The theory of homogenization for equations with random coefficients is now quite well-developed. What is less studied is the theory for the correctors to homogenization, which asymptotically characterize the randomness in the solution of the equation and as such are important to quantify in many areas of applied sciences. I will present recent results in the theory of correctors for elliptic and parabolic problems and briefly mention how such correctors may be used to improve reconstructions in inverse problems. Homogenized (deterministic effective medium) solutions are not the only possible limits for solutions of equations with highly oscillatory random coefficients as the correlation length in the medium converges to zero. When fluctuations are sufficiently large, the limit may take the form of a stochastic equation and stochastic partial differential equations (SPDE) are routinely used to model small scale random forcing. In the very specific setting of a parabolic equation with large, Gaussian, random potential, I will show the following result: in low spatial dimensions, the solution to the parabolic equation indeed converges to the solution of a SPDE, which however needs to be written in a (somewhat unconventional) Stratonovich form; in high spatial dimension, the solution to the parabolic equation converges to a homogenized (hence deterministic) equation and randomness appears as a central limit-type corrector. One of the possible corollaries for this result is that SPDE models may indeed be appropriate in low spatial dimensions but not necessarily in higher spatial dimensions.

After a short introduction presenting our group and our main research topics, I will address the problem of audio-visual speech recognition, i.e. a typical example of multimodal signal analysis, when we want to extract and exploit information coming from two different but complementary signals: an audio and a video channel. We will discuss two important aspects of this analysis. We will first present a new feature extraction algorithm based in information theoretical principles, and show its performances, compared to other classical approaches, in our multimodal context. Then we will discuss multimodal information fusion, i.e. how to combine information from those two channels for optimal classification.

TBD

Reliable information transmission between two sites of a network naturally leads to a percolation problem. When the information to be transmitted is quantum an exciting possibility arises: transform the network performing well chosen measurements to enhance the transmission probability. This idea, introduced recently by Acin, Cirac and Lewenstein is now systematically and successfully applied to a variety of two-dimensional networks, but open questions show that a complete theory is missing. The talk will involve some quanta, some network geometry, some percolation and, hopefully, some fun. No knowledge of quantum theory or percolation theory is assumed. Graduate students are encouraged to attend.

A 'correspondence' between two manifolds is a submanifold in the product. This generalizes the notion of a map (whose graph is a correspondence) ... and is of little use in general since the composition of correspondences, though naturally defined, can be highly singular.

Lagrangian correspondences between symplectic manifolds however are highly useful (and will be defined carefully). They were introduced by Weinstein in an attempt to build a symplectic category that has morphisms between any pair of symplectic manifolds (not just symplectomorphic pairs).

In joint work with Chris Woodward we define such a cateory, in which all Lagrangian correspondences are composable morphisms. We extend it to a 2-category by constructing a Floer homology for generalized Lagrangian correspondences. One of the applications is a general prescription for constructing topological invariants. We consider e.g. 3-manifolds or links as morphisms (cobordisms or tangles) in a topological category. In order to obtain a topological invariant from our generalized Floer homology, it suffices to

(i) decompose morphisms into simple morphisms (e.g. by cutting between critical levels of a Morse function)

(ii) associate to the objects and simple morphisms smooth symplectic manifolds and Lagrangian correspondences between them (e.g. using moduli spaces of bundles or representations)

(iii) check that the moves between different decompositions are associated to 'good' geometric composition of Lagrangian correspondences

After a brief introduction of modular forms on the upper half plane and vector-space valued differential forms, we will explore a very classical result (independently due to Eichler and Shimura) which relates certain cohomology groups to cusp forms on the upper half plane of corresponding weight. We will then put our algebraic hat on, and recast this result in modern light, using the theory of Automorphic forms developed by (among others) Harish-Chandra and Langlands. I hope to make the talk accessible to most graduate students. Though the topics we will talk about are related to my research, it is not a research talk, more of an exposition. The first part of the talk should be a breeze for anyone with understanding of some fundamental concepts in Complex analysis and Algebraic Topology (holomorphic functions, differential forms, deRham cohomology.) A course in Representation Theory would be helpful in relating to the concepts in the second part of the talk.

In geometry, there are certain structures that are "rigid" (like Riemannian manifolds) and others that are "flexible" (like topological manifolds). Symplectic geometry lies in between these two extremes and incorporates some attractive features of both. One consequence is that symplectic techniques have recently been used, to great effect, to give combinatorial approaches to questions in topology that previously required difficult gauge-theoretic and analytic techniques. I will introduce symplectic structures and describe some recent developments linking them to the study of three-dimensional manifolds and knots. No real background will be assumed.

