## Programs & Events

##### Dimension and Dynamics

Feb 1 - May 6, 2016

Since its introduction by Felix Hausdorff in 1919, the concept of the Hausdorff dimension of sets and measures has been a versatile and powerful tool in classical analysis, geometry and geometric measure theory, mathematical physics and their numerous applications. However, there has been a particularly important symbiosis between dynamical systems and dimension theory. This connection arises both from application of dimension theory to the classification and geometric analysis of dynamical systems (and their invariant sets and measures), and the fact that many classical objects of study in mathematics arise from (sometimes implicit) dynamical systems, which often play a role in the dimension theory of said objects.

Recently, there has been substantial progress on a number of central problems in dimension theory, and while many old problems remain, many new ones have also presented themselves. These include a deeper understanding of the relationship between dimension, entropy and... (more)

##### Organizing Committee

- Dmitry Dolgopyat
- Michael Hochman
- Izabella Laba
- Stefano Luzzatto
- Yakov Pesin
- Mark Pollicott
- Jörg Schmeling
- Boris Solomyak
- Warwick Tucker

##### Computational Aspects of the Langlands Program

Sep 9 - Dec 4, 2015

In the late 1960s, Robert Langlands discovered a unifying principle in number theory providing a vast generalization of class field theory to include nonabelian extensions of number fields. This principle gives rise to a web of conjectures called the * Langlands program * which continues to guide research in number theory to the present day. For example, an important first instance of the Langlands program is the modularity theorem for elliptic curves over the rational numbers, an essential ingredient in the proof of Fermat's last theorem.

Despite its many successes, the Langlands program remains vague in many of its predictions, due in part to an absence of data to guide a precise formulation away from a few special cases. In this thematic program, we will experiment with and articulate refined conjectures relating arithmetic-geometric objects to automorphic forms, improve the computational infrastructure underpinning the Langlands program, and assemble additional supporting... (more)

##### Organizing Committee

- Alina Bucur
- John Conrey
- David Farmer
- John Jones
- Kiran Kedlaya
- Michael Rubinstein
- Holly Swisher
- John Voight

##### Phase Transitions and Emergent Properties

Feb 2 - May 8, 2015

Emergent phenomena are properties of a system of many components which are only evident or even meaningful for the collection as a whole. A typical example is a system of many molecules, whose bulk properties may change from those of a fluid to those of a solid in response to changes in temperature or pressure. The basic mathematical tool for understanding emergent phenomena is the variational principle, most often employed via entropy maximization. The difficulty of analyzing emergent phenomena, however, makes empirical work essential; computations generate conjectures and their results are often our best judge of the truth.

The semester will include three workshops that will concentrate on different aspects of current interest, including unusual settings such as complex networks and quasicrystals, the onset of emergence as small systems grow, and the emergence of structure and shape as limits in probabilistic models. The workshops will (necessarily) bring in researchers in... (more)

##### Organizing Committee

- Mark Bowick
- Béatrice de Tilière
- Richard Kenyon
- Charles Radin
- Peter Winkler

##### High-dimensional Approximation

Sep 8 - Dec 5, 2014

The fundamental problem of approximation theory is to resolve a possibly complicated function, called the target function, by simpler, easier to compute functions called approximants. Increasing the resolution of the target function can generally only be achieved by increasing the complexity of the approximants. The understanding of this trade-off between resolution and complexity is the main goal of approximation theory, a classical subject that goes back to the early results on Taylor's and Fourier's expansions of a function.

Modern problems in approximation, driven by applications in biology, medicine, and engineering, are being formulated in very high dimensions, which brings to the fore new phenomena. One aspect of the high-dimensional regime is a focus on sparse signals, motivated by the fact that many real world signals can be well approximated by sparse ones. The goal of compressed sensing is to reconstruct such signals from their incomplete linear information. Another aspect... (more)

##### Organizing Committee

- Dmitriy Bilyk
- William Chen
- Frances Kuo
- Michael Lacey
- Vladimir Temlyakov
- Rachel Ward
- Henryk Wozniakowski

##### Network Science and Graph Algorithms

Feb 3 - May 9, 2014

The study of computational problems on graphs has long been a central area of research in computer science. However, recent years have seen qualitative changes in both the problems to be solved and the tools available to do so. Application areas such as computational biology, the web, social networks, and machine learning give rise to large graphs and complex statistical questions that demand new algorithmic ideas and computational models. A wide variety of techniques are emerging for addressing these challenges: from semidefinite programming and combinatorial preconditioners.

In addition to three international conferences, the program will support several research clusters, concentrated periods of activity organized around a specific and timely approach to graph algorithms.

