## Programs & Events

##### WORKSHOP ONLY OFFERED VIRTUALLY: Lattice Point Distribution and Homogeneous Dynamics

Jun 22 - 26, 2020

In the last decade, there have been several important breakthroughs in Number Theory, where progress on long-standing open problems has been achieved by utilizing ideas originated in the theory of dynamical systems on homogeneous spaces, and their application to lattice point counting and distribution.

The aim of this workshop is to expose young researchers to these fields and provide them with the necessary background from dynamics, number theory, and geometry to allow them to appreciate some of the recent advancements, and prepare them to make new original contributions.

The workshop will include four mini-courses on the topics

1) Dynamics and lattice point counting 2) Thermodynamic formalism 3) Diophantine approximation 4) Fine-scale statistics in number theory and dynamics

In addition, there will be a number of research and expository talks. The talks will emphasize the role that computation and experiment have thus far played in stating key conjectures and establishing key... (more)

##### Organizing Committee

- Dubi Kelmer
- Alex Kontorovich
- Min Lee

##### WORKSHOP ONLY OFFERED VIRTUALLY: Workshop on Arithmetic Geometry, Number Theory, and Computation

Jun 1 - 5, 2020

This program will host 9 collaborative groups led by invited project leaders, who will propose guiding research questions in consultation with the organizers. Individuals interested in contributing to a project or recommended by its leaders may apply via ICERM's online application system (Cube) to join the group.

In their personal statements, applicants should rank in order their top three choices of projects. Applications are currently under review.

##### Organizing Committee

- Jennifer Balakrishnan
- Noam Elkies
- Brendan Hassett
- Bjorn Poonen
- Andrew Sutherland
- John Voight

##### WORKSHOP ONLY OFFERED VIRTUALLY: Competitive Equilibrium with Gross Substitutes, with Applications to Problems in Matching, Pricing, and Market Design

May 11 - 12, 2020

**A short history of equilibrium computation**. The computation of economic equilibrium is making a
spectacular comeback in economics, mathematics and computer science. The availability of massive
real-time datasets and the affordability of computing power, including parallel computation, has made it
possible to implement and build on an effort that had been stalled since the end of the 1970s. But even
more than the new technical possibilities, it is the novel applications to online platforms and market
design tools that led to the surge of interest in computation. Pricing engines like Uberâ€™s, matchmakers
like OkCupid, allocation mechanisms like those that are used by public school districts â€“ all need to
compute an equilibrium problem.

While the problem of equilibrium computation is hard in general, a particular instance of the problem, namely the gross substitutes property, makes it analytically tractable and computable in practice, while able to cover a large number of... (more)

##### Organizing Committee

- Gabrielle Demange
- Alfred Galichon
- Robert Mccann
- Larry Samuelson

##### WORKSHOP ONLY OFFERED VIRTUALLY: Variable Precision in Mathematical and Scientific Computing

May 7 - 8, 2020

From its introduction in the 1980s, the IEEE-754 standard for floating-point arithmetic has ably served a wide range of scientists and engineers. Even today, the vast majority of numerical computations employ either IEEE single or IEEE double, typically one or the other exclusively in a single application. However, recent developments have exhibited the need for a broader range of precision levels, and a varying level of precision within a single application. There are clear performance advantages to a variable precision framework: faster processing, better cache utilization, lower memory usage, and lower long-term data storage. But effective usage of variable precision requires a more sophisticated mathematical framework, together with corresponding software tools and diagnostic facilities.

At the low end, the explosive rise of graphics, artificial intelligence, and machine learning has underscored the utility of reduced precision levels. Accordingly, an IEEE 16-bit "half"... (more)

##### Organizing Committee

- David Bailey
- Neil Burgess
- Jack Dongarra
- Alyson Fox
- Jeffrey Hittinger
- Cindy Rubio-González

##### WORKSHOP ONLY OFFERED VIRTUALLY: Computational Statistics and Data-Driven Models

Apr 20 - 24, 2020

The advancement in computing and storage capabilities of modern computational clusters fosters use of novel statistical techniques in machine learning and deep networks. Such data-driven techniques allow one to learn model features and characteristics that are difficult for mathematical methods alone to reveal. Many computational methods achieve model and complexity discovery using methods that lie at the nexus of mathematical, statistical, and computational disciplines. Statistical methods often employ “big data” approaches that glean predictive capability from diverse and enormous databases of information. Emerging methods in machine learning and deep networks can provide impressive results. This workshop gathers researchers at the frontier of large-scale statistical computation, data science, tensor decompositions and approximations, and data-driven model learning, to focus on modern challenges that use data to reduce complexity of models.