Viewed forwards in time, a population reproducing according to some random mechanism can be thought of as a branching process. What if it is viewed backwards? We can take a sample of individuals from the current generation and trace their genealogy backwards, and for instance find their most recent common ancestor; this is known as a coalescent process. If we know a population's random mating process, but have no actual data as to what the phylogenetic tree looks like, how do we derive the distribution of the time until its most recent common ancestor? I will discuss a variant on the classical Wright-Fisher reproductive model and deduce some parameter thresholds for emergence of different qualitative features of the tree. An isomorphic problem may also be useful in computer science for bounding the running time of certain random sampling algorithms.

Finding globally stable states can provide useful information about the behavior of solutions to PDEs: for any initial condition, the solution will eventually approach such a state. However, in some cases, the solution can exhibit long transients in its approach to the state. If the transient is long enough, it may be this behavior, rather than the limiting behavior, that is observed in numerical simulations or experiments. This is referred to as "metastability" and has been found, for example, in the 2D Navier-Stokes equations with small viscosity. A similar phenomenon has been seen in Burgers equation, which can be explained using global invariant manifolds. More precisely, it is shown that in terms of similarity, or scaling, variables there exists a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold. Metastability corresponds to a fast transient in which solutions approach this 'metastable' manifold, followed by a slow decay along this manifold, and, finally, convergence to the globally stable state.

The efficient parallelization of numerical methods for ordinary or partial differential equations in the temporal direction is an intriguing possibility that has of yet not been fully realized despite decades of investigation. For partial differential equations, virtually all large scale computations now employ parallelization across space, and there are freely available computational tools and libraries to aid in the development of spatially parallelized codes. Conversely, parallelization in the temporal direction is rarely even considered. I will discuss a relatively recent parallel strategy called the parareal algorithm that has generated a renewed wave of interest in time parallelization. I will show how the iterative structure of the parareal algorithm can be interpreted as a particular form of deferred corrections and then present a modified parareal strategy based on spectral deferred corrections that can significantly reduce the computational cost of the method. Finally I will make some observations as to why parallel in time methods may be attractive in the future.

Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant. Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site). This process is highly non-Markovian, due to the interaction between the walk and the environment. We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains and show how this can be transferred to a result for the above process (called random walk in random environment). This is joint work with Timo Sepp\"al\"ainen.

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $\sigma|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. If this "characteristic foliation" has compact leaves, then the space of leaves Y/F is a holomorphic symplectic manifold of dimension 2n-2. This construction also works when Y is a coisotropic submanifold of higher codimension, and is known as "coisotropic reduction". In this talk we will consider when the characteristic foliation has compact leaves, and look at some applications of coisotropic reduction.

Essential dimension is an invariant introduced by Buhler and Reichstein to measure how many parameters are needed to define an algebraic object such as a field extension or an algebraic curve over a field. I will describe joint work with Vistoli and Reichstein which studies essential dimension in the case where the algebraic objects are represented by a stack. I will also give examples of applications in the theory of quadratic forms.

TBD

For Markov random fields temporal mixing, the time it takes for the Glauber dynamics to approach it's stationary distribution, is closely related to phase transitions in the spatial mixing properties of the measure such as uniqueness and the reconstruction problem. Such questions connect ideas from probability, statistical physics and theoretical computer science. I will survey some recent progress in understanding the mixing time of the Glauber dynamics as well as related results on spatial mixing. Partially based on joint work with Elchanan Mossel

The combinatorial complexity of the union of a set of geometric objects is the total number of faces of all dimensions that lie on the boundary of the union. We review some recent results on the complexity of the union of geometric objects in 2d and 3D satisfying various natural conditions and on computing the union. We then discuss the critical roles they play in computing an epsilon net and a hitting set of a set system.

Matroids are combinatorial devices designed to encoded the combinatorial structure of hyperplane arrangements. Combinatorialists have developed many invariants of matroids. I will explain that there is reason to believe that most of these invariants are related to computations in the K-theory of the Grassmannian. In particular, I will explain work of mine limiting the complexity of Hacking, Keel and Tevelev's "very stable pairs", which compactify the moduli of hyperplane arrangements. This talk should be understandable both to those who don't know matroids, and to those who don't know K-theory.

A real matrix is totally nonnegative if every minor in it is nonnegative. The classical Edrei-Thoma theorem classifies totally nonnegative infinite Toeplitz matrices, and is related to problems in representation theory, combinatorics and probability. I will discuss progress towards two variations on this theorem to block-Toeplitz matrices, and to finite Toeplitz matrices. Both of these variations connect the classical theory to loop groups.

In this talk we will discuss a proof of the path-connectedness of the space of positive scalar curvature metrics on the three-sphere. The proof uses the Ricci flow with surgery and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. If time permits we will also discuss an application to general relativity.

In order to understand the nonlinear stability of many types of time-periodic traveling waves on unbounded domains, one must overcome two main difficulties: the presence of zero eigenvalues that are embedded in the continuous spectrum and the time-periodicity of the associated linear operator. I will outline these issues and show how they can be overcome in the context of time-periodic Lax shocks in systems of viscous conservation laws. The method involves the development of a contour integral representation of the linear evolution, similar to that of a strongly continuous semigroup, and detailed pointwise estimates on the resultant Greens function, which are sufficient for proving nonlinear stability under the necessary assumption of spectral stability.

