A few years ago – following a suggestion by I. M. Gelfand – I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity.
I will present recent developments and open problems. I will also discuss new estimates for the degree of maps from $S^n$ into $S^n$, leading to unusual characterizations of Sobolev spaces. I will also discuss the continuity of the Jacobian in the sense of distributions. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model occuring in superconductivity.

We establish a connection between observed scale-free topology and hidden hyperbolic geometry of complex networks. The topologies of many complex networks in nature and society (biological networks, the Internet, social networks, etc.) share two common properties:
(1) strong clustering, i.e., high concentration of triangular subgraphs, and (2) heterogenous (scale-free) node degree distributions, which often closely follow power laws. We show that these two common topological properties of complex networks can be explained by the existence of hidden spaces, which are: (1) metric, and (2) hyperbolic. Strong clustering in a network appears as a reflection of the triangle inequality in its hidden space, while the negative curvature of this hidden space affects the heterogeneity of the degree distribution. We also discuss implications for transport phenomena on networks, such as routing. Embedding a real scale-free network into an appropriate hyperbolic space allows for efficient geometric routing without global knowledge of the network topology, which holds a significant promise to find a variety of practical applications, such as improving the performance of Internet routing.

How, when and where does one apply for a job? What should you expect?
What are employers looking for? What is "mathjobs"? Tips on the presentation of your CV, research statement and teaching statement. How to ask for letters of recommendation. How serious are deadlines?
The panel welcomes your questions about the job process and we will try to share experiences.

While solitary waves and other coherent phenomena dominate the long-time behavior of solutions of nonlinear, dispersive wave equations, real phenomena usually features dissipation as well. In the absence of continual forcing, the coherent structures no longer exist and finite energy disturbances decay back to the rest state.

Motivated by some water wave experiments, we pose some delicate questions about the long-time asymptotics of this decay. Theory pertaining to this question has been worked out in a number of contexts, one or two of which will be discussed in a bit more depth.

For a finite metric space X or a smooth manifold M, one can define, in a canonical manner, a Laplace Operator. One then obtains Laplacian eigenfunctions. The study of Laplacian eigenfunctions is quite old and plays a central role in many areas of mathematics. The point of view of Diffusion Geometry is to use these eigenfunctions to provide local (or global) coordinate systems. The point here is that the object under study (X or M) might be easier to work with if one had new coordinate systems for it. This point of view has become popular recently in areas of applied mathematics. It is unclear however, why using eigenfunctions in this fashion should come with any guarantee of providing "good" representations of the original object. In this lecture we explain a theorem (joint work with Mauro Maggioni and Raanan Schul) that provides strong guarantees for local coordinate charts. For example, on a D dimensional, smooth manifold M of finite volume, we can choose D eigenfunctions to provide "good local charts". Perhaps surprisingly, in the case of simply connected, planar domains, this proof is closely related to the Riemann Mapping Theorem.
Before presenting the theorem we will show some applications of diffusion geometry to various applied problems.

In recent work, we have proposed that human speech communication can be modeled as a combinatorial system (phonology), in which the primitive elements are articulatory gestures, ie., constriction
actions of the vocal organs that can be modeled as point attractor dynamical systems in a constriction task space.
A small number of gestures (about 25 in English) recombine to form the speech patterns of all English words. Articulatory analysis of running speech shows that the gestural regimes constituting a word are not activated in a simple sequence. Rather the gestures are organized into temporal patterns in which some are synchronous, some are sequential, some are
partially overlapping in time.
In order to explain how speakers produce these stable patterns of relative timing, we have developed a
speech planning model in which each gesture is associated with an internal planning oscillator whose task is to trigger the activation of its gesture. Pairs of oscillators are coupled to one in stable
modes (in-phase and anti-phase), so as to insure stable relative timing between the production of that pair of gestures. Thus, a word (or longer stretch of speech) is modeled as network of coupled oscillators that can be represented as a coupling graph.
The topology of these graphs correctly accounts for regularities in the relative timing of gestures and in stochastic variability of their relative timing. It also provides an explanation of the internal combinatorial structure of syllables that linguistic research has uncovered over the years.

Shortwave instabilities of ideal fluids are instabilities that appear in the form of a highly oscillating localized wavelet.
Discovered some 30 years ago they were first recognized as an important stage in the transition process from a laminar to fully turbulent flow in a pipe as well as in flows with exponential stretching of trajectories. Mathematically, shortwave instabilities unlike various classes of modal instabilities are linked to the Fredholm spectrum of the linearized Euler equation. Finding the structure of the Fredholm spectrum is in fact equivalent to finding the range of all possible exponential growth rates for geometric optics solutions. In this talk we give an overview of new tools and developments that appeared recently in the area. In particular, we will show via the use of the hair ball theorem that the Fredholm spectrum of 3D Euler equation consists of a solid single annulus centered at the origin. We will also discuss the problem of nonlinear instability of ideal fluids and the role the Fredholm spectrum plays in its understanding.

