Graduate Student Analysis Seminar

This page maintained by Érik Amorim and last modified Dec 19, 2018.

Next seminar

None currently scheduled.

Previous seminars since Fall 2016 (in reverse chronological order):

Fall 2018

  • Date: Dec 12
    Speaker: Chloe Wawrzyniak
    Affiliation: Rutgers
    Title: A Crash Course in CR Geometry and a New Result on the Stability of the Polynomial Hull of the n-Sphere in C^n.
    Abstract: I will spend the first half of this talk discussing some background in CR geometry including CR manifolds and singularities, holomorphic and polynomial hulls, and attached analytic discs. Throughout this discussion, we will focus on a few key examples: real hypersurfaces, the unit polydisc, and the standard n-sphere S^n. In the second half of the talk, I will present joint work with Purvi Gupta on the construction of the polynomial hull of a small perturbation of the standard n-sphere S^n in C^n using attached analytic discs. We first utilize existing results about local hulls of holomorphy near nondegenerate elliptic CR singularities. We then solve a Riemann-Hilbert problem to construct attached discs on the totally real portion of the perturbed sphere. Finally, we utilize uniqueness results to patch the two constructions together and show that the resulting smooth Levi-flat manifold, which is diffeomorphic to the hull of S^n, is the polynomially convex hull of the perturbed sphere. It is worth stating that although this is new research, I will avoid assuming prior knowledge of concepts in the field. My goal is that the main ideas of this talk is accessible to everyone with an interest in analysis, including first year students.
  • Spring 2018

  • Date: Apr 19
    Speaker: Chloe Wawrzyniak
    Affiliation: Rutgers
    Title: Hörmander’s Sum of Squares Operator.
    Abstract: In 1967, Lars Hörmander proved that second order PDEs of a certain form with smooth real coefficients have smooth solutions, given smooth data. We will discuss this result, though we will not follow Hörmander’s original proof. Rather, we will follow the proof of J. J. Kohn in 1973 using pseudo-differential operators and fractional Sobolev spaces to compute subelliptic estimates. Time permitting, I will mention some results related to this one with either complex coefficients or real-analytic data.

  • Date: Apr 2
    Speaker: Francis Seuffert
    Affiliation: University of Pennsylvania
    Title: The Hunt for the Sharp Constant and Extremals of Morrey's Inequality
    Abstract: The seminorm form of Morrey's Inequality can be summarized as follows: Let $f \in L^\infty (\mathbb{R}^N)$ be such that $Df \in L^p (\mathbb{R}^N)$ and $p > N$. Then there is some $C>0$ depending only on $N$ and $p$ such that C \| Df \|_p \geq [ f ]_{C^{0,1-N/p}} where $[ \cdot ]_{C^{0,1-N/p}}$ is the $C^{0,1-N/p}$-Holder seminorm given by [ f ]_{C^{0,1-N/p}} := sup_{x \neq y} \frac{ | f(x) - f(y) | }{ | x - y | }. This inequality was proven at least 50 year ago by C. B. Morrey Jr. However, to the best of our knowledge, there is little quantitative information on Morrey's Inequality. In particular, the value of the sharp constant and a precise formula is unknown. In fact, existence of extremals had not been established previously. In a recent project, we are endeavoring to find the value of the sharp constant in Morrey's Inequality and what formulas for extremals would be if extremals should exist. We have not figured out what the sharp constant for Morrey's Inequality is or a precise formula for extremals except when $N=1$. However, we have been able to get some good qualitative results, e.g. that extremals of Morrey's Inequality exist and that they must be cylindrically symmetric about a pair of points that achieve its $C^{0,1-N/p}$-Holder seminorm. These and other results as well as their proofs will be recounted in our talk.

    Fall 2017

    Date: December 11
    Speaker: Jeaheang Bang
    Affiliation: Rutgers
    Title: Existence and Uniqueness of a Generalized Solution to the Stationary Navier-Stokes Equations
    Abstract: Before getting into the main topic, I will first briefly explain the Navier-Stokes equations and remind you of some basic tools that we need, such as Sobolev inequality, the Sobolev embedding theorem, Poincare's inequality and the Leray-Schauder theorem. Then we will establish, as main theorems, existence and uniqueness of a generalized solution to the stationary Navier-Stokes equations with homogenous boundary condition on a bounded or unbounded domain. This talk relies on Ladyzhenskaya’s book on the Naiver-Stokes equations.

