In this talk, we present some new results on the long time existence and scattering behavior of rough solutions to nonlinear Schrödinger equations in one dimensions with algebraic nonlinearities, $$i \partial_t u + \Delta u = |u|^{2k}u,\quad k\in \mathbb{N},\ k \geq 3,$$ with initial data in a Sobolev space $H^s(\mathbb{R})$ with index $0<s<1$ lying below the energy exponent. We improve on previous results of Colliander, Holmer, Visan and Zhang by proving new smoothing estimates on the nonlinear part of the solution and new almost-Morawetz estimates. Our main tool is the I-method of Colliander, Keel, Staffilani, Takaoka and Tao. We take advantage of the gained regularity using a linear-nonlinear decomposition that is better able to estimate the energy increment of a modified solution on long time intervals; on short intervals, we use a spacetime $L^2_{t,x}$ bilinear estimate to capture cancellations in the energy increment between low and high frequency factors that appear from doing a Littlewood-Paley frequency decomposition. This is joint work with Xueying Yu (Oregon State University).

For systems of conservation laws the typical approach to establish existence is to create approximate piecewise constant solutions and then imposing a smallness constraint to handle the nonlinearity (such as in the front tracking and Glimm schemes.) These methods lead to solutions for initial data with a sufficiently small BV norm. Today I will present a method to get existence for arbitrary bounded initial data by considering a measure valued limit of bounded approximations, then turning to the div-curl lemma to show that this measure valued solution corresponds to a bounded weak solution. 

In the first part of the talk we will introduce the Landau equation and some of its basic properties, including its weak formulation and conserved moments. Then, we will study the particular case of Maxwell molecules, in which one can show monotonicity of the relative $L^2$ norm with respect to the limiting distribution. This is based on current work with Maria Gualdani and Matias Delgadino.

The description of quantum statistical mechanics near absolute zero temperature has a long history dating back to the 1920s. The experimental verification of theoretical predictions, like Bose-Einstein condensation was only realized in the early 2000s; this motivated much mathematical investigations related to the ground state energy of large Bose gases. More recently, Bose-Fermi mixtures have gained attention in the physics community and novel experiments have been realized. In this talk, I will present some new rigorous results regarding the mathematical understanding of such experiments. In particular, I will present a Theorem capturing a stability-instability transition, observed experimentally. Based on joint work with J.K Miller, D. Mitrouskas, N. Pavlovic (to appear soon *maybe before Friday*).

I will talk about the stability of traveling waves of viscous scalar conservation laws via the anti-derivative method.

I’ll introduce temporal and spatial point processes. Then, we’ll discuss spatial-temporal point processes for the purpose of modeling particle dynamics and define Markov Random Fields as well as give some examples.

From a broad perspective, the theory of degenerate elliptic equations is concerned with extending the results from the uniformly elliptic case, such as hypoellipticity or de Giorgi–Nash–Moser. A particularly important class of degenerate elliptic equations are those where the second-order term is the sum of squares of vector fields. In many cases, one can define a special metric on R^n in terms of these vector fields, allowing one to apply the theory of analysis on metric measure spaces. This talk will be an overview of these topics, with the hope of explaining why people care about metric measure spaces.

Percolation starts by taking a lattice of points, and assigning them as open or closed with probability $p$ or $1−p$. The primary question is whether or not an infinite path of open vertices extending from the origin exists, and if so, with what probability. I will show that for the case of the triangular lattice, this probability is positive for $p>1/2$ and zero otherwise. As some especially weird things happen at the critical probability $1/2$, if there is time I’ll talk about some of what happens there too.

The classical isoperimetric inequality has a noticeable rigidity, which characterizes the equality case as balls. Stability is the natural follow up question — if E almost saturates the inequality, then is E close to a ball in some sense? In this talk we will introduce and survey some results which make this relationship precise. Then I will discuss three works, one joint with M. Caroccia and F. Maggi, and another joint with M. Pozzetta, which prove similar results in three different settings. This is a practice talk for my defense.

The Nash-Kuiper theorem states that a Riemannian manifold can be $C^1$ isometrically embedded in an arbitrarily small volume of Euclidean space. The proof of the theorem introduced an iteration method known as convex integration which has recently been applied to construct non-unique weak solutions to PDEs from fluid mechanics. I will walk through the proof of the Nash-Kuiper Theorem and give an overview of how to construct continuous weak solutions to the Euler equations with an arbitrary kinetic energy profile $e(t)$

We’ll discuss existence and uniqueness of optimal trading strategies in abstract financial markets. Instead of using dynamic programming and HJB equations, we will use martingale and duality techniques that do not require Markovian assumptions. The talk will be a blend of stochastic analysis and convex analysis. Parts of it will still be accessible for those who haven’t taken Probability 2.