## Programs & Events

##### Nonlinear Algebra

Sep 5 - Dec 7, 2018

The theory, algorithms, and software of linear algebra are familiar tools across mathematics, the applied sciences, and engineering. This ubiquity of linear algebra masks a fairly recent growth of *nonlinear algebra* in mathematics and its applications to other disciplines. The proliferation of nonlinear algebra has been fueled by recent theoretical advances, efficient implementations of core algorithms, and an increased awareness of these tools.

The benefits of this nonlinear theory and its tools are manifold. Pushing computational boundaries has led to the development of new mathematical theories, such as homotopy methods for numerical algebraic geometry, tropical geometry and toric deformations, and sums of squares methods for polynomial optimization. This uncovered many concrete nonlinear mathematical objects and questions, many of which are ripe for computer experimentation. In turn, resulting mathematical breakthroughs often lead to more powerful and efficient algorithms... (more)

##### Organizing Committee

- Dan Bates
- Sandra Di Rocco
- Jonathan Hauenstein
- Anton Leykin
- Frank Sottile
- Mike Stillman
- Cynthia Vinzant

##### Point Configurations in Geometry, Physics and Computer Science

Feb 1 - May 4, 2018

The arrangement of point configurations in metric spaces, whether deterministic or random, is a truly interdisciplinary topic of great interest in mathematics, physics and computer science. Mathematical aspects involve optimization, discretization of manifolds, best packing and cubature, among others. For physics, such configurations arise in the study of crystallization, point processes connected with random matrices, self-assembling materials, jammed states, hyperuniformity and phase transitions. For computer science, extremal point configurations play a fundamental role in coding and information theory, and lattice-based protocols in cryptography and related computational complexity issues are of growing importance. Furthermore, there has been recent and substantial progress on related age-old problems (such as the Kepler conjecture).

The investigation of the above topics often evolves from the development of efficient computational methods that enable extensive numerical... (more)

##### Organizing Committee

- Christine Bachoc
- Henry Cohn
- Peter Grabner
- Doug Hardin
- Edward Saff
- Achill Schürmann
- Sylvia Serfaty
- Salvatore Torquato
- Robert Womersley

##### Mathematical and Computational Challenges in Radar and Seismic Reconstruction

Sep 6 - Dec 8, 2017

Inversion and imaging with waves is of fundamental importance in both radar and seismic reconstruction. Mathematics provides the key technology in both areas and, despite differing in many important respects, they have much in common in their underlying mathematical frameworks, approaches, and challenges. This semester program will focus on advancing their common mathematical and computational methodologies, as well as selected subjects distinct to each area, in the context of new challenges and opportunities that have arisen in recent years. Both theory and applications will be of interest. Participants will be drawn from academia, industry, and governmental laboratories in order to broadly address theory, applications, and their synergy.

The program will be influenced by recent developments in wave propagation and imaging, data acquisition and analysis, and high-performance computing. Driven by the ongoing need for more realistic mathematical models and simulations, recent advances... (more)

##### Organizing Committee

- Alexandre Aubry
- Liliana Borcea
- Margaret Cheney
- Armin Doerry
- Vladimir Druskin
- Albert Fannjiang
- Alison Malcolm
- Eric Mokole
- Frank Robey
- Knut Solna
- Chrysoula Tsogka
- Lexing Ying
- Edmund Zelnio

##### Singularities and Waves In Incompressible Fluids

Jan 30 - May 5, 2017

Incompressible fluids are an abundant source of mathematical and practical problems. The question of global-in-time regularity versus finite-time singularity formation for incompressible fluids, governed by the Navier-Stokes or Euler equations, has been one of the most challenging outstanding problems in applied PDE. There have also been new developments in the study of the onset of turbulence due to linear and nonlinear instabilities in incompressible fluids. Interfacial and surface water waves are physical phenomena that, in addition to the challenges outlined above, involve the evolution of free boundaries. These problems embody many of the mathematical challenges found in studies of nonlinear PDEs.

Progress on these topics is possible because of advances in analysis, numerical computations and physical experiments. In addition, ocean field observations provide a reality test to all conclusions and invite new problems to be addressed. In this program, we provide a venue for... (more)

##### Organizing Committee

- Bernard Deconinck
- Yan Guo
- Diane Henderson
- Govind Menon
- Paul Milewski
- Helena Nussenzveig Lopes
- Walter Strauss
- Jon Wilkening

##### Topology in Motion

Sep 6 - Dec 9, 2016

This thematic semester aims at exploring those areas of topology where the research challenges stem from scientific and engineering problem and computer experiments rather than the intrinsic development of the topology proper. In this context, topology is a toolbox of mathematical results and constructions which impacts and inspires developments in other areas. Born as a supporting discipline, aimed at creating a foundation of intuitive notions immensely useful in differential equations and complex analysis, algebraic topology remains indispensable in many disciplines.

Our goal here is to concentrate on relatively recent areas of research enabled, in particular, by the computational revolution in mathematical discovery.

The past decade has seen a surge of interest in applied topology, a vaguely defined area that became a focus of attraction for several communities neighboring mathematics proper - biology, engineering, computer science, data analysis, to name a few. The central lure... (more)

##### Organizing Committee

- Yuliy Baryshnikov
- Fred Cohen
- Matthew Kahle
- Randall Kamien
- Sayan Mukherjee
- Igor Pak
- Ileana Streinu
- Rade Zivaljevic

##### 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