Incompressible fluids are an abundant source of mathematical and practical problems. The question of global-in-time regularity versus finite-time singularity formation for incompressible fluids, governed by the Navier-Stokes or Euler equations, has been one of the most challenging outstanding problems in applied PDE. There have also been new developments in the study of the onset of turbulence due to linear and nonlinear instabilities in incompressible fluids. Interfacial and surface water waves are physical phenomena that, in addition to the challenges outlined above, involve the evolution of free boundaries. These problems embody many of the mathematical challenges found in studies of nonlinear PDEs.

Progress on these topics is possible because of advances in analysis, numerical computations and physical experiments. In addition, ocean field observations provide a reality test to all conclusions and invite new problems to be addressed. In this program, we provide a venue for... (more)

The purpose of the topical workshop is to gather leading experts, postdoctoral scholars, and graduate students, to present exciting new developments in the field of mathematical fluid dynamics. The focus of the meeting will be placed on current research on regularity, instabilities, and the onset of turbulence in fluid flow, from a theoretical and from a computational perspective. Despite their long and fruitful history, to date these topics continue to enchant and inspire mathematicians, physicists, and computational scientists: in part due to their ubiquitous applications in areas from aeronautical engineering to medicine, and in part because the basic mathematical questions are still open. Among these are global in time existence of solutions to the equations describing motion of inviscid and viscous fluids in three spatial dimensions, and the conjectured relation between the phenomenological theories of turbulence and the statistical properties of solutions to the underlying... (more)

This program will focus on phase retrieval, a research area introduced by Pete Casazza (one of the organizers of this program) and others. Phase retrieval originates from harmonic analysis, where one wants to recover a function from the magnitude of its Fourier transform without any phase information. The phase retrieval problem has a natural generalization to finite dimensional Hilbert spaces. A finite dimensional signal is sought to fit the magnitudes of its linear measurements. Phase retrieval in this finite dimensional setting has become one of the growing research areas in recent years. The techniques from the finite dimensional setting are promising to become indispensable in many imaging techniques such as x-ray crystallography, electron microscopy, diffractive imaging, astronomical imaging, x-ray tomography etc. It also has other important applications in optics, communication, audio signal processing, and more.

Many challenging and fundamental problems in phase retrieval... (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)

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 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)

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 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)

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)