Junior Analysis Seminar
Welcome to Junior Analysis. If you would like to give a talk this academic year (Fall 2024 - Spring 2025) please contact me.
We will meet in PMA 10.176 from 3-4 pm Friday. Weekly reminders are sent via the Junior Analysis email list. You can subscribe here.
Spring Schedule
Zach L. - Global well-posedness and scattering for defocusing algebraic NLS in one dimension via new smoothing and almost Morawetz estimates. (January 24th)
Abstract
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). No Speaker (January 31st)
Cooper - Large solutions to conservation laws via Young measures and compensated compactness (February 7th)
Abstract
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. Unai - Homogeneous Landau equation and monotonicity of relative $L^2$ norm in the case of Maxwell molecules. (February 14th)
Abstract
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.Esteban - On the stability-instability transition in large Bose-Fermi mixtures(February 21st)
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).
Jeffrey (February 28th)
Luisa (March 7th)
Justin (March 14th)
Jake (March 28th)
Erisa (April 4th)
Ken (April 11th)
Mark (April 18th)
Zach R. (April 25th)
Fall Schedule
Abstract
I will introduce systems of conservation laws and their stability. In particular, I will highlight the Kruzkov theory, which gives $L^1$ stability for scalar laws, and the Dafermos-DiPerna relative entropy method, which gives $L^2$ stability for systems as long as the solution in question is Lipschitz. We then turn to the theory of a-contraction with shifts, a method developed by Alexis Vasseur and collaborators to extend these results. To this point, most of the work of a-contraction has been done with scalar laws, systems of two equations, and the extremal families of systems (consisting of the shocks with the highest speeds.) I will motivate the difficulties of extending these results the intermediate shock case and present necessary and sufficient conditions for a-contraction to hold for these families. *This is a mock candidacy talk.*Esteban - Surprise Talk (September 16th)
Abstract
“I do not have a title or abstract. Could you send the email saying there will be a ’surprise talk’?” – Esteban
*Surprise - Esteban will speak at a later date. Please attend the Senior Analysis seminar in place of the Junior seminar.*Justin - Singular Integrals (September 23rd)
Abstract
Singular integrals have played a fundamental role in the development of the modern theory of harmonic analysis. At the same time, the subject's technical nature can make it seem off-puttingly arcane without motivation. The goal of this talk is to describe the motivation before giving an overview of the off-puttingly arcane technicalities. Luisa - The Six-Wave Kinetic Equation (September 30th)
Abstract
The attention of the wave turbulence community has been mainly focused on derivations of wave kinetic equations (WKE) from dynamics governed by nonlinear dispersive equations which model the evolution of a system of interacting waves. The inhomogeneous six-wave kinetic equation is one such effective equation derived from the quintic nonlinear Schrödinger equation in the mesoscopic limit. However, the exploration of this WKE has started just recently. In that context, we will show global well-posedness and existence of nonnegative solutions to the WKE in exponentially weighted $L^\infty$ spaces. Moreover, these arguments motivate our analysis of long-time behavior of solutions and we show that they scatter and that the corresponding wave operators are bijective. This is based on a joint work with N. Pavlović and M. Taskovic.Flavio - Interacting Diffusions (October 7th)
Abstract
Stochastic Differential Equations (SDEs) are an important probabilistic object, but they are also a good modeling tool, and they exhibit deep connections with partial differential equations. In this talk, we will try to cover some preliminary material on stochastic analysis and SDEs; and then explore some questions related to interacting diffusions, such as convergence to a McKean - Vlasov limit in the case of mean field interactions, propagation of chaos, and derivation of the Fokker Planck equation.
Esteban - Weyl’s Law in Quantum Mechanics (October 14th)
Abstract
I will give an introductory talk to Weyl's law, regarding the asymptotic distribution of eigenvalues of certain self-adjoint operators. We will mostly talk about its application to study the semi-classical limit of Schrödinger operators. At the end we will also touch upon the connection with Thomas-Fermi theory in many-particle fermonic systems.Jeffrey - $L^2$-Stability and Uniqueness for Scalar Conservation Laws with Concave-Convex Fluxes (October 21st)
Abstract
We study stability properties of solutions to $1-d$ scalar conservation laws with a class of non-convex fluxes. Using the theory of a-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations, and give estimates on the weight coefficient a in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under reduced entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2 \times 2$ system setting by Chen, Golding, Krupa, and Vasseur.
*Note* Jeffrey actually did not present this and instead discussed the regularizing effect of the nonlinearity for scalar conservation laws. Martha Hartt - An ergodic continued fraction algorithm on $\mathbb{R}^{1,1}$ (October 28th)
Abstract
In this talk, we will explore continued fraction algorithms as dynamical systems. We will discuss a proof of why regular continued fractions are ergodic and how this proof can be extended to a certain continued fraction algorithm on 2D Minkowski space. Time permitting, we will also discuss the extended even continued fractions and their higher dimensional analog. Jake - Sets of Finite Perimeter (November 11th)
Abstract
Sets of finite perimeter are useful in solving certain geometric variational problems. We will develop the theory of sets of finite perimeter up to the first and second variation of perimeter and time permitting use that to solve some of these problems.Unai - Interactions between Optimal Transport Theory and Partial Differential Equations (November 18th)
Abstract
The aim of this talk is not to dive deep into any particular topic or theorem, but to motivate the study of optimal transport (OT) by outlining its connections with partial differential equations (PDEs). We will begin by reviewing basic OT theory, including the Monge problem, the Kantorovich problem, the dual problem, and Brenier's theorem. We will then briefly discuss the use of PDEs to study the regularity of optimal solutions to the Monge problem. Finally, I will outline some applications of OT theory to the study of PDEs. If time permits, I will talk about the appearance of the continuity equation in Wasserstein spaces and the JKO scheme.Ken - A geometric approach to calculus on metric measure spaces (December 9th)
Abstract
The space of Riemannian manifolds satisfying a lower bound on the Ricci curvature and an upper bound on the dimension is pre-compact in the Gromov-Hausdorff convergence. A natural question is: what are the limit spaces, and how can we characterize them? One way to do this is to search for properties which are “stable” with respect to Gromov-Hausdorff convergence. In particular, in this talk I will explain how the Sobolev spaces converge in a certain sense, and the limiting space coincides with a typical way to define Sobolev spaces on metric measure spaces.
