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)

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.

The purpose of this workshop is to bring together experts in representation theory, categorification, low-dimensional topology, mathematical physics, and combinatorics, in other 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.

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.

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.

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)

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 Uber’s, matchmakers like OkCupid, allocation mechanisms like those that are used by public school districts – 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)

This program will host 11 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.

PROJECT DESCRIPTIONS

In their personal statements, applicants should rank in order their top three choices of projects. APPLICATION DEADLINE: March 15, 2020

The faculty advisers will present a variety of interdisciplinary research topics utilizing large-scale linear algebra, model reduction, randomized algorithms, and deep learning. Participants will have the opportunity to learn the theoretical underpinnings of these research topics in applied and computational mathematics and will help develop open-source software tools that accomplish data-driven scientific predictions.

The faculty will begin the program with brief introductory talks. Throughout the eight-week program, students will work on assigned projects in groups of two to four, supervised by faculty advisors and aided by teaching assistants. Students will meet daily, give regular talks about their findings, attend mini-courses, guest talks, and professional development seminars, practice coding, version control, and Tex typesetting. Students will learn how to collaborate mathematically, and they will work closely in their teams to write up their research into a poster and/or... (more)

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)

This workshop brings together two distinct streams of mathematics - on the one hand, the classical rigidity theory of bar-joint frameworks in combinatorics and discrete geometry, and on the other the theory of generalized circle packings that arose from the study of 3-manifolds in geometric topology.

Combinatorial and Geometric rigidity theory is concerned with the local and global uniqueness of congruence classes of frameworks as solutions to their underlying geometric constraint system.

The focal point of circle packing theory is the Koebe-Andre'ev-Thurston Theorem that gives conditions that guarantee the existence and rigidity of circle packings on closed surfaces in the pattern of a given triangulation of the surface.

A scattering of results in recent years has started to forge connections between these research areas. The main aim of the workshop is to develop a cross-fertilization of such ideas, with particular emphasis on the rigidity of inversive distance packings. As well... (more)