Mean field game theory describes continuum limits of symmetric large-population games. These games can often be seen as competitive extensions of classical models of interacting particle systems, where the particles are now "controlled state process" (with application-specific interpretation, such as position, income, wealth, etc.). The coupled optimization problems faced by each process are typically resolved by Nash equilibrium, and there is a large and growing literature on solvability problems (both theoretical and computational). On the other hand, relatively little is known on how to rigorously pass from a finite population to a continuum, especially for dynamic stochastic games. The basic question is: Given for each N a Nash equilibrium for the N-player game, do the equilibria (more precisely, the empirical distributions of state processes) converge as N tends to infinity? This talk is an overview of the known probabilistic limit theorems in this context (law of large numbers, fluctuations, and large deviations), the ideas behind them, and some open problems.

The terrestrial water balance is forced by highly intermittent and unpredictable pulses of rainfall. This in turn impacts several related hydrological and ecological processes, such as plant photosynthesis, soil biogeochemistry and has feedbacks on the local climate. We treat the rainfall forcing at the daily time scale as a of marked (Poisson) point processes, which is then used the main driver of the stochastic soil water balance equation. We analyze the main nonlinearities in the soil water losses and discuss the probabilistic dynamics of soil water content as a function of soil-plant and vegetation characteristics. Crossing and mean-first-passage-time properties of the stochastic soil moisture process define the statistics of plant water stress, which in turn control plant dynamics, as shown in application to tree-grass coexistence in the Kalahari transect.
In the second part of this overview, we briefly illustrate: i) the propagation of soil moisture fluctuations through the nonlinear soil carbon and nitrogen cycles, ii) the possible emergence of persistence and preferential states in rainfall occurrence due to soil moisture feedback, and iii) the impact of inter-annual rainfall variability in connection to recent theory of ‘superstatistics’.

REFERENCES:
Rodriguez-Iturbe I. and A. Porporato, Ecohydrology of water controlled ecosystems: plants and soil moisture dynamics. Cambridge University Press, Cambridge, UK. 2004.
Laio F., Porporato A., Ridolfi L., and Rodriguez-Iturbe I. (2001) Plants in water controlled ecosystems: Active role in hydrological processes and response to water stress. II. Probabilistic soil moisture dynamics. Advances in Water Research, 24, 707-723.
Porporato A., Laio F., Ridolfi L., and Rodriguez-Iturbe I. (2001) Plants in water controlled ecosystems: Active role in hydrological processes and response to water stress. III. Vegetation water stress. Advances in Water Research, 24, 725-744.
Porporato A., DÂ’Odorico P., Phase transitions driven by state-dependent Poisson noise, Phys. Rev. Lett. 92(11), 110601, 2004.
DÂ’Odorico P., Porporato A., Preferential states in soil moisture and climate dynamics, Proc. Nat. Acad. Sci. USA, 101(24), 8848-8851, 2004. Manzoni S., Porporato A., DÂ’Odorico P. and I. Rodriguez-Iturbe. Soil nutrient cycles as a nonlinear dynamical system. Nonlin. Proc. in Geophys. 11, 589-598, 2004.
Porporato A., G. Vico, and P. Fay, Interannual hydroclimatic variability and Ecosystem Superstatistics. Geophys. Res. Lett., 33, L5402, 2006.
Daly, E., and A. Porporato, Inter-time jump statistics of state-dependent Poisson processes, Phys. Rev. E, 75, 011119, 2007.

Some of the old problems in algebraic geometry, as well as relatively new problems in the theory of quantization, were solved using topological sigma models. The sigma models describe maps of a manifold M to a target space X. It is very well-known that no sensible theory exists when the dimension of M is greater than two. In my talk I will try to argue in favor of the existence of an interesting theory of maps in the case where M is a four-dimensional Riemannian manifold and X is a classifying space of some compact Lie group (or its finite-dimensional model). To get there we will need to introduce & develop certain aspects of Donaldson theory and higher-dimensional analogues of Whitman hierarchies. No knowledge of Donaldson theory or Whitman hierarchies is necessary.

In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy.

Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in plant leafs and fracture networks in rocks. We present and analyze a PDE (Continuum) framework to model transportation networks in nature, consisting of a reaction-diffusion gradient-flow system for the network conductivity constrained by an elliptic equation for the transported commodity (fluid).

