# Seminars & Events for Department Colloquium

##### Harmonic Analysis and Geometries of Digital Data Bases

Given a matrix (of Data) we describe methodologies to build two multiscale (inference) Geometries/Harmonic Analysis one on the rows , the other on the columns . The geometries are designed to simplify the representation of the data base . We will provide a number of examples including; matrices of operators , psychological questionnaires, vector valued images, scientific articles, etc.

##### Entropy in Measurable Dynamics

In 1958, Kolmogorov defined the entropy of a probability measure preserving transformation. Entropy has since been central to the classification theory of measurable dynamics. In the 70s and 80s researchers extended entropy theory to measure preserving actions of amenable groups (Kieffer, Ornstein-Weiss). My recent work generalizes the entropy concept to actions of sofic groups; a class of groups that contains for example, all subgroups of GL(n,C). Applications include the classification of Bernoulli shifts over a free group, answering a question of Ornstein and Weiss.

##### Symplectic Embeddings and Continued Fractions

It has been known since the time of Gromov that questions about symplectic embeddings lie at the heart of symplectic geometry. This talk will mostly be about some recent work with Schlenk in which we work out precisely when a four dimensional ellipsoid embeds symplectically in a ball. This problem turns out to have unexpected relations with the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.

##### Some variants on the flows of suspensions: Diffusion, dispersion, and biofilms

In this talk I will present several fluid mechanics problems that concern the flow of particles and suspensions. This topic has many variants, which I will introduce to provide breadth and perspective for the listener (most of you) who has not studied the topic. After the introduction I will highlight (i) shear-enhanced diffusion, as studied in a microfluidic device, (ii) axial dispersion due to shear-enhanced diffusion, and (iii) unusual structures formed when bacteria flow, and biofilms grow, in curved channels. Some answers will be given and pen questions will be indicated.

##### Geometry and Analysis of point sets in high dimensions

The analysis of high dimensional data sets is useful in a large variety of applications, from machine learning to dynamical systems: data sets are often modeled as low-dimensional, noisy data sets embedded in high-dimensional spaces; dynamical systems often have very high-dimensional state spaces but sometimes interesting dynamics occurs on low-dimensional sets.

##### A model-theoretic approach to certain diophantine problems

I will describe some results and problems about the distribution of rational points on certain non-algebraic sets in real space. They find their natural setting in the model-theoretic notion of an 'o-minimal structure over the real numbers.' I will describe a result joint with Alex Wilkie in this setting. A surprising strategy, proposed by Umberto Zannier, uses this result to approach diophantine problems in the Manin-Mumford-Andre-Oort circle of conjectures. I will describe some implementations of this strategy highlighting the use of o-minimality at different junctures, and discuss its prospects.

##### Testable New Theory about Early-Universe Density Fluctuations and Origins of Solar Systems: Applied-Probability and Quantum-Physics Aspects

The talk will summarize, with a focus on applied-probability aspects, the main findings, testable predictions and research opportunities stemming from a new probabilistic model of how complex patterns of energy-density fluctuations may have arisen during the inflation phase of the Big Bang. Based on first (quantum-physical) principles and requiring a minimum number of (observationally-accessible) parameters, the "embryonic inflation model" yields a coherent set of testable (hence falsifiable) hypotheses about the formation, evolution, composition, internal structure and cosmic environment of galaxies, stars and planets, and is consistent with key findings from observations of the Cosmic Microwave Background (CMB).

##### Abelian sandpile model and self-similar groups

The sandpile model was introduced in 1987 by physisists Bak, Tang and Wiesenfeld as a tool to study what they called *the self-organized criticality*—spontaneous appearance of power laws or fractal interfaces, observed in some natural phenomena. The mathematical study of the model was initiated a couple of years later by Deepak Dhar. It begins with a simple cellular automaton (also known in combinatorics under the name of *chip-firing game* on a finite graph, and leads to interesting long time and large volume limit dynamics when considered on increasing sequences of graphs.