##### Organizing Committee

- Andrea Bertozzi
- Jonathan Kelner
- Philip Klein
- Claire Mathieu
- David Shmoys
- Eli Upfal

##### Low-dimensional Topology, Geometry, and Dynamics

Sep 9 - Dec 6, 2013

The program focuses on the recent impact of computation and experiment on the study of the pure mathematics sides of topology, geometry, and dynamics. Specific areas include 3-dimensional topology, the study of locally symmetric spaces, low-dimensional dynamics, and geometric group theory. Included are areas where computation has not yet had an impact, but might do so in the near future.

##### Organizing Committee

- Marc Culler
- Nathan Dunfield
- Walter Neumann
- Richard Schwartz
- Caroline Series
- Dylan Thurston
- Genevieve Walsh
- Anton Zorich

##### Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series

Jan 28 - May 3, 2013

L-functionsâ€”vast generalizations of the Riemann zeta functionâ€” are fundamental objects of study in number theory. In the 1980's the idea emerged that it could be useful to tie together a family of related L-functions in one variable to create a "double Dirichlet series," which could be used to study the average behavior of the original family of L-functions. Double Dirichlet series soon became multiple Dirichlet series. It has gradually emerged that the local structure of these multiple Dirichlet series shows a rich connection to combinatorial representation theory.

This program will explore this interface between automorphic forms and combinatorial representation theory, and will develop computational tools for facilitating investigations. On the automorphic side, Whittaker functions on p-adic groups and their covers are the fundamental objects. Whittaker functions and their relatives are expressible in terms of combinatorial structures on the associated L-group, its flag... (more)

##### Organizing Committee

- Sara Billey
- Ben Brubaker
- Daniel Bump
- Gautam Chinta
- Solomon Friedberg
- Dorian Goldfeld
- Jeffrey Hoffstein
- Anne Schilling
- Nicolas Thiéry

##### Computational Challenges in Probability

Sep 5 - Dec 7, 2012

Modern explorations in science, technology and medicine increasingly demand complex stochastic models. Computational and theoretical advances are needed in order to formulate, analyze, apply and interpret these models. Recent years have witnessed a remarkable interplay between computation and probability. On the one hand, probabilistic techniques have led to powerful computational methods such as Markov chain Monte Carlo algorithms, while on the other hand the calculation of probabilistic quantities such as modes and marginals of high-dimensional distributions and the analysis of data from random samples has posed several computational challenges.

The Fall 2012 Semester on "Computational Challenges in Probability" aims to bring together leading experts and young researchers who are advancing the use of probabilistic and computational methods to study complex models in a variety of fields. The goal is to identify common challenges, exchange existing tools, reveal new application areas... (more)

##### Organizing Committee

- Jose Blanchet
- Paul Dupuis
- Roger Ghanem
- George Karniadakis
- Kavita Ramanan
- Boris Rozovsky
- Eric Vanden-Eijnden

##### Complex and Arithmetic Dynamics

Jan 30 - May 4, 2012

The goal of this program is to bring together researchers in complex dynamics, arithmetic dynamics, and related fields, with the purpose of stimulating interactions, promoting collaborations, making progress on fundamental problems, and developing theoretical and computational foundations on which future work will build. Complex dynamics is the study of iteration of holomorphic self-maps of a complex space. Fundamental examples of such maps arise as algebraic self-maps of algebraic varieties. Starting with the fundamental results of Fatou and Julia, complex dynamics has evolved into a well established field with many deep theorems and many important unresolved questions. Arithmetic dynamics refers to the study of number theoretic phenomena arising in dynamical systems on algebraic varieties. Many global problems in arithmetic dynamics are analogues of classical problems in the theory of Diophantine equations or arithmetic geometry, including for example uniform bounds for rational... (more)

##### Organizing Committee

- Rob Benedetto
- Laura DeMarco
- Mikhail Lyubich
- Juan Rivera-Letelier
- Joseph Silverman
- Lucien Szpiro
- Michael Zieve

##### Kinetic Theory and Computation

Sep 7 - Dec 9, 2011

Kinetic theory plays a central role in many areas of mathematical physics, from nanoscales to continuum mechanics. It is an indispensable tool in the mathematical description of applications in physical and social sciences, from its origin in dilute gases, to wide applications such as semi-conductors, polymers, cells, plasma, galaxies, traffic networking, and swarming. The number of particles is typically more than 10^{20}.

On the one hand, kinetic models provide more detailed and accurate description of regimes where hydrodynamic equations are either invalid or simply not available. On the other hand, because modern computers are still inadequate in simulating the molecular or even quantum dynamics in emerging industrial needs in micro- and nanotechnology, kinetic equations provide models that can capture important features of microscopic or quantum phenomena with a manageable computational cost. Kinetic theory is at the core of multiscale modeling, which connects fundamental... (more)

##### Organizing Committee

- Jose Blanchet
- Francis Filbet
- Irene Gamba
- Yan Guo
- Chi-Wang Shu
- Walter Strauss