## Programs & Events

##### Illustrating Geometry and Topology

Sep 16 - 20, 2019

This workshop will focus on the interaction between visualization, computer experiment, and theoretical advances in all areas of research in geometry and topology. Fruitful interactions of this type have a long history in the field, with physical models and computer images and animations providing both illustration of existing work and inspiration for new developments. Emerging visualization technologies, such as virtual reality, are poised to further increase the tools available for mathematical illustration and experimentation. By bringing together expert practitioners of mathematical visualization techniques and researchers interested in incorporating such tools into their research, the workshop will give participants a clear picture of the state of the art in this fast-moving field while also fostering new collaborations and innovations in illustrating geometry and topology.

##### Organizing Committee

- Keenan Crane
- David Dumas

##### Illustrating Number Theory and Algebra

Oct 21 - 25, 2019

The symbiotic relationship between the illustration of mathematics and mathematical research is now flowering in algebra and number theory. This workshop aims to both showcase and develop these connections, including the development of new visualization tools for algebra and number theory. Topics are wide-ranging, and include Apollonian circle packings and the illustration of the arithmetic of hyperbolic manifolds more generally, the visual exploration of the statistics of integer sequences, and the illustrative geometry of such objects as Gaussian periods and Fourier coefficients of modular forms. Other topics may include expander graphs, abelian sandpiles, and Diophantine approximation on varieties. We will also focus on diagrammatic algebras and categories such as Khovanov-Lauda-Rouquier algebras, Soergel bimodule categories, spider categories, and foam categories. The ability to visualize complicated relations diagrammatically has led to important advances in representation theory... (more)

##### Organizing Committee

- Joel Kamnitzer
- Ellen Eischen
- Alex Kontorovich
- Katherine Stange

##### 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.

##### Organizing Committee

- Jayadev Athreya
- Alexander Holroyd
- Sarah Koch

##### 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

- Apala Majumdar
- Shawn Walker
- Jan Lagerwall

##### 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

##### 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

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