Rutgers Graduate Student Analysis Seminar, Fall 2011

The Rutgers graduate student analysis seminar began as a supplementary seminar to Professor Brezis’s functional analysis course, but it gradually evolved into a general-topics seminar in mathematical analysis.

Location: Hill 525
Time: 3:20 – 4:40 on Thursdays
Organizer: Mark Kim (markhkim [at] eden [dot] rutgers [dot] edu)

Past Talks

Tuesday, September 13

Speaker: Mark Kim
Title: The Hahn-Banach Theorem and Symmetry Invariance I
Location: Hill 425
Time: 3:20 – 4:20
Abstract: Eleven years after Hahn’s original proof of the celebrated Hahn-Banach theorem, Agnew and Morse proved a generalization that also preserves invariance under certain commuting operators. I will present the full proof of the theorem, along with a few simple examples.

Wednesday, September 14

Speaker: Mark Kim
Title: The Hahn-Banach Theorem and Symmetry Invariance II
Location: Hill 425
Time: 3:20 – 4:20
Abstract: Eleven years after Hahn’s original proof of the celebrated Hahn-Banach theorem, Agnew and Morse proved a generalization that also preserves invariance under certain commuting operators. I will present the full proof of the theorem, along with a few simple examples.

Thursday, September 22

Speaker: Katy Craig
Title: Small Sets and Why Good Estimates Are Good
Time: 3:20 – 4:20
Abstract: I will talk about types of small sets and why good estimates can give great qualitative information about linear operators. I’ll explore these themes in the context of the Baire Category Theorem and its three important corollaries: the Uniform Boundedness Principle, the Open Mapping Theorem, and the Closed Graph Theorem.

Thursday, September 29

Speaker: Manuel Larenas
Title: On the Delta Functions/Distributions
Time: 3:20 – 4:20
Abstract: I will briefly describe the space of tempered distributions: its topology, operations and fundamental examples (including the well-known Dirac delta). The sentence: ”this is true in the sense of distributions” is dense in the context of PDE’s; let’s unravel the mysteries of this obscure assertion.

Thursday, October 6

Speaker: Frank Seuffert
Title: The Sharp Hardy-Littlewood-Sobolev Inequality I
Time: 3:20 – 4:20
Abstract: The sharp Hardy-Littlewood-Sobolev inequality states that

$\displaystyle \frac{\displaystyle \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} f(x) |x-y|^{-r} \, dx \, dy}{\|f\|^2_{p(r)}} \leq \frac{\displaystyle \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} h(x) |x-y|^{-r} h(y) \, dx \, dy}{\|h\|^2_{p(r)}}$

for all nonnegative $f$ in $L^p(\mathbb{R}^d)$ , where $r \in [0,d)$ , $p(r) = 2d/(2d - r)$ , and $h(x) = (1/(1 + |x|^2))^{d/p}$ . Despite being a very obviously analysis type theorem, it has a very beautiful proof by showing that if we take any nonnegative $f$ in $L^p(\mathbb{R}^d)$ and then form a sequence, ${f_n}$ , from that $f$ by successively projecting it onto $\mathbb{S}^d$ , rotating, and then projecting back down to $\mathbb{R}^d$ , that $f_n$ tends to a constant multiple of $h$ . My talk will outline the fundamental parts of this proof in hopefully an hour.

Thursday, October 20

Speaker: Frank Seuffert
Title: The Sharp Hardy-Littlewood-Sobolev Inequality II
Time: 3:20 – 4:20
Abstract: The sharp Hardy-Littlewood-Sobolev inequality states that

Thursday, October 27

Speaker: Jianguo Xiao
Title: Spectral theorem for self-adjoint operators on Hilbert space
Time: 3:20 – 4:20
Abstract: There is a beautiful theorem in Linear Algebra tells us that: every Hermitian matrix is unitarily equivalent to a diagonal matrix. Similarly in the realm of Hilbert space, the spectral theorem states: every self-adjoint operator is unitarily equivalent to a multiplication. In this talk, we will go over different versions of this theorem and some related concepts.

Thursday, November 3

Speaker: Zhuohui Zhang
Title: The Kakutani Fixed-point Theorem
Time: 3:20 – 4:20
Abstract: In this talk I will first introduce some good properties of convex sets, and will use the typical arguments on convexity to prove the Kakutani’s fixed point theorem, which says an equicontinuous group action of “linear” maps on a compact convex set in a locally convex space has a fixed point. I will use this fixed point theorem to find a left-and-right invariant probability measure on a compact topological group, though for Lie groups this may be obvious.

Thursday, November 10

Speaker: Thomas Sznigir
Title: An Introduction to Integral Equations
Time: 3:20 – 4:20
Abstract: No abstract.

Thursday, November 17

Speaker: Po-Lam Yung
Title: An Introduction to Singular Integrals
Time: 3:20 – 4:20
Abstract: Singular integrals are integral operators that naturally arise in the study of partial differential equations and complex analysis. In this talk we will discuss some classical examples of singular integrals, and then turn to some aspects of their $L^2$ theory. The lemma of Cotlar-Stein on almost orthogonality will be proved, and some applications will be given. (notes)

Thursday, December 1

Speaker: Mark Kim
Title: Interpolation Theorems I
Time: 3:20 – 4:20
Abstract: The first of this two-part talk will begin with a quick overview the $L^1$ and $L^2$ theory of the Fourier transform. The Hausdorff-Young inequality will serve as a motivation to the method of complex interpolation, and the Riesz-Thorin interpolation theorem will be proved. We will then use Riesz-Thorin to prove Hausdorff-Young inequality and Young’s inequality.

Thursday, December 8

Speaker: Mark Kim
Title: Interpolation Theorems II
Time: 3:20 – 4:20
Abstract: The second part of the talk will begin with a discussion of the Lebesgue differentiation theorem. The Hardy-Littlewood maximal function will serve as a motivation to the method of real interpolation, and the Marcinkiewicz interpolation theorem will be proved. We will then use Marcinkiewicz to prove the $L^p$ -boundedness of the Hardy-Littlewood maximal function and the Hilbert transform.