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.

PROJECT DESCRIPTIONS

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

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)

Experimental geometry labs create an environment ripe for students and faculty to treat mathematics as a laboratory science. Visualization and computational pattern discovery help guide research, formulate conjectures and develop ideas in proofs. In addition to research, experimental geometry labs foster community engagement via grassroots outreach activities in local schools, libraries, and museums. These activities spread the wonder and excitement of mathematics to people both within and outside the academy.

This workshop is partially supported by the Department of Mathematical Sciences and the College of Science at George Mason University.

For this workshop, ICERM welcomes applications from undergraduates, graduates, postdocs, and faculty who wish to participate. Undergraduate students and graduate students who apply should ask their advisor to submit a statement of support. We will ask students to present their work in a poster.

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)

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)

The structure of free resolutions plays an important role in analyzing singularities of varieties of low codimension. Codimension 2 Cohen-Macaulay varieties (resp. codimension 3 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 3 and Gorenstein varieties of codimension 4.

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.

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 – 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)

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)

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.