A moduli space parameterizes geometric objects with alike structures and encodes in itself the geometry of all possible families of such objects. This workshop will focus on three aspects of moduli spaces: Cycles, Geometric Invariant Theory, and Dynamics. One of our main goals is to synthesize the recent progress on moduli of abelian differentials on algebraic curves motivated by dynamics and in the GIT constructions of related moduli spaces, with the view towards better understanding of geometric cycles on these moduli spaces.

In many cases, computer programming and experiments are important tools to discover new phenomena, both in dynamics and in the study of cycles on moduli spaces. Hence many talks will emphasize computational and experimental aspects of these fields and the workshop will feature a computational problem session whose goal is to disseminate computational techniques and problems to a wider body of researchers.

An integral part of the workshop is a series of three... (more)

This workshop is a one-week program aimed at 20-25 researchers interested in the opportunity to shape the future of research on the mathematics of crime. Small teams will come together to work on real problems with real crime and policing data provided by the Providence Police Department. Five teams will be assembled, each with a technical advisor who will share their expertise and serve as an anchor point and leader for hands-on research that will take place over the course of the week. This will be a truly hands-on experience in which groups will spend time brainstorming mathematical methods and models to approach the problem at hand, analyzing data provided, and creating code to implement ideas as necessary. There will also be research presentations from the technical advisors throughout the week, as well as closing presentations by each team to present their ideas and progress at the end of the workshop. We fully anticipate that lasting collaborations will be formed, and that work... (more)

GirlsGetMath is a weeklong mathematics summer day-program for 9th and 10th grade high school girls in the Providence, RI area.

GirlsGetMath occurs in an encouraging environment that builds young women's 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 will become a replicable national model of mathematical outreach for high school girls, with an emphasis on mathematical experimentation.

This five-day non-residential mathematics program is open to high school girls from the greater Providence, RI area who will be entering the 10th or 11th grade in the fall of 2016.

GirlsGetMath@ICERM encourages 20-25 young women to explore,... (more)

This thematic semester aims at exploring those areas of topology where the research challenges stem from scientific and engineering problem and computer experiments rather than the intrinsic development of the topology proper. In this context, topology is a toolbox of mathematical results and constructions which impacts and inspires developments in other areas. Born as a supporting discipline, aimed at creating a foundation of intuitive notions immensely useful in differential equations and complex analysis, algebraic topology remains indispensable in many disciplines.

Our goal here is to concentrate on relatively recent areas of research enabled, in particular, by the computational revolution in mathematical discovery.

The past decade has seen a surge of interest in applied topology, a vaguely defined area that became a focus of attraction for several communities neighboring mathematics proper - biology, engineering, computer science, data analysis, to name a few. The central lure... (more)

This workshop will bring together researchers interested in a panoply of unusual configuration spaces, arising in applied fields or in plausible models, to look for similarities or creative tensions between them.

Classical configuration spaces in mechanics are spaces of tuples of material points in Euclidean 3D space, sometimes constrained by the mutual distances or just precluding coincidences of any two particles. The no-k-equal configuration spaces and their generalizations relax the no-coincidence conditions allowing some number (greater than 1) of points to coincide. Linkages and origami (hinged structures) represent another well-established class of configuration spaces. Despite many spectacular results on the topology and geometry of the configuration (or moduli) spaces of such mechanical constructions, many natural questions are hard and remain unanswered. In general, neither the topology (even for generic structures), nor singularities of the configuration spaces are known... (more)

This workshop will explore topological properties of random and quasi-random phenomena in physical systems, stochastic simulations/processes, as well as optimization algorithms. Practitioners in these fields have written a great deal of simulation code to help understand the configurations and scaling limits of both the physically observed and computational phenomena. However, mathematically rigorous theories to support the simulation results and to explain their limiting behavior are still in their infancy.

Randomness is inherent to models of the physical, biological, and social world. Random topology models are important in a variety of complicated models including quantum gravity and black holes, filaments of dark matter in astronomy, spatial statistics, and morphological models of shapes, as well as models appearing in social media. The probabilistic method, theory of point processes, and ideas from stochastic and integral geometry have been central tools for proofs and efficient... (more)

While the semester program as a whole is inspired by the original view of topology as analysis situs fueled by applications in natural sciences and engineering, this workshop emphasizes the impact of topology and geometry on discrete structures.

Combinatorially inspired configuration spaces, such as arrangements of points, lines, hyperplanes, polytopes, and the like, provide intricate material and ongoing challenge for topological and geometric techniques. The latter have often gone through a process of adjustment towards their discrete, stratified objects, as in the case of discrete Morse theory or application of Fourier analysis. Notably, the recent solution of the log-concavity conjecture for matroids by Adiprasito, Huh and Katz was achieved by developing Hodge theory for combinatorial geometries which opens up most exciting perspectives on further applications. The construction of higher dimensional expanders is yet another promising direction. Inspired by the rich theory of graph... (more)

This program will focus on integral equation methods, fast algorithms and their applications to fluid dynamics and materials science. Integral equation methods have been used for more than a century to establish existence and uniqueness results for a variety of elliptic, parabolic and hyperbolic partial differential equations (PDEs). From a computational perspective, they have been used most extensively in the elliptic (steady state or time harmonic) case, because of their ability to handle complex geometry, unbounded domains and radiation conditions and because of the availability of fast algorithms to reduce the cost of handling the dense matrices that arise from their discretization. These algorithms include fast multipole methods (FMM), methods based on the Fast Fourier Transform (FFT) or the non-uniform FFT (“NUFFT”), and hierarchical compression-based methods (wavelet and SVD-based schemes, H-matrices, HSS-matrices, etc.). The fundamental issue is that discretization of an... (more)