In this expository talk, we prove strong unique continuation for J-holomorphic curves by first giving a simple proof of Aronszajn's theorem in the special case of the two-dimensional flat Laplacian. The key tool is a Carleman/Aronszajn-type weighted integral estimate. Based on this paper.

The RAGE theorem provides a connection between the dynamics of solutions of a linear evolution equation (such as the Schroedinger equation) and the spectral properties of the corresponding Hamiltonian (the generator of the dynamics). In the first part of a series of two talks, I plan to review the spectral theorem, explain different types of spectra and give a detailed proof of the RAGE-type theorems and their applications. I will also explain how one can interpret the RAGE theorem in the case non-linear dynamics, despite the absence of a notion of spectrum in the non-linear case. This leads to "Soliton Resolution Conjectures", a series of major outstanding conjectures in the theory of non-linear dispersive equations. I will state recent results of T. Tao, which I plan to review in more detail in a future talk. The talk will be elementary.

Kolmogorov complexity measures the algorithmic information content of individual binary strings. It can also be used to define randomness for individual infinite binary sequences, with respect to a wide range of measures. In particular, if the underlying measures are Hausdorff measures, we obtain a notion of dimension for individual points in a Euclidean space.

I will introduce the basic notions of algorithmic information theory and randomness, before I will introduce some basic questions of the effective theory and compare them (and their solutions) to questions from the "classical" realm. No prior knowledge of logic or computability theory is required.

In this expository talk, I will introduce the fruitful topic of wave packet analysis, a creative extension of Littlewood-Paley theory. I will use this wave packet decomposition to prove (a toy version of) Carleson's Theorem on pointwise convergence of truncations of the Fourier transform. No background is required, beyond familiarity with the Fourier transform in L² of the real line.

The distribution of antibound states, the resonances lying on the imaginary axis, has been a recurring theme in scattering theory since the works of Lax and Philips. In this talk I will consider a simple special one-dimensional case and prove a symmetry property for bound and antibound states first observed in numerical experiments by Bindel and Zworski.

The Lebesgue differentiation theorem states that the family of sets $\{(-r,r)\}_{r>0}$ differentiates $L^1_{\text{loc}}(\R)$, in the sense that for every $f\in L^1_{\text{loc}}(\R)$ $$\lim\limits_{r\to 0}\frac{1}{2r}\int_{x+(-r,r)}f(y)dy=f(x)$$ for almost every $x\in\R$. In this talk we will explore differentiation theorems in $\R$ in the case when we replace the family $\{(-r,r)\}_{r>0}=\{r(-1,1)\}_{r>0}$ by one of the form $\{rS_k\}_{r>0,k\in\N}$, where $\{S_k\}_k$ is a decreasing sequence of sets whose limit set has Lebesgue measure zero and prescribed Hausdorff dimension (greater than $2/3$). We will prove (or try to prove) the existence of such a sequence of sets by considering the corresponding maximal operator and applying a probabilistic method.

Reference: Maximal operators and differentiation theorems for sparse sets, by I. Laba and M. Pramanik.

Originally developed in 1950s to prove results about automorphic forms, the Selberg Trace Formula has since been greatly generalized and is now of enormous importance in number theory. It is also an analytical tool, however, and in this talk we will focus entirely on its applications to hyperbolic manifolds. In this incarnation, the formula relates certain geometric features of a Riemannian manifold $X$ to its spectral properties. We will give a complete proof of the formula in the simple case when $X$ is compact. We'll also discuss some of its consequences including Weyl's law, the prime geodesic theorem and the analyticity of the Selberg zeta function.

We consider multilinear oscillatory integrals of the form $$I_\lambda(f_1,\ldots, f_n)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^n f_j\circ\pi_j(x)\eta(x)dx$$ and explore algebraic conditions on the polynomial phase $P$ and geometric conditions on the projections $\{\pi_j\}_j$ that ensure decay estimates $$|I_\lambda (f_1,\ldots,f_n)|\lesssim \langle\lambda\rangle^{-\epsilon}\prod_{j=1}^n \|f_j\|_{L^\infty}.$$

In this talk, we will show that the GP equation can be used to descibe the macroscopic property of dynamics of Bose-Einstein condenstaes. In particular, we want to show that the solution to BBGKY hierarchy converges in some suitable space to a solution of the infinite hierarchy, which is unique.

Reference: Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau: Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate

Thomas Chen, Nataša Pavlovic: A short proof of local well-posedness for focusing and defocusing Gross-Pitaevskii hierarchies

Mathematically, quantum decay rates appear as imaginary parts of poles of the meromorphic continuation of Green’s functions. As energy grows, decay rates are related to properties of geodesic flow and to the structure at infinity. For a cusp, infinity is “small”, which typically slows decay. However, I will present a class of examples for which decay rates go to infinity with energy even in the presence of a cusp. This is part of a more general investigation of resonances on manifolds with hyperbolic ends.

Let $(M, g)$ be a closed Riemann surface, $(N, h)$ a Riemannian manifold, and let $\phi : M \to N$ be a homeomorphism. There is a functional on the homotopy class of $\phi$, the energy functional, that takes a map $f$ and computes the integral of the square norm of its differential. Critical points of this functional are called harmonic maps. In the case that $ M $ and $ N $ are 2-dimensional, compact surfaces, minimizers of this functional are smooth, and given a homeomorphism $ f : M \to N $, there is a harmonic diffeomorphism that is homotopic to $ f $. If $ N $ is negatively curved, or if the genus of $ N $ is greater than $ 1 $, this harmonic map is the unique one in the homotopy class of $ f $. In this talk, I'll discuss how to extend these results to the setting of target manifolds with conic singularities. The argument is a perturbative one, so linear PDE is the main tool. In the conic setting, the relevant PDE lies in a class of linear operators called $b$-operators, whose study, originally undertaken by Melrose and Mendoza in 1983, provides us with a complete set of tools for solving this problem. We will discuss these tools and how they apply in this situation.

The so-called spherical harmonics are the eigenfunctions of the Laplacian on the sphere S^2. I will show how to use the representations of SU(2) to derive the spherical harmonics. The approach will be global in the sense that there will be no coordinate chart arguments.

The Schwarzschild space-time represents a stationary black hole, and is the only stationary spherically symmetric solution for the vacuum Einstein equations. Around 1974 a physicist named Robert Price, studying linear waves on the Schwarzschild space-time, heuristically derived a $t^{-3}$ local pointwise decay rate. This became known as Price's Law, and several partial results have been obtained in recent years. In the talk I will describe a proof of Price's Law, which applies to a larger class of asymptotically flat stationary space-times.