The Trace Formula can be understood roughly as an equation relating spectral data to geometric information. It is obtained via expansion of the trace of certain operators that are associated to the Representation Theory of an affine algebraic group, justifying its name. Therefore, the spectral side of the expansion by nature contains data of arithmetic interests. However, the spectral side is generally less accessible. Meanwhile, the geometric side consists of terms that can be written in a more explicit fashion. The computation of the geometric side, which is now referred to as the Orbital Integrals, thus come on the scene. In this talk, we plan to briefly introduce the general derivation of the (vaguely described) Trace Formula, and demonstrate a few concrete examples of it.

PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.

I will talk about geometric transitions on the moduli space of four dimensional, N=1 supersymmetric, string compactifications. In particular, I shall discuss some recent results on geometric transitions of Calabi-Yau's in the presence of branes and flux.

In this talk I will review the basic ideas of the regularity structure theory developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint work with Hairer on the sine-Gordon equation and central limit theorems for stochastic PDEs.

The dynamics of a set of chemical reactions are usually modeled by mass action kinetics as a set of algebraic ordinary differential equations. This model sees the state space of the system as a continuum, whereas chemical reactions represent interactions of a discrete set of molecules. We study large fluctuations of the stochastic mass action kinetics model through Freidlin-Wentzell theory. The application of such a theory to this framework requires justification, in particular because of the non-uniformily Lipschitz character of the model. We therefore find, using tools of Lyapunov stability theory, a set of sufficient conditions for the applicability of large deviations theory to this framework, and prove that such conditions are satisfied by a large class of chemical reaction networks identified exclusively on the base of their topological structure.

A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $S^{2}$. It associates to a Zoll metric on $S^{2}$ a family of holomorphic disks in $CP_{2}$ with boundary in a totally real submanifold $P\subset CP_{2}$. For a fixed $P\subset CP_{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.

In certain situations global topology may force singularities. For example, the topology of the Klein bottle forces self-intersections when mapped into 3-space. Any map of the projective plane must have at least cusp singularities when mapped into the plane. The topology of a manifold may force any differential form on it to degenerate at certian points. In a family of vector bundles over a complex curve some must degenerate to a non-stable bundle (in the GIT sense), depending on the topology of the family. In a family of vector bundle maps---arranged according to a directed graph (quiver)---some may be forced to degenerate. In families of linear spaces some have special incidence with some other fixed ones (Schubert calculus). These degenerations are governed by a unified notion in equivariant cohomology, the Thom polynomial of "singularities". In the lecture I will review Thom polynomials, computational strategies (interpolation, localization, Grobner basis), show examples and applications.

The cosmic censorship conjecture roughly states that singularities in the evolution of spacetime are always hidden from the outside world by event horizons. As a test for this conjecture Penrose proposed the inequality M >= (A/16pi)^1/2, relating the total ADM mass M of a spacetime to the area A of an event horizon. For time symmetric initial data sets of Einstein's equations this inequality has been confirmed, independently by Huisken and Ilmanen (for one black hole) and by Bray (for multiple black holes). The purpose of this talk is to show how the time symmetric proofs can be generalized to apply to general initial data, assuming existence for a canonical degenerate elliptic system of equations. This is joint work with Hubert Bray.

Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.

A well-known theorem of Waldspurger relates central values of automorphic L-functions for GL(2) to automorphic period integrals over non-split tori. His result was reproved by Jacquet via the comparison of relative trace formulae. Guo-Jacquet’s conjecture aims to generalise Waldspurger’s result as well as Jacquet’s approach to higher dimensions. In this talk, we shall first recall the background of Guo-Jacquet trace formulae. Then we shall focus on an infinitesimal variant of these formulae and try to explain several results on the local comparison of most terms. Our infinitesimal study is expected to be relevant to the study of geometric sides of the original Guo-Jacquet trace formulae.

Abstract: Gopakumar and Vafa have proposed a formula for the Gromov-Witten invariants of Calabi-Yau 3 folds in terms of numbers of BPS states. Part of this formula is explained by higher genus multiple covers and collapsed contributions in Gromov-Witten theory. The mathematical analysis leads to a direct generalization of Gopakumar and Vafa's formula to an integral structure for arbitrary 3 folds. I will discuss a mathematical approach to these questions via integrals over the moduli space of stable curves.

