## Next seminar

None currently scheduled.

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

### Fall 2018

• Date: Dec 12 Chloe Wawrzyniak Rutgers A Crash Course in CR Geometry and a New Result on the Stability of the Polynomial Hull of the n-Sphere in C^n. 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 Chloe Wawrzyniak Rutgers Hörmander’s Sum of Squares Operator. 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 Francis Seuffert University of Pennsylvania The Hunt for the Sharp Constant and Extremals of Morrey's Inequality 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.

Date: Mar 22 Matthew Charnley Rutgers An energy lemma and an application to thin inhomogeneities. For elliptic PDEs, the solution (at least in a weak form) can usually be found via the minimization of a certain energy functional. In this talk, we will discuss a result that says that if two energy functionals are close, in an appropriate sense, then their resulting minimizers are also close. We will then talk about an application of this idea to analyzing the solution to conductivity problems with thin inhomogeneities.

Date: Mar 1 Jeaheang Bang Rutgers On Laplace equation in an exterior domain with a nonzero boundary condition at infinity. We will discuss Laplace equation in an exterior domain in Euclidean space, which is by definition the complement of a compact subset. In addition, we impose boundary conditions on it: a solution vanishes on the “actual” boundary, and a solution converges to a given nonzero constant at infinity. In this setting, particularly we will investigate: 1) existence, 2) regularity and 3) behavior at infinity of a weak solution.

Date: February 15 Erik Amorim Rutgers Classical topics on nonclassical solvability for linear PDE A PDE can have classical solutions, which means differentiable functions that solve it, and generalized solutions, which means distributions that solve it in some generalized sense. I plan to talk about two classical results on the topic of generalized solvability of linear PDE's: one is Hans Lewy's example of a very simple linear operator with nonconstant coefficients that is not solvable, and the other is the very simple Malgrange-Ehrenpreis theorem which asserts that linear operators with constant coefficients are always solvable. These two topics have been partially covered in this seminar in the past; this time the focus will be less on detailed calculations for completeness and more on the general ideas of the proofs. They involve basic tools from Hilbert space theory and functional inequalities, which the audience will be tested on. I mean, reminded about.

### Fall 2017

Date: December 11 Jeaheang Bang Rutgers Existence and Uniqueness of a Generalized Solution to the Stationary Navier-Stokes Equations 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 Chloe Wawrzyniak Rutgers Polynomial Convexity 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 Francis Seuffert University of Pennsylvania 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 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 Matt Charnley Rutgers Some dumb equalities, and a non-trivial application 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 Luochen Zhao Rutgers Some problems in elementary complex analysis 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 Zhuolun Yang Rutgers Best representative of Sobolev functions 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 Chloe Wawrzyniak Rutgers The One vs The Many: Domains of Holomorphy 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 Luochen Zhao Rutgers Construction of Haar measures 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 Erik Amorim Rutgers Continuous symmetries of differential equations and what they're good for 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

• Date: May 01, 2017 Jeaheang Bang Rutgers Some concrete examples of elliptic equations In this talk, instead of learning theories, we will take some concrete examples of elliptic equations. And we will discuss which methods we should apply and which one cannot work. We will figure out which method should be most elementary to solve each example.

• Date: April 17, 2017 Jeaheang Bang Rutgers L^2 and L^p theory for Poisson’s equation The main goal of this talk is to discuss L^p theory for Poisson’s equation and to compare this with L^2 theory. For this purpose, we will also briefly talk about Sobolev Space; weak and strong solution; Riesz representation theorem; Calderon Zygmund estimate; and maximum principle. (This talk is for preparation of my oral qualifying exam. I would love to be interrupted whenever you want to ask me questions.)

• Date: April 04, 2017 Zhuolun Yang Rutgers Existence and regularity theories of Poisson's equation In this talk I will show the Schauder estimate (C^{2, alpha} regularity) of Possion’s equation with potential. Combining Perron’s method, it will give us the existence of solution of the Dirichlet problem. This material will be one of the main topics in my oral exam, so everyone will be appreciated to embarrass me with challenging questions.

• Date: March 27, 2017 Cole Franks Rutgers TBA

• Date: March 07, 2017 Chloe Urbanski Rutgers Geometric Proofs of Picard's Theorems In this talk, we will use tools from Complex Geometry to prove Picard's Little Theorem and Great Theorem. I will not assume any Geometry background - Complex or otherwise. We don't need the geometric tools in their full generality, so we will build any tools we need from the ground up.

• Date: February 21, 2017 Matt Charnley Rutgers A Linear Sampling Method for Through-the-Wall Radar Detection In this talk, I will discuss the so-called Linear Sampling method which can be used to solve inverse problems for the Helmholtz equation. I will then explain how this can be modified and used to solve inverse problems in the area of Through-the-Wall imaging. This was work done with the Air Force Institute of Technology, and is a talk that I will be giving there on March 2.

### Fall 2016

• Date: November 09, 2016 Jared Speck Massachusetts Institute of Technology An introduction to the formation of shock singularities In the last two decades, there has been remarkable progress on the formation of shock singularities in solutions to various evolution PDEs. All of the proofs adhere to the following paradigm: i) construct a dynamic, geometrically motivated coordinate system relative to which the solution remains singular; ii) prove that the geometric coordinate system degenerates relative to the standard one; and iii) show that the degeneracy causes a singularity. In this talk, I will provide a detailed analysis of several model problems in order to explain the main ideas behind implementing this approach. I will also describe some of the history of the subject. This will serve as an introduction to my forthcoming talks on the formation of shock singularities in solutions to the compressible Euler equations with vorticity.

• Date: October 17, 2016 Matt Charnley Rutgers The Linear Sampling Method The inverse scattering problem, trying to locate an obstacle from either the far-field or near-field scattering data, is a highly non-linear problem. If the wavelength of the source is similar to the dimensions of the obstacle, the problem is inherently non-linear, and neither the high- or low-frequency approximations work. In this talk, I will discuss the so-called "linear sampling method" for solving this inverse problem. It reduces the problem to solving a linear integral equation at each point, and using the result to determine whether or not that point is contained in the obstacle. At the end, we will discuss the potential for applying this method in physical situations.

• Date: October 10, 2016 Katie McKeon Rutgers Hausdorff Dimension I'll discuss a new technique (due to Professors Falk and Nussbaum) for efficiently estimating Hausdorff dimension of certain sets.

• Date: October 03, 2016 Erik Amorim Rutgers Distributions, Fourier Transform, L²-Sobolev Spaces and the Elliptic Regularity Theorem I will talk about the things in the title, assuming no prior knowledge, and show how they can all be used together to prove the Elliptic Regularity Theorem. This theorem basically says that an elliptic PDE has the property that its solutions are as regular as you can expect, given the order of the equation, for an appropriate meaning of "regular".