If a polynomial map from $C^n$ to itself is injective, then it is also surjective. The first proof of this elegant result actually used a passage to characteristic p. I'll introduce the relevant notions from model theory and explain the proof, which will naturally yield the following much stronger result: Every statement in the first order logic of rings is true in the algebraic closure of $F_p$ for almost all p iff it is true in C.

The Lovasz Local Lemma is a subtle and far-reaching probability lemma. Roughly, given a large number of bad events which are "mostly" independent it sieves out an instance when none of the bad events occur. As an illustrative example, consider a huge family of 10-element sets, where each set overlaps at most 100 others. We color randomly red and blue and for each set e there is a bad event that e is monochromatic. The conclusion is the existence of a coloring where none of the e are monochromatic. In theory. But if there are n sets a random coloring only has exponentially small chance of succeeding. Finding a polynomial time algorithmic implementation has proven elusive. There were some results of Jozsef Beck and Noga Alon in the early 1990s but they had limited application. Very recently Robin Moser (ETH) has made a breakthrough, giving an amazingly simple probabilistic algorithm to find the coloring and a not quite so simple proof that the algorithm indeed works.

We shall rephrase Moser's argument and give the entire argument.

Given an L^2-normalized cusp form f on a modular curve X_0(N), what can be said about pointwise bounds for f? For Hecke eigenforms, we will prove the first non-trivial bound in terms of the level N as well as hybrid bounds in terms of the level and the Laplacian eigenvalue. Similar techniques work for functions on other spaces, e.g. quotients of quaternion algebras. This is joint work with R. Holowinsky.

In part I, I will describe background material and a new proof for the uniqueness of solutions to the Gross-Pitaevskii hierarchy. This is joint work with S. Klainerman and is a simplification, based on space-time estimates, of an older proof of Erdös, Schlein and Yau.

In Part II (joint work with M. Grillakis and D. Margetis) I will discuss a new, highly non-linear but explicit NLS in two space variables, whose solutions, if they exist, provide a second order correction to the usual tensor product approximation, which works in the Fock space norm. This is inspired by recent work of Rodnianski and Schlein, as well as older work of Wu.

The intersection form on the group of line bundles on a complex algebraic surface always has signature (1,n) for some n. So the automorphism group of an algebraic surface always acts on hyperbolic n-space. For a class of surfaces including K3 surfaces and many rational surfaces, there is a close connection between the properties of the variety and the corresponding group acting on hyperbolic space. (In fancier terms: the Morrison-Kawamata cone conjecture holds for klt Calabi-Yau pairs in dimension 2.)

A Kakeya set is a set in (F_q)^n (the n dimensional vector space over a field of q elements) which contains a line in every direction. In this talk I will present a recent result which gives a lower bound of (q/2)^n on the size of such sets. This bound is tight to within a multiplicative factor of two from the known upper bounds. The proof extends the polynomial method used in [Dvir 08, Saraf Sudan 08] and uses polynomials of unbounded degree. If time allows I will also discuss the applications to the explicit construction of mergers that follow from our techniques.

Joint work with Kopparty, Saraf and Sudan (arxiv paper: "Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers").

Recent work in the emerging field of compressed sensing indicates that, when feasible, judicious selection of the type of image transformation induced by imaging systems may dramatically improve our ability to perform reconstruction, even when the number of measurements is small relative to the size and resolution of the final image. The basic idea of compressed sensing is that when an image is very sparse (i.e. zero-valued at most locations) or highly compressible in some basis, relatively few incoherent observations suffice to reconstruct the most significant non-zero basis coefficients. These theoretical results have profound implications for the design of new imaging systems, particularly when physical and economical limitations require that these systems be as small, mechanically robust, and inexpensive as possible.

In this talk I will describe the primary theory underlying compressed sensing and discuss some of the key mathematical challenges associated with its application to practical imaging systems. In particular, I will explore several novel imaging system designs based on compressed sensing, including compressive coded aperture and hyperspectral imagers, and examine the interplay between compressed sensing theory and the practical physical constraints which must be considered to effectively exploit this theory.

We will discuss the problem of extending the construction of the classical Abel maps for smooth curves to the case of singular curves. The construction of degree-1 Abel maps will be shown, together with an approach for constructing higher degree Abel maps.

A corner stone of the theory of random matrices is Wigner's semi-circle law, obtained in the 1950s, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random hermitian matrix with iid (upper diagonal) entries follows the semi-circle law. The non-hermitian case is the famous Circular Law Conjecture, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random matrix with iid entries is uniform in the unit circle.

Despite several important partial results (Ginibre-Mehta, Girko, Bai, Edelman, Gotze-Tykhomirov, Pan-Zhu etc) the conjecture remained open for more than 50 years. Last year, T. Tao and I confirmed the conjecture in full generality. I am going to give an overview of this proof, which relies on rather surprising connections between various fields: combinatorics, probability, number theory and theoretical computer science. In particular, ideas from ADDITIVE COMBINATORICS play crucial roles.