##### Solving High-Dimensional Stochastic Optimization Problems using Approximate Dynamic Programming

There are many stochastic resource allocation problems arising in transportation, energy and health that involve high-dimensional state and action variables in the presence of dierent forms of uncertainty. These might involve discrete or continuous resources, and generally involve vectors of random variables that preclude exact computation of expectations. I will also describe our research into the important \exploration vs. exploitation" problem that arises in approximate dynamic programming, where we have the ability to choose the next state we will visit.

##### Concentration Compactness for critical Wave Maps

This talk will discuss a recent result on global regularity and asymptotic behavior of large critical wave maps with hyperbolic target, obtained jointly with W. Schlag. The proof relies on an adaptation of the recently developed method of Kenig-Merle to the case of systems of wave equations, as well as the harmonic analytic methods devised by Klainerman-Machedon, Tataru and Tao.

##### Generic singularities of mean curvature flow

Mean curvature flow (or MCF) is a nonlinear heat equation for hyper-surfaces, where the surface evolves by moving in the direction where volume locally decreases the fastest. The simplest non-static examples are round concentric spheres, where the radius shrinks until it becomes zero at "extinction" (a singularity of the flow). Singularities are unavoidable as the flow contracts any closed surface and thus one of the most important problems in MCF is understanding the singularities. Matt Grayson, Mike Gage and Richard Hamilton proved that this is the only singularity for simple closed curves in the plane. However, many examples were discovered in higher dimensions.

I will describe recent work with Toby Colding, MIT, where we:

##### Imaging Techniques and the Rejuvenation of Artwork

Advances in digital imaging within the visible spectrum enable the accurate color rendering of artwork. It is possible to generate a colorimetric image with high spatial resolution and high image quality (appropriate sharpness and low noise). When the number of sensor channels exceeds three, it is also possible to generate spectral images. Spectral images can be used to calculate colorimetric images for any illuminant and observer pair, to evaluate color inconstancy, as an aid in retouching (i.e., restorative inpainting), for pigment mapping, and to improve printed reproductions. These digital images, of course, record the color and spectra of the artwork in its current condition. Depending on how the artwork has aged, its color may bear little resemblance to its appearance when first executed.

##### Random Matrices: Universality of Local Eigenvalues Statistics

One of the main goals of the theory of random matrices is to establish the limiting distributions of the eigenvalues. In the 1950s, Wigner proved his famous semi-cirle law (subsequently extended by Anord, Pastur and others), which established the global distribution of the eigenvalues of random Hermitian matrices. In the last fifty years or so, the focus of the theory has been on the local distributions, such as the distribution of the gaps between consecutive eigenvalues, the k-point correlations, the local fluctuation of a particular eigenvalue, or the distribution of the least singular value. Many of these problems have connections to other fields of mathematics, such as combinatorics, number theory, statistics and numerical linear algebra.

##### Ideas around Symplectic Field Theory

Symplectic Field Theory (SFT) is the study of (pseudo-)holomorphic curves in symplectic cobordisms and contains Gromov-Witten theory and symplectic Floer theory as special cases. The algebraic invariants of SFT are obtained by a simultaneous study of infinitely many interdependent first order elliptic systems which exhibit compactness and transversality issues. A treatment of SFT with classical (nonlinear) Fredholm theory, though possible, would be extremely cumbersome. This lead to the development of a new generalized Fredholm theory in a new class of general spaces called polyfolds. In the talk a certain number of ideas are described which might be also useful in different contexts.

##### A Phase transition for a model of Random band matrices

Miniconference on Dynamical Systems at Princeton

##### Zero temperature limits of Gibbs states

Miniconference on Dynamical Systems at Princeton

##### Escape rates and variational principles for dynamical systems with holes

Miniconference on Dynamical Systems at Princeton

##### Progress on Affine Sieves

Miniconference on Dynamical Systems at Princeton

##### The Mobius function, randomness and dynamics

Miniconference on Dynamical Systems at Princeton

##### The Full Renormalization Horseshoe for Unicritical Maps, revisited

Miniconference on Dynamical Systems at Princeton