Blind source separation is a statistical inverse problem aiming to recover source signals and mixing filters (discrete Green's functions) without detailed knowledge of the environment. Cocktail party problem is an example of how humans perform this task by paying attention. Yet little is known of the computation inside human brain for this task.
For sound mixtures, source signals viewed as time series are much more independent of each other than their mixtures. The separation is formulated mathematically as minimization of generalized cross correlations. We derive iterative methods from statistical principles, however, the resulting dynamics are nonlinear and solutions may blow up. We devise a class of discrete ordinary integral differential equations to impose soft constraints, and control the scaling behavior of iterations. The solutions then exist globally and converge in some weak sense to the desired separation conditions.

El Nino Southern Oscillation (ENSO) is a global coupled ocean-atmosphere phenomenon, associated with floods, droughts, and other disturbances in a range of locations around the world. ENSO is the most prominent known source of inter-annual variability in weather and climate around the world (about 3 to 8 years). In spite of its importance and a long history of studies, the understanding of its nature and mechanism is still lacking, and a careful fundamental level examination of the problem is crucial.

We present in this talk a new mechanism of the ENSO, as a self-organizing and self-excitation system, with two highly coupled processes. The first is the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and the second is the spatiotemporal oscillation of the sea surface temperature (SST) field.
The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO. The new mechanism is rigorously derived using a dynamic transition theory developed recently by the authors, which has also been successfully applied to a wide range of problems in nonlinear sciences. This is joint work with Tian Ma.

The 2D surface quasi-geostrophic equation (SQGE) has recently been a focus of much research. This equation appears naturally in geophysics.
It has a simple structure yet its solutions exhibit rich and complex behaviors. The SQGE is perhaps the simplest looking evolutionary equation of fluid dynamics for which the fundamental question of global existence of smooth solutions remains open. I will review some recent results on the regularity and roughening of solutions of the SQGE. In particular, I will talk about nonlocal maximum principle for SQGE, a new idea which found wider applicability, and about search for monotone quantities which can help understand energy transfer to higher modes.

We discuss properties of a shell type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which is an exponential global attractor. Every solution blows up in H5/6 in finite time . After this time, all solutions stay in Hs, s<5/6, and "turbulent" dissipation occurs. Onsager's conjecture is confirmed for the model system. We discuss the augmented viscous system and show that Kolmogorov's law for turbulence holds in the limit of vanishing viscosity. This is joint work with Alexey Cheskidov and Nataša Pavlović.

An m-sequence of degree n is a binary sequence of 0's and 1's of period $2^d-1$ generated by an n-stage linear feedback shift register. These sequences have several pseudo-randomness properties that make them useful in many communications applications (cryptography, radar, CDMA wireless, etc). There is a bijection between m-sequences and primitive polynomials over GF(2). Numerous solved and unsolved problems about m-sequences and their randomness will be discussed.

In Kolmogorov theory of turbulence there is a "cascade" of energy from large to small scales where energy is dissipated by viscosity. In two-dimensional flows (which have been used for example to model geophysical flows if rotation can be neglected), the flow is dominated by the formation of small stable vortices and what cascades to small scales is not energy but enstrophy. Informally, enstrophy can be thought of as the energy associated to vortices. In both cases, there must be a finite rate of dissipation as viscosity vanishes, as observed experimentally and in simulations. Mathematically, this is the case if there are irregular solutions to the Euler equations, modeling inviscid fluid flow, which do not conserve energy in 3D or enstrophy in 2D. We will discuss some results concerning the behavior of enstrophy in 2D Euler solutions, in particular how to reconcile turbulence with the fact Euler solutions conserve enstrophy exactly. This is joint work with Milton Lopes and Helena Nussenzveig Lopes.

The usual process of "carries" when adding numbers turns out to have interesting mathematics hidden in it. It begins with an "amazing" matrix discovered by Holte, which has close connections to the usual way of mixing cards by riffle shuffling. The connections give new results for addition and for shuffling. This is joint work with Jason Fulman.

We know that voting rules experience all sorts of difficulties, e.g., as shown in this lecture, the problems are sufficiently bad that we should seriously worry whether the person elected is who the voters really want. The central issue is to understand why this is so and whether any rule has reliable outcomes. Because voting rules serve as a prototype for a wide class of aggregation rules, ranging from statistics, much of what is done in the social sciences, to even engineering multiscale processes, answers could be of wide value. In this lecture, Donald Saari will outline the mathematical structure of voting rules, and show how these structures completely explain all of those many troubling paradoxes.

In this talk we report recent progress based on the seminal work of M. Waterman on the combinatorics of RNA secondary structures. We first discuss a new idea to obtain the generating function of pseudoknot RNA via the reflection principle. Then we discuss the singularity analysis of the latter and extend the results to the biologically relevant canonical RNA structures which satisfy certain minimum stack-length conditions. Then we observe that these ideas can be generalized for tertiary interactions and introduce the concept of tangles. Finally we introduce a new diagram algebra which is related to the Brauer centralizer algebra.