Wavelets are widely used in signal processing (e.g. analysis of sounds and music) and imaging, for tasks such as denoising and compression (ever wondered how jpeg works?). In harmonic analysis they have been used to understand and solve problems involving integral operators motivated by PDEs. In numerical PDEs they lead to fast algorithms for solving certain types of integral equations and PDEs. I will give a gentle introduction to wavelets and some of their motivating applications, accompanied by live demos. If time allows, I will discuss shortcomings and how they have been addressed in more recent developments and generalizations.

The superprocess or Dawson-Watanabe process is one of the most intensively studied stochastic processes of the last quarter century. It arises as a limit of population processes, and includes information about the physical location of individuals. Usually the superprocess is measure valued, but In one dimension it has a density that satisfies a parabolic stochastic PDE. For a long time uniqueness for this equation was unknown. In joint work with Barlow, Mytnik, and Perkins, we show that nonuniquess holds for the superprocess equation and several related equations.

We shall review the Penrose process for extracting mass and angular momentum from the Kerr black hole solution of the Einstein equations. We will show that Christodoulou's bound on the maximal energy gain by the classical Penrose process can be realized by choosing suitable wave packet initial data for the scalar wave equation in Kerr geometry, thereby putting super-radiance on a rigorous mathematical footing. This is joint work with Felix Finster, Joel Smoller and Shing-Tung Yau.

In this talk, we plan to explain some existence results for complete Ricci-flat \kahler metrics on crepant resolutions of singularities. The method allows us to provide a wider class of examples of complete Ricci-flat Kahler metrics with richer topoplogy at infinity.

I will present results on constructing an estimator of a function on a compact manifold for the purpose of recovering its "topology". What this means will be explained in detail. The talk will conclude with an application to brain imaging.

Laplace gave the simplest early statement of reductionism. His Demon, if supplied with the positions and momenta of all the particles in the universe, could, using Newton's laws, calculate the entire future and past of the universe. Add fields, quantum mechanics, and General Relativity and you have, roughly, modern physics. There are four features to Laplace's reductionism: (I) Everything that happens is deterministic, called into question a century later by quantum mechanics and the familiar Copenhagen interpretation and Born rule. (ii) All that is ontologically real are "nothing but" particles in motion. (iii) All that happens in the universe is describable by universal laws. (iv) There exists at least one language able to describe all of reality. Quantum mechanics is evidence against (i). I will argue that biological evolution, the coming into existence in the universe of hearts and humming birds co-evolving with the flowers that feed them and that they pollenate, cannot be deduced or simulated from the basic laws of physics. In Weinberg's phrase, they are not entailed by the laws of physics. I will then claim that at levels above the atom, the universe will never make all possible proteins length 200 amino acids, all possible organisms, or all possible social systems. The universe is indefinitely open upwards in complexity. More, proteins, organisms, and social systems are ontologically real, not just particles in motion. Most radically, I will contest (iii). I will try to show that we cannot pre-state Darwinian pre-adaptations, where a pre-adaptation is a feature of an organism of no use in the current selective environment, but of use in a different environment, hence selected for a novel function. Swim bladders are an example. Let me define the "adjacent possible" of the biosphere. Once there were the lung fish that gave rise to swim bladders, swim bladders were in the adjacent possible of the biosphere. Before there were multi-celled organisms, swim bladders were not in the adjacent possible of the biosphere. What I am claiming is that we cannot pre-state the adjacent possible of the biosphere. How could we pre-state the selective conditions? How could we pre-specify the features of one or several organisms that might become pre-adaptations? How could we know that we had completed the list? The implications are profound, if true. First, we can make no probability statement about pre-adaptations, for we do not know the sample space, so can formulate no probability measure. Most critically, if a natural law is a compact description before hand and afterward of the regularities of a process, then there can be no natural law sufficient to describe the emergence of swim bladders. Thus, the unfolding of the universe is partially lawless! This contradicts our settled convictions since Descartes, Galileo, Newton, Einstein and Schrödinger. It says that (iii) is false. In place of law is a ceaseless creativity, a self consistent self construction of the biosphere, the economy, our cultures, partially beyond law. Were reductionism sufficient, the existence of swim bladders in the universe would be entailed by physical law, hence "explained". But it appears that physics, as stated, is not sufficient in its reductionist version. Then we must explain the existence in the universe of swim bladders and humming birds pollenating flowers that feed them, on some different ground. We need a post-reductionist science. Autocatalytic mutualisms of organisms, the biosphere, and much of the economy, may be part of the explanation we seek. In turn this raises profound questions about how causal systems can coordinate their behaviors, let alone the role of energy, work, power, power efficiency, in the self-consistent construction of a biosphere. There is a lot to think about.