## 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

##### 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

##### 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

##### Variable Precision in Mathematical and Scientific Computing

May 6 - 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
- Nicholas Burgess
- Jack Dongarra
- Alyson Fox
- Jeff Hittinger
- Cindy Rubio Gonzalez

##### Competitive Equilibrium with Gross Substitutes, with Applications to Problems in Matching, Pricing, and Market Design

May 11 - 15, 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

##### 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

##### MAA & TRIPODS Advanced Workshop in Data Science for Mathematical Sciences Faculty

Jul 20 - 24, 2020

The MAA & TRIPODS Advanced Workshop in Data Science for Mathematical Sciences Faculty is a 4-day hands-on workshop for mathematical sciences faculty who have had some exposure to and experience with data science but who are not themselves data science experts. Participants of the 2017 or 2019 PIC Math Data Science Workshops that were held at BYU qualify and those who have experience coding in Python and applying basic statistical techniques to a large data set. The goal of the workshop is to bring together faculty from a range of institutions and expand the knowledge of the participants so that they are better armed to prepare students for the data science workforce.

Participants will learn more advanced techniques in the fields of data science, statistical learning, and machine learning. They will collaborate on data science projects that will involve accessing and cleaning large data sets and... (more)

##### Organizing Committee

- Michael Boardman
- Michael Dorff
- Rachel Levy
- Suzanne Weekes

##### Women in Algebraic Geometry

Jul 27 - 31, 2020

The Women in Algebraic Geometry Collaborative Research Workshop will bring together researchers in algebraic geometry to work in groups of 4-6, each led by one or two senior mathematicians. The goals of this workshop are: to advance the frontiers of modern algebraic geometry, including through explicit computations and experimentation, and to strengthen the community of women and non-binary mathematicians working in algebraic geometry. This workshop capitalizes on momentum from a series of recent events for women in algebraic geometry, starting in 2015 with the IAS Program for Women in Mathematics on algebraic geometry.

Successful applicants will be assigned to a group based on their research interests. The groups will work on open-ended projects in diverse areas of current interest, including moduli spaces and combinatorics, degenerations, and birational geometry. Several of the proposed projects extensively involve experimentation and computation, which will increase the likelihood... (more)

##### Organizing Committee

- Melody Chan
- Antonella Grassi
- Rohini Ramadas
- Julie Rana
- Isabel Vogt

##### Free Resolutions and Representation Theory

Aug 3 - 7, 2020

The structure of free resolutions plays an important role in analyzing singularities of varieties of low codimension. Codimension two Cohen-Macaulay varieties (resp. codimension three Gorenstein varieties) come from rank conditions on an n x (n+1) matrix (resp. a skew-symmetric (2n+1) x (2n+1) matrix).

This workshop seeks to push such results to Cohen-Macaulay varieties of codimension three and Gorenstein varieties of codimension four.

This problem turns out to be related to the classification of semi-simple Lie algebras. These new methods allow one to create a ‘map’ of free resolutions of a given format. The calculations that arise are very demanding and require new computational methods involving both commutative algebra and representation theory.

##### Organizing Committee

- Lars Christensen
- Claudia Miller
- Steven Sam
- Jerzy Weyman

##### Symmetry, Randomness, and Computations in Real Algebraic Geometry

Aug 24 - 28, 2020

Real algebraic (and semi-algebraic) geometry studies subsets of R^n defined by a finite number of polynomial equalities and inequalities. Such sets occur ubiquitously in practice both inside and outside of mathematics. While being easy to define, semi-algebraic sets can be complicated topologically, which restricts the application of many algorithms. In recent years, there has been progress in proving much stronger results â€“ both quantitative and algorithmic -- when the problem under consideration involves the invariance under some group action.

In this workshop, we plan to focus on two situations where this phenomenon happens.

The first one is the statistical study of the topology of random real algebraic varieties as well as semi-algebraic sets, where the polynomials defining these objects are picked from a distribution invariant under the action of a certain group (usually the orthogonal group) acting on the space of variables. The behavior of the set of zeros (or more... (more)

##### Organizing Committee

- Saugata Basu
- Antonio Lerario
- Annie Raymond
- Cordian Riener

##### 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

##### Advances and Challenges in Computational Relativity

Sep 14 - 18, 2020

This kick-off workshop will seek to provide an overview of both the state-of-the-art and open challenges drawing from multiple themes (theory, analysis of the equations, computation, and data analysis) within the broad context of Einstein’s general relativity theory. The workshop will also feature a code bootcamp on the last day. The bootcamp participants will be given both an overview of the key pieces of software used in the field as well as practical instructions on installing and running example cases.

##### Organizing Committee

- Douglas Arnold
- Scott Field
- Gaurav Khanna
- Deirdre Shoemaker