##### Organizing Committee

- Lexin Li
- Youssef Marzouk
- Shari Moskow
- Benjamin Peherstorfer
- Abel Rodriguez
- Daniele Venturi
- Rachel Ward

##### WORKSHOP ONLY OFFERED VIRTUALLY: Algorithms for Dimension and Complexity Reduction

Mar 23 - 27, 2020

Mathematical advances that reduce the complexity of models are complemented by algorithms that achieve the desired reduction in computational effort. This workshop focuses on the synthesis and development of algorithmic approaches to model order reduction. These methods tackle fundamental problems in structure- and topology-preserving reductions, low-rank models and dimension reduction, multi-level approaches, and empirical interpolation and approximations, etc. Complementary approaches that target computational efficiency include strategies with offline and online phases and divide-and-conquer algorithms.

##### Organizing Committee

- Kevin Carlberg
- Yanlai Chen
- Francisco Chinesta
- Misha Kilmer
- Yvon Maday
- Gianluigi Rozza

##### Soergel Bimodules and Categorification of the Braid Group

Feb 28 - Mar 1, 2020

The purpose of this workshop is to bring together experts in representation theory, categorification, low-dimensional topology, mathematical physics, and combinatorics, in order to understand how categorifications of the braid groups and Hecke algebras allow one to compute and understand link invariants. Our concrete goals are to:

(a) develop and compare various algebro-geometric models for link homology, and use them to explicitly compute Khovanov-Rozansky homology of various links

(b) categorify various structures in the Hecke algebra (center, cocenter, Kazhdan-Lusztig cells, Jones-Wenzl projectors) using Soergel bimodules

(c) compare the geometric and algebraic constructions above, and understand the connection between the (co)center of the Soergel category and the Hilbert scheme of points on the plane

This workshop is fully funded by a Simons Foundation Targeted Grant to Institutes.

##### Organizing Committee

- Ben Elias
- Eugene Gorsky
- Andrei Negut

##### Mathematics of Reduced Order Models

Feb 17 - 21, 2020

Mathematical models of scientific applications often involve simulations with a large number of degrees of freedom that strain even the most efficient of algorithms. A clear need is the rigorous development of models with reduced complexity that retain fidelity to the application. Mathematics-based reduced-order modeling applies techniques in nonlinear approximation, projection-based discretizations, sparse surrogate construction, and high-dimensional approximation, in order to construct a model surrogate with near-optimal approximation properties. This workshop focuses on theoretical and algorithmic advances in mathematics-based model order reduction of various types: reduced basis methods, projection-based methods for dynamical systems, and sparse and low-rank approximations in high dimensions.

##### Organizing Committee

- Peter Benner
- Albert Cohen
- Serkan Gugercin
- Olga Mula
- Akil Narayan
- Karen Veroy-Grepl

##### 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

##### Numerical Methods and New Perspectives for Extended Liquid Crystalline Systems

Dec 9 - 13, 2019

Liquid crystals (LCs) are classic examples of partially ordered materials that combine the fluidity of liquids with the long-range order of solids, and have great potential to enable new materials and technological devices. A variety of LC phases exist, e.g. nematics, smectics, cholesterics, with a rich range of behavior when subjected to external fields, curved boundaries, mechanical strain, etc. Recently, new systems came into focus, such as bent-core LC phases, twist-bend-modulated nematics, chromonics and polymer-stabilized blue phases, with more to be discovered.

Best known for applications in displays, LCs have recently been proposed for new applications in biology, nanoscience and beyond, such as biosensors, actuators, drug delivery, and bacterial control (related to active matter). Indeed, it is believed that the LC nature of DNA once enabled the mother of all applications, namely life itself. New numerical methods and scientific computation is needed to guide new theory and... (more)

##### Organizing Committee

- Jan Lagerwall
- Apala Majumdar
- Shawn Walker

##### Illustrating Dynamics and Probability

Nov 11 - 15, 2019

This workshop will focus on the theoretical insights developed via illustration, visualization, and computational experiment in dynamical systems and probability theory. Some topics from complex dynamics include: dynamical moduli spaces and their dynamically-defined subvarieties, degenerations of dynamical systems as one moves toward the boundary of moduli space, and the structure of algebraic data coming from a family of dynamical systems. In classical dynamical systems, some topics include: flows on hyperbolic spaces and Lorentz attractors, simple physical systems like billiards in two and three dimensional domains, and flows on moduli spaces. In probability theory, the workshop features: random walks and continuous time random processes like Brownian motion, SLE, and scaling limits of discrete systems.

This workshop is partially funded by the Alfred P. Sloan Foundation award G-2019-11406 and supported by a Simons Foundation Targeted Grant to Institutes.

##### Organizing Committee

- Jayadev Athreya
- Alexander Holroyd
- Sarah Koch

##### Math + Art Panel

Nov 11, 2019

This panel discussion will explore the different ways in which artists and mathematicians approach mathematical concepts. We expect a dynamic conversation that will spark continued dialogue and future collaborations.

##### Organizing Committee

- Jayadev Athreya
- Allison Paschke
- Masha Ryskin
- Richard Schwartz