## Programs & Events

##### Numerical PDEs: Analysis, Algorithms, and Data Challenges

Jan 29 - May 3, 2024

This Semester Program will bring together both leading experts and junior researchers to discuss the current state-of-the-art and emerging trends in computational PDEs. While there are scores of numerical methodologies designed for a wide variety of PDEs, the program will be designed around three workshops each centered around a specific theme: PDEs and Geometry, Nonlocal PDEs, and Numerical Analysis of Multiphysics problems. This grouping of topics embodies a broad representation of computational mathematics with each set possessing its own skill set of mathematical tools and viewpoints. Nonetheless, all workshops will have the common theme of using rigorous mathematical theory to develop and analyze the convergence and efficiency of numerical methods. The diversity of the workshop topics will bring together historically distinct groups of mathematicians to interact and facilitate new ideas and breakthroughs.

##### Organizing Committee

- Marta D'Elia
- Johnny Guzman
- Brittany Hamfeldt
- Michael Neilan
- Maxim Olshanskiy
- Sara Pollock
- Abner Salgado
- Valeria Simoncini

##### Math + Neuroscience: Strengthening the Interplay Between Theory and Mathematics

Sep 6 - Dec 8, 2023

The goal of this Semester Program is to bring together a variety of mathematicians with researchers working in theoretical and computational neuroscience as well as some theory-friendly experimentalists. However, unlike programs in neuroscience that emphasize connections between theory and experiment, this program will focus on building bridges between theory and mathematics. This is motivated in part by the observation that theoretical developments in neuroscience are often limited not only by lack of data but also by the need to better develop the relevant mathematics. For example, theorists often rely on linear or near-linear modeling frameworks for neural networks simply because the mathematics of nonlinear network dynamics is still poorly understood. Conversely, just as in the history of physics, neuroscience problems give rise to new questions in mathematics. In recent years, these questions have touched on a rich variety of fields including geometry, topology, combinatorics,... (more)

##### Organizing Committee

- Carina Curto
- Brent Doiron
- Robert Ghrist
- Kathryn Hess
- Zachary Kilpatrick
- Matilde Marcolli
- Konstantin Mischaikow
- Katie Morrison
- Elad Schneidman
- Tatyana Sharpee

##### Discrete Optimization: Mathematics, Algorithms, and Computation

Jan 30 - May 5, 2023

Discrete optimization is a vibrant area of computational mathematics devoted to efficiently finding optimal solutions among a finite or countable set of possible feasible solutions.

A famous and classical example of a problem in discrete optimization is the *traveling salesperson problem*: For given cities and distances of traveling from one city to another, we seek to find the shortest route that visits each city once and returns to the starting city. Discrete optimization problems naturally arise in many kinds of applications including bioinformatics, telecommunications network design, airline scheduling, circuit design, and efficient resource allocation. The field also connects to a variety of areas in mathematics, computer science, and data analytics including approximation algorithms, convex and tropical geometry, number theory, real algebraic geometry, parameterized complexity theory, quantum computing, machine learning, and mathematical logic.

The semester program... (more)

##### Organizing Committee

- Jesús De Loera
- Antoine Deza
- Marcia Fampa
- Volker Kaibel
- Jon Lee
- Laura Sanità

##### Harmonic Analysis and Convexity

Sep 7 - Dec 9, 2022

In recent years, the interaction between harmonic analysis and convex geometry has dramatically increased, which resulted in solutions to several long-standing problems. The program will bring together leading mathematicians in both areas, along with researchers working in related applied fields, for the first-ever long-term joint program.

The main directions of the program will include: the Fourier approach to Geometric Tomography, the study of geometric properties of solids based on information about their sections and projections, Volume and Duality, Bellman technique for extremal problems of harmonic analysis, and various types of convexity of solutions of corresponding Hamilton–Jacobi–Bellman equation, as well as numerical computations and computer-assisted proofs applied to the aforementioned problems. The computational part will cover theoretical aspects (optimal algorithms, and why they work) as well as more applied ones (implementation).

##### Organizing Committee

- Javier Gomez Serrano
- Irina Holmes Fay
- Bo'az Klartag
- Alexander Koldobskiy
- Sergei Treil
- Alexander Volberg
- Artem Zvavitch

##### Braids

Feb 1 - May 6, 2022

Braid groups were introduced by Emil Artin almost a century ago. Since then, braid groups, mapping class groups, and their generalizations have come to occupy a significant place in parts of both pure and applied mathematics. In the last 15 years, fields with an interest in braids have independently undergone rapid development; these fields include representation theory, low-dimensional topology, complex and symplectic geometry, and geometric group theory. Braid and mapping class groups are prominent players in current mathematics not only because these groups are rich objects of study in their own right, but also because they provide organizing structures for a variety of different areas. For example, in modern representation theory, important equivalences of categories are organized into 2-representations of braid groups, and these same 2-representations appear prominently in parts of geometry and mathematical physics concerned with mirror dualities; in low-dimensional topology,... (more)

##### Organizing Committee

- Marc Culler
- Ben Elias
- John Etnyre
- Benson Farb
- Juan González-Meneses
- Matthew Hedden
- Keiko Kawamuro
- Anthony Licata
- Joan Licata

##### Hamiltonian Methods in Dispersive and Wave Evolution Equations

Sep 8 - Dec 10, 2021

Dispersive equations are ubiquitous in nature. They govern the motion of waves in plasmas, ferromagnets, and elastic bodies, the propagation of light in optical fibers and of water in canals. They are relevant from the ocean scale down to atom condensates. There has been much recent progress in different directions, in particular in the exploration of the phase space of solutions of semilinear equations, advances towards a soliton resolution conjecture, the study of asymptotic stability of physical systems, the theoretical and numerical study of weak turbulence and transfer of energy in systems out of equilibrium, the introduction of tools from probability and the recent incorporation of computer assisted proofs. This semester aims to bring together these new developments and to explore their possible interconnection.

