Rutgers Graduate Student Analysis Seminar, Spring 2012
This is a general-audience seminar in mathematical analysis, aiming to provide an informal speaking environment for graduate students interested in analysis at large. Everyone is welcome to give a talk—even you, Mrs. Lovett, even I!—but we ask that each speaker keep the background to a minimum, preferably assuming no more than basic real and complex analysis covered in Math 501 and Math 503. If you are interested in giving a talk, please contact me or Katy—our contact information can be found below. If you are really interested in giving a talk, you might want to consider tips from senior mathematicians as to what constitutes a good talk: for example, here is Terence Tao’s advice. Please stay awhile and enjoy!
Location: Hill 525
Time: 4:50 – 5:50 on Mondays
Organizers: Katy Craig (katycc [at] math [dot] rutgers [dot] edu) and Mark Kim (markhkim [at] eden [dot] rutgers [dot] edu)
Past Talks
Monday, January 30
Speaker: Mark Kim
Title: The Kakeya Problem in Harmonic Analysis
Time: 4:50 – 5:50
Abstract: What is the minimum area required to spin a needle around? Yes? Did you say, “That’s so 1920s”? Fear not! In this talk, we will present a bird’s-eye view of the problems known collectively as the Kakeya problem, ranging from theorems from 1970s to still-unsolved conjectures. The focus will be on the harmonic-analytic approach, though it should be noted that there are plenty of geometry, number theory, and combinatorics floating around in the scene. (notes)
Monday, February 6
Speaker: Katy Craig
Title: When an inf is also a sup: Fenchel-Rockafellar and Kantorovich duality
Time: 4:50 – 5:50
Abstract: Kantorovich duality says that the least amount of effort required to rearrange one pile of dirt to look like another is equal to the largest amount an enterprising friend could charge to perform the task for you (even with some constraints on his pricing structure). Thinking of probability measures as piles of dirt, the least amount of effort required to make one look like another provides a way to measure their distance, and Kantorovich duality tells you how to approximate this distance from above and below. I will reformulate the (discrete) Kantorovich problem as the minimization of a sum of two convex functions and prove Kantorovich duality as a consequence of Fenchel-Rockafellar. (notes)
Monday, February 13
Speaker: Po Lam Yung
Title: A “random theorem” in harmonic analysis
Time: 4:50 – 5:50
Abstract: In this talk, we will discuss a “random theorem” about randomness in harmonic analysis. The result dates back to Marcinkiewicz and Zygmund, and we will use this to prove Fefferman’s ball multiplier theorem. (notes)
Monday, February 20
Speaker: Moulik Kallupalam
Title: What’s Schrichartz Estimate?
Time: 4:50 – 5:50
Abstract: In the world where Fourier Transformed functions live, life is sweet, simple and fun. Going back and forth between this and our world, is often useful, and technical. We’ll talk a little about Fourier Transform, how it helps define spaces with fractional derivatives, and then talk about ” Strichartz estimates” for the wave equation in 3 space dimensions. (notes)
Monday, February 27
Speaker: Tom Sznigir
Title: Laplace’s Equation on Manifolds
Time: 4:50 – 5:50
Abstract: In this talk, we’ll look at PDEs on manifolds. With a few modifications, we can go from solving Laplace’s equation on Euclidean space to trying to solve it on a compact manifold. We would like to know when one differential form can be written as the Laplacian of another differential form. The answer to this question is given by the Hodge Decomposition theorem. As it turns out, knowing the solutions to Laplace’s equation also gives us information about the global topology of the manifold. (notes)
Monday, March 5
Speaker: Jianguo Xiao
Title: Hermite Functions in Schwartz Space
Time: 4:50 – 5:50
Abstract: Schwartz space is very important in the realm of Fourier analysis. In this talk, we will look at some interesting functions in Schwartz space, called Hermite functions. And we will introduce a representation theorem for Schwartz space based on these special functions.
Monday, March 19
Speaker: Xukai Yan
Title: Some Properties of Viscosity solutions to elliptic equations
Time: 4:50 – 5:50
Abstract: In this talk, we will introduce the concept of viscosty solutions to PDEs. We are going to discuss some properties of viscosity solutions to elliptic equations, including a sufficient condition of removable singularities for viscosity solutions, a strong maximal principle and hopf lemma. The proof of the last two rely on the use of comparison functions, which is a useful method in PDE.
Monday, March 26
Speaker: Doug Schultz
Title: Minkowski’s Existence Theorem for Convex Bodies
Time: 4:50 – 5:50
Abstract: Given K, a compact convex set with interior in R^n, one can construct a natural measure on the unit sphere by dictating that the measure of a set W is the N-1 dimensional Hausdorff measure of the set of points on the boundary of K which have a vector in W as a “supporting normal vector” . On the other hand, one might absent-mindedly ask: “Is this the totality of measures on S^n-1?” Under some mild assumptions, the answer is actually yes, and we give a survey and an outline of the proof of the existence (and uniqueness, if time/energy) of a convex body which induces a prescribed measure. Disclaimer: Not analysis per se, but interesting (and tending toward elementary).
Monday, April 2
Speaker: Liming Sun
Title: Rearrangement and Riesz inequality
Time: 4:50 – 5:50
Abstract: Rearrangements manipulate the shape of a geometric set while preserving its size. They are used in calculus of variations to find extremals of geometric functionals or prove geometric inequality. We will begin with introducing the radial decreasing rearrangement f^\ast of a measurable function f. Then I will state the Riesz inequality. Riesz inequality implies Polya-Szego inequality and Brunn-Minkowski inequality. After showing its application, I will try to prove it if time possible. In order to prove it. I will introduce the Steiner symmetrization and Schwarz symmetrization and compactness of these two sequences.
Monday, April 9
Speaker: Bence Borda
Title: The Haar Measure
Time: 4:50 – 5:50
Abstract: On every locally compact Hausdorff topological group one can define a translation invariant Borel measure with some extra regularity properties called the Haar measure. In the talk we will first examine a few topological groups and see how we can find a Haar measure on them. Then we will give a combinatorial proof of the existence of a Haar measure on compact Hausdorff topological groups.
Monday, April 16
Speaker: Mark Kim
Title: Fourier Analysis on Finite Groups (and Other Nice Groups)
Time: 4:50 – 5:50
Abstract: In this talk, we will survey the basic theory of Fourier analysis on groups other than R. We will focus on finite cyclic groups, although other groups will be considered as well. (notes)
Monday, April 23
Speaker: Tom Sznigir
Title: The Hamilton-Jacobi Equation
Time: 4:50 – 5:50
Abstract: The usual way of solving a first-order PDE is to transform it into a system of ODEs. The Hamilton-Jacobi equation allows us to reverse the process, transforming certain systems of ODEs into a single first-order PDE. It also has applications to the calculus of variations as well as mathematical physics. We will also look at some interesting geometric interpretations of the Hamilton-Jacobi Equation.