A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. Such tensors are interesting because their decomposition can be found efficiently. We study their spectral properties and give a formula for all of their eigenvectors. We also give equations defining all real symmetric orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. In an attempt to extend the definition to a larger set of tensors, we define tight-frame decomposable tensors and show that for certain equiangular tight frames they too can be decomposed via the tensor power method.

In this talk we present two intimately related approaches in speeding-up molecular simulations via Monte Carlo simulations. First, we discuss coarse-graining algorithms for systems with complex, and often competing particle interactions, both in the equilibrium and non-equilibrium settings, which rely on multilevel sampling and communication. Second, we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting of the underlying high-dimensional generator, as means of decomposing efficiently and systematically the computational load and communication between multiple processors. The common theme in both methods is the desire to identify and decompose the particle system in components that communicate minimally and thus local information can be either described by suitable coarse-variables (coarse-graining), or computed locally on a individual processors within a parallel architecture.

Hirschsprung's Disease is a relatively common human congenital defect where the nervous system supporting our gut (the enteric nervous system) fails to develop properly. During embryonic development, the enteric nervous system forms as a result of neural crest cell invasion. Neural crest cells migrate from the hindbrain to the anal end of the gastrointestinal tract. This is one of the longest known cell migration paths, both spatially and temporally, occurring during vertebrate embryogenesis. Neural crest cell invasion is complicated by the simultaneous expansion of underlying tissues and the influence of several growth factors. This presentation outlines a combined experimental and mathematical approach used to investigate and deduce the mechanisms responsible for successful neural crest cell colonization. This approach enables previously hypothesized mechanisms for neural crest cell colonization of the gut tissues to be refuted and refined. The current experimental and mathematical results are focused on population-scale approaches. Further experimental details of cell-scale properties thought to play an important role will be presented. Preliminary discrete modelling results aiming to realize the cell-scale phenomena will also be discussed and outlined as future work.

In this talk, we investigate modular forms and their many connections to number theory. Modular forms are analytic functions on the upper half complex plane that satisfy certain functional equations. They arise in many contexts in number theory: from partitions of integers, to arithmetical divisor functions, to cutting edge research on special values of the Riemann zeta function. We discuss both classical and modern examples, with a view toward illustrating the profound connections between analysis, topology, and number theory.

We describe an automatic chaos verification scheme based on set oriented numerical methods, which is especially well suited to the study of area and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, unlike existing methods, does not require the computation of chain recurrent sets. We give several example computations in dimension two and three.

This work was done in close collaboration with M. Khovanov. The Witten-Reshetikhin-Turaev invariant Z(M,L;r) of a link L in a 3-manifold M is a seemingly random function of an integer r. However, for a small class of 3-manifolds constructed by identical gluing of two handlebodies (e.g., for S3 and for S2 x S1) and for sufficiently large values of r the ratio Z(M,L;r)/Z(M;r) is equal to a rational function J(M,L;q) of q evaluated at the first 2r-th root of unity. If M = S3, then J is the Jones polynomial. Khovanov categorified J(S3,L), that is, to a link L in S3 he assigned a homology H(L) with an extra Z-grading such that its graded Euler characteristic equals J(S3,L). We extend Khovanov's construction to links in S2 x S1 thus categorifying J(S2xS1,L). In his work on categorification of the Jones polynomial, Khovanov introduced special algebras H_n and assigned a H_m x H_n module to every (2m,2n)-tangle. We show that if a link L in S2 x S1 is presented as a closure of a (2n,2n)-tangle, then the Hochschild homology of its H_n bimodule is determined by the link itself and serves as a categorificaiton of J(S2xS1,L). Moreover, we show that this Hochschild homology can be approximated by Khovanov homology of the circular closure of the tangle within S3 by a high twist torus braid, thus providing a practical method of its computation.

We study the conformal field theory describing the extreme low-energy excitations of parallel D3-branes located at a singular point of the transverse space. For quotient singularities such a description is known. Using the fact that partial resolutions of quotient singularities contain other kinds of singular points, as well as a mapping of the moduli space of a singularity onto the parameter space of the corresponding field theory, we compute the worldvolume field theory for branes at more general singularities. We compare our results to the predictions of an extended version of Maldacena's AdS/CFT correspondence as presented last week by D. Morrison.