Speaking at a general level, a major goal of the p-adic Langlands program (from a global, rather than local, perspective) is to find a p-adic generalization of the notion of automorphic eigenform, the hope being that every p-adic global Galois representation will correspond to such an object. (Recall that only those Galois representations that are motivic, i.e. that come from geometry, are expected to correspond to classical automorphic eigenforms).

In certain contexts (namely, when one has Shimura varieties at hand), one can begin with a geometric definition of automorphic forms, and generalize it to obtain a geometric definition of p-adic automorphic forms. However, in the non-Shimura variety context, such an approach is not available. Furthermore, this approach is somewhat remote from the representation-theoretic point of view on automorphic forms, which plays such an important role in the classical Langlands program.

In this talk I will explain a different, and very general, approach to the problem of p-adic interpolation, via the theory of p-adically completed cohomology. This approach has close ties to the p-adic and mod p representation theory of p-adic groups, and to non-commutative Iwasawa theory.

After introducing the basic objects (namely, the p-adically completed cohomology spaces attached to a given reductive group), I will explain several key conjectures that we expect to hold, including the conjectural relationship to Galois deformation spaces. Although these conjectures seem out of reach at present in general, some progress has been made towards them in particular cases. I will describe some of this progress, and along the way will introduce some of the tools that we have developed for studying p-adically completed cohomology, the most important of these being the Poincare duality spectral sequence.

This is joint work with Frank Calegari.

The Empirical Mode Decomposition (EMD) was an empirical one-dimensional data decomposition method invented by Dr. Norden Huang about ten years ago and has been used with great success in many fields of science and engineering. In this talk, I will introduce, from the perspective of a physical scientist, the thinking behind and the algorithm of EMD; and its most recent developments, especially the Ensemble EMD (EEMD), a noise-assisted data analysis method, and the multi-dimensional EMD based on EEMD. I will also outline some open questions that we currently do not have answers, or even clues to the answers, such as how to optimize EMD algorithm, what is the mathematical nature of EMD. To a significant degree, this is a talk intended for obtaining helps from mathematicians.

Consider the following random process. At each round, one is presented with two random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is known in the literature as an Achlioptas process, and has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component.

In our work, we investigate the classical small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is a deterministic online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n,M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than a factor of 2, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles C_t, cliques K_t, and complete bipartite graphs K_{t,t}.

Joint work with Michael Krivelevich and Benny Sudakov.

Compressed sensing (CS) is a signal processing technique that allows for accurate, polynomial time recovery of sparse data-vectors based on a small number of linear measurements. In its most basic form, robust CS can be viewed as a specialized error-control coding scheme in which the data alphabet does not necessarily have the structure of a finite field and where the notion of a “parity-check” is replaced by a more general functionality. It is therefore possible to combine and extend classical CS and coding-theoretic paradigms in terms of introducing new minimum distance, reconstructions complexity, and quantization precision constraints. In this setting, we derive fundamental lower and upper bounds on the achievable compression rate for such constrained compressed sensing (CCS) schemes, and also demonstrate that sparse reconstruction in the presence of noise can be performed via low-complexity correlation-maximization algorithms that operate based on belief propagation iterations. Our problem analysis is motivated by a myriad of applications ranging from compressed sensing microarray designs, reliability-reordering decoding of linear block-codes, identification in multi-user communication systems, and fault tolerant computing. This is a joint work with Wei Dai and Vin Pham Hoa from the ECE Department at UIUC.

In combinatorial geometry one frequently wants to select a point or a set of points that meets many simplices of a given family. The two examples are choosing a point in many simplices spanned by points of some P in R^d, and choosing a small set of points which meets the convex hull of every large subset of P (the weak epsilon-net problem). I will present a new class of constructions that yield the first nontrivial lower bound on the weak epsilon-net problem, and improve the best bounds for several other selection problems. Joint work with Jiří Matoušek and Gabriel Nivasch.

To simulate supernova explosions, one must solve simultaneously the non-linear, coupled partial differential equations of radiation hydrodynamics. What's more, due to a variety of instabilities and asymmetries, this must eventually be accomplished in 3D. The current state-of-the-art is 2D, plus rotation and magnetic fields (assuming axisymmetry). Nevertheless, with the current suite of codes, we have been able to explore the evolution of the high-density, high-temperature, high-speed environment at the core of a massive star at death. It is in this core that the supernova explosion is launched. However, the complexity of the problem has to date obscured the essential physics and mechanisms of the phenomenon, making it indeed one of the "Grand Challenges" of 21st century astrophysics. Requiring forefront numerical algorithms and massive computational resources, the resolution of this puzzle awaits the advent of peta- and exa-scale architectures and the software to efficiently use them. In this talk, I will review the current state of the science and simulations as we plan for the fully 3D, multi-physics capabilities that promise credibly to crack open this obdurate astrophysical nut.