A new era of astronomical observation was announced in 2016 when the first-ever detection of gravitational waves from a binary black hole system occurred. Gravitational waves encode detailed information about the astrophysical systems they emerge from and complement what can be learned through traditional light-based observation.

Gravitational wave science requires high-fidelity numerical simulations of the expected merger events. The Simulating eXtreme Spacetimes (SXS) collaboration has managed the development of two distinct codes for this purpose: (i) the Spectral Einstein Code (SpEC) based on pseudospectral methods, and (ii) an open-source code SpECTRE, an hp-adaptive discontinuous Galerkin scheme that also includes a sub-cell finite volume scheme in regions of strong shock formation that is ideally suited for multi-scale, multi-physics problems. SpECTRE targets problems in multi-messenger astrophysics, including neutron star mergers, core-collapse supernovae, and gamma-ray... (more)

GirlsGetMath@ICERM is a five-day non-residential mathematics program that is open to high schoolers, regardless of gender, who live in or near greater Rhode Island and who will be entering the 10th or 11th grade in the fall of 2024.

GirlsGetMath occurs in an encouraging environment that builds young students' confidence in math and science.

GirlsGetMath expands participants' understanding and knowledge of mathematics through computations and experimentations.

GirlsGetMath provides expert mathematical training and mentoring.

GirlsGetMath@ICERM encourages 20-25 high schoolers to explore topics such as cryptography, the mathematics of voting, image processing, prime numbers and factoring, fractals, matrices, vectors, and data science.

The goals of the program are:

• to show young adults that the... (more)

The central theme of this workshop is the analysis and computation of Schrödinger operators and applications to nonlinear problems in several areas of Mathematical Physics, Analysis of Partial Differential Equations, Quantum Chemistry and more. The simplest, most basic example, of such an operator is of the form H = −∆+V on an appropriate Hilbert space, and their Dirac analogues.

Many problems in Quantum Physics and Chemistry require a precise understanding of the spectra of Schrödinger operators, H = −∆ + V , for various classes of potentials V (x), and in various regimes, especially in the semi-classical and adiabatic ones. The analysis entails determining eigenvalues and eigenvectors and more generally the evolution generated by H, the study of wave operators, and of the “distorted Fourier transform” and its mapping properties. All of these can be interpreted as diagonalization procedures which are especially delicate for non-selfadjoint operators that can arise as... (more)

The goal of this reunion meeting is to bring together the participants from the spring 2023 program “Discrete Optimization: Mathematics, Algorithms, and Computation” uniting experts in combinatorial optimization, mixed-integer linear and non-linear optimization, with the aim of having an environment to discuss the latest advances and to catalyze new collaborations.

A fundamental challenge that spans nearly all areas of evolutionary biology is the development of effective techniques for analyzing the unprecedented amount of genomic data that has become readily available within the last decade. Such data present specific challenges for the area of phylogenetic inference, which is concerned with estimating the evolutionary relationships among collections of species, populations, or sequences. These challenges include development of evolutionary models that are sufficiently complex to be biologically realistic while remaining computationally tractable; deriving and implementing algorithms to efficiently estimate phylogenetic relationships that use models whose theoretical properties are well-understood and therefore interpretable; and devising ways to scale novel methodology developed to handle datasets that are increasingly large and complex.

This semester program brings together mathematicians, statisticians, computer scientists, and experimental... (more)

The program will explore different perspectives on uncertainty quantification, efficient simulation, and the analysis of complex stochastic systems. The topics covered will include exciting recent developments on efficient methods for approximating quasi-stationary distributions, simulation of equilibrium distributions, information divergences in sensitivity analysis of rare events, large deviations methods for efficient importance sampling, accelerated Monte Carlo via nonlinear PDE, and complex probabilistic models including reflected diffusions and high-dimensional dynamics arising in chemistry and physics.

Computational phylogenetic methods have become essential tools for understanding the evolutionary relationships that underlie much life science research. Motivated by biological questions and insights, built on a broad spectrum of mathematical and statistical ideas and approaches, and implemented through novel and sophisticated algorithmic design, their development draws from multiple fields. Bringing together researchers spanning disciplinary perspectives and techniques, this workshop will present a diverse sample of work addressing current challenges in phylogenetics, with an eye toward future progress.

The unprecedented amount of genomic data that has become readily available presents specific challenges for the area of phylogenetic inference, which is concerned with estimating the evolutionary relationships among collections of species, populations, or sequences. These challenges include development of evolutionary models that are sufficiently complex to be biologically realistic while remaining computationally tractable; deriving and implementing algorithms to efficiently estimate phylogenetic relationships that use models whose theoretical properties are well-understood and therefore interpretable; and devising ways to scale novel methodology developed to handle datasets that are increasingly large and complex.

This workshop focuses on statistical modeling and the scaling of phylogenetic methods. Topics will include modeling (e.g. multispecies coalescent model with extension to networks; diversification models) and inference with speed to scale to genomic datasets,... (more)

Inferring phylogenetic relationships requires complex mathematical models. As advances are made in modeling complex evolutionary processes, we need practical algorithms that translate the mathematical advances into software tools. This translation of theory to usable tools is more challenging than it may appear. Phylogenetic problems are often NP-hard, necessitating heuristic solutions that can compromise accuracy. The accuracy and scalability of such heuristics are often established only empirically, creating a need for careful simulation and testing. Moreover, software tools are used within complicated pipelines, so the input to tools may be impacted by errors from prior data processing steps. In addition, the output has many aspects, from the discrete-spaced topology to continuous-spaced branch lengths and other numeric parameters, and measures of uncertainty and visualizations. Furthermore, software tools need to be evolvable, allowing the incorporation of new features and new... (more)

In recent years, there has been a significant increase in the interaction between harmonic analysis and convex geometry, leading to solutions for several longstanding open problems, discovery of new phenomena, as well as many new intriguing open questions. These connections were studied during the Fall 2022 ICERM semester on “Harmonic Analysis and Convexity”. The objective of this workshop is to revisit and review the results produced during the semester and the subsequent year.

The primary areas of focus for the workshop will encompass: The Fourier approach to Geometric Tomography; Volume and Duality; Bellman technique for extremal problems in harmonic analysis; Convexity of solutions to Hamilton–Jacobi–Bellman equations; as well as numerical computations and computer-assisted proofs: Exploring the use of computational methods for theoretical aspects, including optimal algorithms, as well as practical implementation.

Biological systems are typically highly interconnected and complex. With technological advances, it is possible to collect massive amounts of data from these systems, but it is not always clear how to organize the information to draw conclusions and make predictions. In such cases, mathematical formulations are powerful tools allowing researchers to frame questions, explore patterns, and synthesize information. Augmenting and expanding computational algorithms, machine learning algorithms, and data science techniques is necessary to keep pace with the complexity of the models needed for predictive modeling. The interdisciplinary nature of mathematical biology requires a variety of skills and facilitating interaction among research groups and institutions is important to moving the discipline forward.

The workshop aims to build research collaboration among researchers in mathematical biology. Participants will spend a week making significant progress on a research project and foster... (more)