9/9: "The wave equation and Fourier analysis" by Chris Sogge.

Abstract: Many problems in harmonic analysis involve the wave equation, and one can use Fourier analysis and Fourier integral operators to solve wave equations. We shall discuss several of these problems, including spherical maximal estimates, local smoothing bounds and Kakeya problems. We shall also go over recent decoupling estimates of Bourgain and Demeter that were inspired by the work of Wolff on regularity estimates for the wave equation.

9/16: "Linear algebra in the tropics" by Aurelien Sagnier.

Abstract: In undergrad, my teacher always incited us to learn proofs of linear algebra with in mind the goal of being able to rewrite all the theory if we somehow ended up by accident on a desert island. But what happens if you land on a tropical island and instead of doing linear algebra over a field, you do it on the tropical (idempotent) semifield (real numbers with minus infinity with laws "maximum"and "addition") ? In this lecture, I will present an quick overview of what happens to a selection of classical theorems of linear algebra in this tropical context and then try to convince you this exotic linear algebra is very useful even in daily life.

9/23: "Holomorphic rigidity of Teichmuller space" by Chikako Mese.

Abstract: We will describe the Teichmuller space and the mapping class group of a compact surface with the goal of understanding the holomorphic rigidity theorem of Teichmuller space.

9/30: "Which powers of a holomorphic function are integrable?-- Shokurov's ACC conjecture for log canonical thresholds" by Jingjun Han.

Abstract: The log canonical threshold of a holomorphic function f is the largest positive real number s, such that |f|^{-2s} is integrable on some neighborhood of the origin. In birational geometry, a remarkable conjecture due to Shokurov predicts that the set of log canonical thresholds satisfies the ascending chain condition. In this talk, I will introduce this conjecture and related topics.

10/7: "Well-posedness questions for nonlinear dispersive equations." by Ben Dodson.

Abstract: In this talk we will discuss the theory for the nonlinear wave, Schrodinger, and Korteweg de Vries equations. We will discuss important examples of phenomena that can occur.

10/14: "Families of Galois representations" by David Savitt.

Abstract: Mathematical objects are often studied by arranging them in families and studying entire families rather than the individual objects. We will explain some of the difficulties in doing this for representations of Galois groups and explain how these might be overcome.

10/21: "Quantization for the Liouville equation" by Ali Hyder.

Abstract: I will talk about concentration phenomena for a sequence of solutions to a Liouville equation, and its relation with the mean-field equation, Nirenberg problem and Moser-Trudinger embedding.

10/28: ''Learning interacting kernels of mean-field equations for systems of interacting particles'' by Fei Lu.

Abstract: Given observation data for a system of interacting particles/agents, can we learn how the particles interact with each other? We address this inverse problem by introducing an efficient nonparametric learning algorithm based on least squares, along with a mathematical learning theory. This talk reviews recent developments, focusing on the function space of learning, the identifiability, and the open questions in analysis. The talk aims to show the connections between analysis (including PDE and probability), statistical learning and computational math, and it welcomes graduate students with interests in these areas.

11/4: ''Harmonic analysis, intersection cohomology, and L-functions'' by Yiannis Sakellaridis.

Abstract: The goal of this lecture will be to describe a link between geometric-topological objects (certain intersection complexes on singular loop spaces), and objects of arithmetic interest (L-functions). The link between the two is by a Fourier/spectral transform. I will begin by giving an overview of Iwasawaâ€“Tate theory, which expresses the Riemann zeta function as the Mellin transform of a certain theta series, and will conclude by describing joint work with Jonathan Wang (MIT), which expresses other L-functions as spectral transforms of functions obtained from intersection complexes on singular arc spaces. No prior familiarity with notions such as L-functions or intersection cohomology will be assumed.

11/11: ''The Alexandrov-Fenchel inequality'' by Yi Wang.

Abstract: The Alexandrov-Fenchel inequality is one of the most important and beautiful inequalities in convex geometry. It relates to differential equations in many ways. In this talk, I will first present the classical theory. After that, I will discuss recent progress using different methods to reprove this inequality and generalize it.

11/18: ''An introduction to the Weil conjecture through examples'' by Xiyuan Wang.

Abstract: In this short talk, I will compute the numbers of solutions of some equations over finite fields. Those numbers will give you a hint about Weil conjecture.

12/2: ''What are geometric flows and what are they good for?'' by Alex Mramor.

Abstract: In this talk aimed at non-geometers, I will discuss what geometric flows are and general tactics for how they are fruitfully applied illustrated through some simple examples, with a focus on the mean curvature flow which is well represented in the department.