial data lead to geodesically incomplete solutions. However, these theorems are “soft” in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is tied to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness stemming from lack of information for how to uniquely continue the solution (this is roughly known as the formation of a Cauchy horizon). In various works, some joint with I. Rodnianski, we have obtained the first results in more than one spatial dimension that eliminate the ambiguity for an open set of initial data: for the solutions that we studied, the incompleteness is tied to the blowup of various spacetime curvature scalars along a spacelike hypersurface. Physically, this phenomenon corresponds to the stability of the Big Bang and/or Big Crunch singularities. From an analytic perspective, the main theorems are stable blowup results for quasilinear systems of elliptic-hyperbolic PDEs. In this talk, I will provide an overview of these results and explain how they are tied to some of the main themes of investigation by the mathematical general relativity community. I will also discuss the role of geometric and gauge considerations in the proofs, as well as intriguing connections to other problems concerning stable singularity formation.

• Speaker: Noam Elkies, Harvard University
• Title: TBA
• Date: 12/13/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.