This workshop will provide a platform for researchers working on localized kernel-based methods to present and discuss their latest developments, as well as the current theoretical and practical challenges in the field. These methods, such as radial basis function-generated finite differences (RBF-FD) or RBF-generated partition of unity methods (RBF-PUM), promise to develop into general-purpose meshless techniques for the numerical solution of partial differential equations that inherit the ease of implementation of the finite difference method, and yet potentially possess a greater ability than the finite element method to fit any geometry or adapt to singularities or other features of the solution.

The numerical evidence collected in recent years by a rapidly growing community of researchers suggests that these methods combine numerical stability on irregular node layouts, high computational speed, high accuracy, easy local adaptive refinement, and excellent opportunities for... (more)

The complex dynamical behavior of large pedestrian crowds has long fascinated researchers from various scientific fields. Academic studies began in earnest in the last century, starting with empirical observations in the early 1950’s and continuing with the development of models in the field of applied physics. In more recent years, applied mathematicians have become increasingly interested in the analytical aspects and computational challenges related to simulation of existing models. With ongoing technical development, more and more data such as pedestrian trajectories and velocities have become available, leading to new questions of calibration of the mathematical models.

Since the inception of the field of study, several scientific communities have been independently working on the challenge of describing and simulating pedestrian dynamics. While mathematicians have mainly focused on the modeling and analytical aspects, physicists have developed experimental setups and methods... (more)

The problem of decomposing a function into a sum of simply structured functions is a classical area of research in Analysis. Exciting recent progress, e.g. the solution to the Kadison-Singer problem, results about exponential frames and Riesz bases in various settings, and results about orthogonal exponential bases for convex polytopes, has re-energized discussion in this area, opened new directions for study, and turned it into an even more active and fruitful area for research. The goal of this workshop is to discuss such new developments in this area. In particular, the workshop will focus on problems regarding exponential systems in weighted spaces and the Fuglede conjecture. Related settings will also be of interest, for example: (i) Systems of vectors obtained by translating, translating and modulating, or translating and dilating a single function over the line; (ii) Sampling and decomposition of functions in the finite dimensional setting; (iii) Sampling and interpolation of... (more)

Imagine spending eight-weeks on the beautiful Brown University campus in historic Providence, RI, working in a small team setting to solve mathematical research problems developed by faculty experts in their fields.

Imagine creating career-building connections between peers, near peers (graduate students and postdocs), and academic professionals.

Imagine spending your summer in a fun, memorable, and intellectually stimulating environment.

Now, imagine having this experience with support for travel within the U.S., room and board paid, plus a $3,500 stipend*.

The 2018 Summer@ICERM program at Brown University is an eight-week residential program designed for a select group of 16-20 undergraduate scholars from around the world.

The faculty advisers will present a variety... (more)

Fractional partial differential equations (FPDEs) are emerging as a powerful tool for modeling challenging multiscale phenomena including overlapping microscopic and macroscopic scales. Compared to integer-order PDEs, the fractional order of the derivatives in FPDEs may be a function of space and time or even a distribution, opening up great opportunities for modeling and simulation of multi-physics phenomena, e.g. seamless transition from wave propagation to diffusion, or from local to non-local dynamics. In addition, data-driven fractional differential operators may be constructed to fit data from a particular experiment or specific phenomenon, including the effect of uncertainties. FPDEs lead to a paradigm shift, according to which data-driven fractional operators may be constructed to model a specific phenomenon instead of the current practice of tweaking free parameters that multiply pre-set integer-order differential operators. This workshop will cover all these areas, including... (more)

Forward simulations of the propagation and scattering of transient electromagnetic (EM) waves in complex media are important in a variety of applications, such as radar, environmental and medical imaging, noninvasive detection of cancerous tumors, design of engineered composites such as metamaterials, communication and computation, and global climate assessment, among others. These applications involve multiple spatial and temporal scales, complex geometries, spatial and temporal heterogeneities, and stochastic effects at small scales.

Biological tissues are complex media with inhomogeneous and frequency dependent (dispersive) properties. Analyses of EM wave interactions with biological media is fundamental in many medical applications, such as noninvasive diagnosis techniques, and for advancing the quality of medical imaging in general. Characterization of EM wave interaction with natural media is of great importance for environmental remote sensing and global climate assessment.... (more)

SageMath (sometimes Sage for short) is an open-source, general purpose mathematical software based on the Python programming language. It was created in 2005 by William Stein as a viable alternative to commercial software with an active and established community. SageMath has a broad library of functions useful to mathematicians in many fields, including combinatorics and representation theory. The welcoming and engaged community of users and contributors helps to create an environment of collaboration in both software development and mathematical research, leading to SageMath being cited in over 300 papers.

The study of the representation theories of certain algebras (e.g., Lie algebras, Hecke algebras, Khovanov–Lauda–Rouquier (KLR) algebras, quantum groups, etc.) also amounts to understanding the associated combinatorics. This has exposed deep connections between the associated representation theory and other areas of... (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 who live in greater Rhode Island and who will be entering the 10th or 11th grade in the fall of 2018.

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

Modern data analysis presents a variety of challenges, including the size, the dimensionality, the complexity, and the multiple-modality of the data. In an attempt to keep pace with these growing challenges, data scientists combine tools inspired from mathematics, from computer science, and from statistics. This TRIPODS Summer Bootcamp will provide attendees a hands-on introduction to emerging techniques for using topology with machine learning for the purpose of data analysis.

Topological and machine learning techniques potentially play complimentary roles for analyzing data. In topological data analysis, one leverages the fact that the shape of the data often reflects important and interpretable patterns within, although topological techniques alone typically cannot match the predictive power of machine learning. By contrast, machine learning algorithms provide state-of-the-art accuracies on predictive tasks, but the manner by which they arrive at a prediction is often difficult to... (more)

Building Community in the Foundations of Data Science

Brown's NSF TRIPODS grant is sponsoring a two-day informal networking workshop for the greater New England Foundations of Data Science community. In a series of informal discussions and short talks, we would like to draw attention to the opportunities to collaborate in foundational questions that lie at the focus of our TRIPODS program:

• structure of large and complex networks
• causal inference
• geometry and topology of data

We invite short talks on how these or other foundational or methodological data science themes appear in ongoing research projects in your work. We will spend the afternoon of the second day engaging in brainstorming for how collaborative structures across institutions can build and strengthen data science activity in the region.

This workshop is part of the Brown Data Science... (more)

Partial differential equations (PDEs) have long played crucial roles in the field of fluid dynamics. These PDE models, including Euler and Navier-Stokes equations for incompressible and compressible flows, kinetic equations for rarefied flows, and equations for more complex flows such as magneto-hydrodynamics flows, have motivated numerous studies from the theory of PDEs to the design and analysis of computational algorithms, and their implementation and application in computational fluid dynamics (CFD). This discipline is continually and dynamically evolving, constantly bringing forward new results in PDE theory, computation, and application to CFD, and also setting up the ground for generalizations to other related applications including electro-magnetics, fluid-structure interactions, cosmology, and computational electronics.

The aim of this workshop is to review the recent progress in the type of PDEs arising from fluid dynamics and other related physical areas, in terms of their... (more)