Dispersive phenomena appear in physical situations, where some energy is conserved, and are naturally related to Hamiltonian systems. This semester proposes to explore... (more)

##### Organizing Committee

- Diego Cordoba
- Erwan Faou
- Patrick Gerard
- Pierre Germain
- Alexandru Ionescu
- Alex Kiselev
- Andrea Nahmod
- Kenji Nakanishi
- Benoit Pausader
- Themistoklis Sapsis
- Gigliola Staffilani

##### Combinatorial Algebraic Geometry

Feb 1 - May 7, 2021

Combinatorial algebraic geometry comprises the parts of algebraic geometry where basic geometric phenomena can be described with combinatorial data, and where combinatorial methods are essential for further progress.

Research in combinatorial algebraic geometry utilizes combinatorial techniques to answer questions about geometry. Typical examples include predictions about singularities, construction of degenerations, and computation of geometric invariants such as Gromov-Witten invariants, Euler characteristics, the number of points in intersections, multiplicities, genera, and many more. The study of positivity properties of geometric invariants is one of the driving forces behind the interplay between geometry and combinatorics. Flag manifolds and Schubert calculus are particularly rich sources of invariants with positivity properties.

In the opposite direction, geometric methods provide powerful tools for studying combinatorial objects. For example, many deep properties of... (more)

##### Organizing Committee

- Anders Buch
- Melody Chan
- June Huh
- Thomas Lam
- Leonardo Mihalcea
- Sam Payne
- Lauren Williams

##### Advances in Computational Relativity

Sep 9 - Dec 11, 2020

The Nobel-Prize-winning detection of gravitational waves from binary black hole systems in 2015 by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the LIGO Scientific Collaboration has opened a new window on the universe. In addition, the 2017 observation of both gravitational and electromagnetic waves emitted by a binary neutron star system marked a new era of multi-messenger astronomy. While these successes are a remarkable experimental feat, they also constitute a significant computational achievement due to the crucial role played by accurate numerical models of the astrophysical sources in gravitational-wave data analysis. As current detectors are upgraded and new detectors come online within an international network of observatories, accurate, efficient, and advanced computational methods will be indispensable for interpreting the diversity of gravitational wave signals. This semester program at ICERM will emphasize the fundamental mathematical and... (more)

##### Organizing Committee

- Stefanos Aretakis
- Douglas Arnold
- Manuela Campanelli
- Scott Field
- Jonathan Gair
- Jae-Hun Jung
- Gaurav Khanna
- Stephen Lau
- Steven Liebling
- Deirdre Shoemaker
- Jared Speck
- Saul Teukolsky

##### Model and dimension reduction in uncertain and dynamic systems

Jan 27 - May 1, 2020

Today's computational and experimental paradigms feature complex models along with disparate and, frequently, enormous data sets. This necessitates the development of theoretical and computational strategies for efficient and robust numerical algorithms that effectively resolve the important features and characteristics of these complex computational models. The desiderata for resolving the underlying model features is often application-specific and combines mathematical tasks like approximation, prediction, calibration, design, and optimization. Running simulations that fully account for the variability of the complexities of modern scientific models can be infeasible due to the curse of dimensionality, chaotic behavior or dynamics, and/or overwhelming streams of informative data.

This semester program focuses on both theoretical investigation and practical algorithm development for reduction in the complexity - the dimension, the degrees of freedom, the data - arising in these... (more)

##### Organizing Committee

- Yanlai Chen
- Serkan Gugercin
- Misha Kilmer
- Yvon Maday
- Shari Moskow
- Akil Narayan
- Daniele Venturi

##### Illustrating Mathematics

Sep 4 - Dec 6, 2019

The Illustrating Mathematics program brings together mathematicians, makers, and artists who share a common interest in illustrating mathematical ideas via computational tools.

The goals of the program are to:

- introduce mathematicians to new computational illustration tools to guide and inform their research;
- spark collaborations among and between mathematicians, makers and artists;
- find ways to communicate research mathematics to as wide an audience as possible.

Mathematical topics include: moduli spaces of geometric structures, hyperbolic geometry, configuration spaces, sphere eversions, apollonian packings, kleinian groups, sandpiles and tropical geometry, analytic number theory, supercharacters, complex dynamics, billiards, random walks, and Schrammâ€“Loewner... (more)

##### Organizing Committee

- David Bachman
- Kelly Delp
- David Dumas
- Saul Schleimer
- Richard Schwartz
- Henry Segerman
- Katherine Stange
- Laura Taalman

##### Computer Vision

Feb 4 - May 10, 2019

Computer vision is an inter-disciplinary topic crossing boundaries between computer science, statistics, mathematics, engineering and cognitive science.

Research in computer vision involves the development and evaluation of computational methods for image analysis. This includes the design of new theoretical models and algorithms, and practical implementation of these algorithms using a variety of computer architectures and programming languages. The methods under consideration are often motivated by generative mathematical models of the world and the imaging process. Recent approaches also rely heavily on machine learning techniques and discriminative models such as deep neural networks.

Problems that will be considered in the program include image restoration, image segmentation, object recognition and 3D reconstruction. Current approaches to address these problems draw on a variety of mathematical and computational topics such as stochastic models, statistical methods,... (more)

##### Organizing Committee

- Yali Amit
- Ronen Basri
- Tamara Berg
- Alex Berg
- Pedro Felzenszwalb
- Benar Fux Svaiter
- Stuart Geman
- Basilis Gidas
- David Jacobs
- Olga Veksler

##### 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