In this talk, I will present joint work with Stefania Patrizi on a fractional Allen-Cahn type equation with a well-prepared initial datum. Our focus is on the behavior of solutions as ε -> 0. In particular, we show that in this limit, the solution develops an interface whose evolution is governed by velocity proportional to the fractional mean curvature of the front. 

With great effort, I have managed to come up with a talk that will be educational and entertaining for everyone in attendance. First years in PDE 1 will get some review on the Lax-Milgram theorem, while Cooper will see a fun application of compensated compactness.

The Strichartz estimates are an important tool in the theory of dispersive equations, particularly in moving from the linear to the semilinear theory. In this talk, we will examine the theory of these estimates in a particular case, the free Schrödinger equation, focusing on their proof and their applications to wellposedness. In parallel, we will discuss the connection between Strichartz estimates and Fourier restriction theory, which was the basis of Strichartz’s original paper on the subject.

The typical isoperimetric cluster problem asks the following: given N disjoint sets (chambers) with volumes v_1,…, v_N what configuration minimizes the perimeter? However, this question leaves a bit to be desired physically. To remedy the issues of this, we assume the boundaries of each chamber themselves have a volume. In this talk I will discuss existence and regularity of minimizes for this new problem. If this is not interesting enough, there will also be snacks.

In recent work, Guillen–Silvestre showed that the Fisher information decreases along solutions to the homogeneous Landau equation. This raises a natural question: are there other functionals that exhibit similar monotonicity or boundedness properties? In this talk, we study the behavior of the 2‑Wasserstein distance, a natural candidate for such a functional. Using the symmetrization technique introduced by Guillen–Silvestre, we establish a new analytic proof of contractive estimates originally obtained by Fournier and Guérin via stochastic methods.

 This is joint work with Matias Delgadino, Maria Gualdani and Maja Taskovic.

 Riemann invariants are special coordinate transforms that diagonalize a system of conservation laws. The new coordinates are invariant along certain integral curves in the state space of the PDE. I will prove that Riemann invariants always exist for 2 conservation laws and show how we can use them to prove some interesting properties of the system.

We define three free boundary problems and describe their physical applications. For one of the problems, we establish full regularity of the solution and the free boundary using tools from geometric measure theory. We conclude by stating some open problems in this area and the current progress made by several authors.

Neural operators are a generalization of neural networks that approximate operators between Banach spaces. I will introduce their architecture, key analytical properties, and applications for numerically solving PDEs. Afterwards, I will give a brief demo of how to implement neural operators.

We will discuss some of the basic concepts and tools of mathematical finance, including Girsanov’s Theorem, risk-neutral measures, and the Martingale Representation Theorem, culminating in the derivation of the Black-Scholes-Merton equation for derivative pricing.  Familiarity with concepts from Probability II and/or Stochastic Processes is assumed, but all are welcome to attend.

When you pour milk into your tea you probably stir it. When you do this, you are exploiting a subtle interaction between the dynamics of transport and viscous diffusion, accelerating the rate at which your milk-tea mixture becomes uniform. In this talk I will elaborate on this phenomenon, primarily trying to answer the simple question: what makes a flow a good mixer?

We introduce compressed sensing from a signal processing (Fourier analysis) perspective. Compressed sensing solves underdetermined systems of linear equations whose solutions are sparse, i.e. they can be described in some basis with few nonzero coefficients. In the real world, compressed sensing is fundamental to machine learning, medical imaging, wireless communication networks, and numerically approximating solutions to PDEs. We go over the foundational papers by EJ Candes and T Tao.

We apply the method of inverse iteration to the Laplace eigenvalue problems with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results.

We also investigate a related problem in the theory of optimal insulation.
In this problem, the goal is to insulate a region of space in an optimal manner, given a fixed amount of insulating material, by choosing an insulation profile that minimizes the appropriate energy functional. We prove that for large enough masses of insulating material, the optimal insulation profile can be computed directly from the optimal insulation profile for a certain critical mass of insulating material.