## Programs & Events

##### 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

##### 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

##### 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

##### Algorithms in Complex Dynamics and Mapping Class Groups

Nov 2 - 3, 2019

Thurston maps are orientation-preserving branched covering maps of the two-sphere to itself for which the forward orbits of the branch points form a finite set. They arise in the classification of complex dynamical systems.

Recent work has shown close connections between the combinatorial, topological, and algebraic theory of Thurston maps and that of mapping class groups. The algorithmic and computational theories of mapping class groups are highly advanced and have reached the point of effective implementation via computer programs. However, such implementations for Thurston maps are significantly less advanced. The aim of the proposed Hot Topic workshop is to bring together researchers in the computational theory of mapping class groups and those in the combinatorial theory of Thurston maps in order to make headway on fundamental problems.

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

##### Organizing Committee

- Dan Margalit
- Kevin Pilgrim
- Becca Winarski

##### An ICERM Public Lecture: The nth Perspective

Oct 30, 2019

##### THIS TALK HAS BEEN CANCELED DUE TO UNFORESEEN CIRCUMSTANCES.

**We will still host the planned reception starting at 6:00pm at Brown University's Granoff Center. Please come enjoy the food and the opportunity to interact with the artists-in-residence featured in the "Math+Art" exhibit now on display in the Atrium Gallery. No tickets required.**

In mathematics, as in art, progress and innovation often come from looking at the world in a new way. These shifts in viewpoint sometimes come from a clear process of deduction, while other times they seem to arise mysteriously. In either case, they are often accompanied by a strong â€œAha!â€ feeling of insight. When first revisiting previous ideas in such a new light, a sense that one finally has â€œit all really rightâ€ emerges. Yet as experience with a new point of view develops, its own shortcomings tend to surface, setting the stage for another shift in perspective.

Through interactive demonstrations and... (more)

##### 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

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

##### Math + Art Panel

Oct 21, 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

##### Math + Art Panel

Oct 7, 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.

*Light refreshments will be served following the panel at 5 PM.*

##### Organizing Committee

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

##### Computational Textiles Working Group

Sep 23 - 27, 2019

The aim of this working group is to bring together theorists and practitioners of computational fiber arts and will have three related themes.

First, we would like to prove theorems about the geometry and topology of knitting. Second, we would like to explore the idea that knitting and textiles can be a physical embodiment of ideas in computational geometry. Third, we will use knitting and other textile arts as a way of visually communicating mathematical ideas to a broader audience.

The working group will take place the week following the Illustrating Geometry and Topology workshop.

##### Organizing Committee

- Sabetta Matsumoto
- Saul Schleimer
- Henry Segerman
- Laura Taalman

##### Illustrating Geometry and Topology

Sep 16 - 20, 2019

**Applications closed. Workshop is at capacity.**

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.

This workshop is partially funded by the... (more)

##### Organizing Committee

- Keenan Crane
- David Dumas