    Date: December 4
    Speaker: Chloe Wawrzyniak
    Affiliation: Rutgers
    Title: Polynomial Convexity
    Abstract: A compact set X in C^n is polynomially convex if for every x outside X, there is a polynomial P such that |P(x)| is strictly larger than the supremum of |P| over X. A natural question is to classify which sets are polynomially convex. In one complex dimension, the answer is well-known. But in dimension 2 and higher, this question is still an area of active research. Many of the proofs in this area are long, technical, and require extra background. So, we will spend most of our time playing with examples while assuming some major results. If there is time and interest, I can also discuss some applications of polynomial convexity and why this question is important.

    Date: November 13
    Speaker: Francis Seuffert
    Affiliation: University of Pennsylvania
    Title: An Extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux’s Generalization of the Sobolev Inequality to Continuous Dimensions and an Application
    Abstract: The Bianchi-Egnell Stability Estimate is a stability estimate or quantitative version of the Sobolev Inequality – it states that the difference of terms in the Sobolev Inequality controls the distance of a given function from the manifold of extremals of the Sobolev Inequality with distance measured in the gradient square or $\dot{H}^1$ norm. In this talk, we present an extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux’s Generalization of the Sobolev Inequality to Continuous Dimensions. We also demonstrate a deep link between the Sobolev Inequality and a one-parameter family of sharp Gagliardo-Nirenberg (GN) inequalities and how this link can be used to derive a new stability estimate on the one-parameter family of sharp GN inequalities from our stability estimate on Bakry, Gentil, and Ledoux’

    Date: October 20
    Speaker: Matt Charnley
    Affiliation: Rutgers
    Title: Some dumb equalities, and a non-trivial application
    Abstract: A paper I recently finished reading by Dapogny and Vogelius uses a not-very-interesting equality that stumped me for a long time, but actually makes the desired problem solvable in a more attractive way. I'll go through the equalities and show the application to the particular PDE problem of interest.

    Date: October 23
    Speaker: Luochen Zhao
    Affiliation: Rutgers
    Title: Some problems in elementary complex analysis
    Abstract: In this supposedly interactive talk I’ll introduce some elementary yet hard (at least to the speaker) problems in complex analysis. The audience are very welcomed to be involved in discussions. The only caveat is that some problems might have already been solved, and thus won’t earn your fame or dollars.

    Date: October 16
    Speaker: Zhuolun Yang
    Affiliation: Rutgers
    Title: Best representative of Sobolev functions
    Abstract: Given a Lebegue integrable function, if we take its average on a ball and shrink the radius to 0, we know the limit exists almost everywhere by Lebesgue differential theorem. This limit gives us a good representative for Lebegue integrable. What if we know its derivatives are also integrable? Should we expect the “singular set” to be smaller? In this talk, I will introduce Capacity measure to measure this “singular set” of Sobolev functions. I will go over some classical arguments in measure theory, and some of which are very useful for PDE. e.g. to get partial regularity.

  • Date: October 09
    Speaker: Chloe Wawrzyniak
    Affiliation: Rutgers
    Title: The One vs The Many: Domains of Holomorphy
    Abstract: Roughly speaking, a domain of holomorphy is an open set in C^n on which we can find a holomorphic function which can not be extended holomorphically past the boundary. Domains of holomorphy are key to solving a lot of problems in Several Complex Variables (SCV), but the question of which sets are domains of holomorphy is completely different in one complex dimension. We'll discuss this difference, why these domains are so important in two or more dimensions, and time permitting, some equivalent characterizations of these sets. Most proofs will be sketched or skipped entirely, but examples will be plentiful. I should also note that this talk is given in preparation for my oral qualifying exam.

  • Date: October 02
    Speaker: Luochen Zhao
    Affiliation: Rutgers
    Title: Construction of Haar measures
    Abstract: The main purpose of this talk is to present Cartan's approach towards the existence of Haar measures on locally compact groups. To avoid absolute abstract nonsense I’ll talk about several examples, highlighting an application in the representation of compact topological groups.

  • Date: September 25
    Speaker: Erik Amorim
    Affiliation: Rutgers
    Title: Continuous symmetries of differential equations and what they're good for
    Abstract: After a brief introduction to manifolds and Lie groups, we will discuss what it means to say that one has a continuous symmetry of a system of differential equations (partial or ordinary), according to the theory first put together by Sophus Lie. We will show examples of useful applications that can be constructed from such a symmetry, and show how to obtain a symmetry from basic algebraic manipulations of the system.
  • Spring 2017

    Fall 2016