The goal of this semester program is to bring together a variety of mathematicians with researchers working in theoretical and computational neuroscience as well as some theory-friendly experimentalists. However, unlike programs in neuroscience that emphasize connections between theory and experiment, this program will focus on building bridges between theory and mathematics. This is motivated in part by the observation that theoretical developments in neuroscience are often limited not only by lack of data but also by the need to better develop the relevant mathematics. For example, theorists often rely on linear or near-linear modeling frameworks for neural networks simply because the mathematics of nonlinear network dynamics is still poorly understood. Conversely, just as in the history of physics, neuroscience problems give rise to new questions in mathematics. In recent years, these questions have touched on a rich variety of fields including geometry, topology, combinatorics,... (more)

One of the fundamental questions in neuroscience is to understand how network connectivity shapes neural activity. Over the last 10 years, tremendous progress has been made in collecting neural activity and connectivity data, but theoretical advances have lagged behind. This workshop will focus on identifying mathematical challenges that arise in studying the dynamics of learning, memory, plasticity, decision-making, sequence generation, and central pattern generator circuits. Mathematical ideas and approaches from dynamical systems, statistical mechanics, linear algebra, graph theory, topology, and traditional areas of applied mathematics are all expected to play an important role.

In the last decade or so, applied topology and algebraic geometry have come into their own as vibrant areas of applied mathematics. At the same time, ideas and tools from topology and geometry have infiltrated theoretical and computational neuroscience. This kind of mathematics has shown itself to be a natural and useful language not only for analyzing neural data sets, but also as a means of understanding principles of neural coding and computation. This workshop will bring together leading researchers at the interfaces of topology, geometry and neuroscience to take stock of recent work and outline future directions. This includes a focus on topological data analysis (persistent homology and related methods), topological analysis of neural networks and their dynamics, topological decoding of neural activity, evolving topology of dynamic networks (e.g., networks that are changing as a result of learning), and analysis of connectome data. Related topics may include the geometry and... (more)

Cracking the neural code is one of the longstanding questions in neuroscience. How does the activity of populations of neurons represent stimuli and perform neural computations? Decades of theoretical and experimental work have provided valuable clues about the principles of neural coding, as well as descriptive understandings of various neural codes. This raises a number of mathematical questions touching on algebra, combinatorics, probability, and geometry. This workshop will explore questions that arise from sensory perception and processing in olfactory, auditory, and visual coding, as well as properties of place field codes and grid cell codes, mechanisms for decoding population activity, and the role of noise and correlations. These questions may be tackled with techniques from information theory, mathematical coding theory, combinatorial commutative algebra, hyperplane arrangements, oriented matroids, convex geometry, statistical mechanics, and more.

This workshop focuses on connections between higher-order statistics and symmetric tensors, and their applications to machine learning, network science and other domains. Higher-order statistics refers to the study of correlations between three or more covariates. This is in contrast to the usual mean and covariance, which are based on one and two covariates.

Higher-order statistics are needed to characterize complex data distributions, such as mixture models. Symmetric tensors, meanwhile, are multi-dimensional arrays. They generalize covariance matrices and affinity matrices, and can be used to represent higher-order correlations. Tensor decompositions extend matrix factorizations from numerical linear algebra to multilinear algebra. Recently tensor-based approaches have been become more practical, due to the availability of bigger datasets and new algorithms.

The workshop brings together applied mathematicians, statisticians, probabilists, machine learning experts and computational... (more)

This semester program will bring together both leading experts and junior researchers to discuss the current state-of-the-art and emerging trends in computational PDEs. While there are scores of numerical methodologies designed for a wide variety of PDEs, the program will be designed around three workshops each centered around a specific theme: PDEs and Geometry, Nonlocal PDEs, and Numerical Analysis of Multiphysics problems. This grouping of topics embodies a broad representation of computational mathematics with each set possessing their own skill set of mathematical tools and viewpoints. Nonetheless, all workshops will have the common theme of using rigorous mathematical theory to develop and analyze the convergence and efficiency of numerical methods. The diversity of the workshop topics will bring together historically distinct groups of mathematicians to interact and facilitate new ideas and breakthroughs.

It is practically rare that a natural phenomenon or engineering problem can be accurately described by a single law of physics. The striking diversity of rules of life forces scientists to continuously increase the complexity of models to address the ever-growing requirements for their prediction capabilities. It remains a formidable challenge to derive and analyze numerical methods which are universal enough to handle complex multiphysics problems with the same ease and efficiency as traditional methods do for textbook PDEs.

The workshop will focus on recent trends in the field of numerical methods for multiphysics problems that include the development of monolithic approaches, structure preserving discretizations, geometrically unfitted methods, data-driven techniques, and modern algebraic methods for the resulting linear and nonlinear discrete systems. The topics of interest include models and discretizations for fluid - elastic structure interaction, non-Newtonian fluids, phase... (more)

The development and analysis of numerical methods for PDEs whose formulation or interpretation is derived from an underlying geometry is a persistent challenge in numerical analysis. Examples include PDEs posed on complicated manifolds or graphs, PDEs that describe interactions across complex interfaces, and equations derived from intrinsically geometric concepts such as curvature-driven flows or highly nonlinear Monge-Ampere equations arising in optimal transport. In recent years, these PDEs have gained significance in diverse areas such as machine learning, optical design problems, meteorology, medical imaging, and beyond. Hence, the development of numerical methods for this class of PDEs is poised to lead to breakthroughs for a wide range of timely problems. However, designing methods to accurately and efficiently solve these PDEs requires careful consideration of the interactions between discretization methods, the PDE operators, and the underlying geometric properties.

This... (more)

This workshop focuses on the modeling, analysis, approximation, and applications of nonlocal equations, which have raised new challenges to mathematical modeling, numerical analysis, and their computational implementation. Recent applications include, but are not limited to: heat and mass diffusion, mechanics, pattern formation, image processing, self-organized dynamics, and population dispersal.

Invited speakers and participants will bring expertise from a wide range of related fields, including mathematical and numerical analysis of nonlocal and fractional equations, numerical methods and discretization schemes, multiscale modeling, adaptivity, machine learning, software implementation, peridynamics modeling of material failure and damage, nonlocal and fractional modeling of anomalous heat and mass diffusion, and several engineering and scientific applications in which nonlocal modeling is useful.

Systems of interacting particles or agents are studied across many scientific disciplines. They are used as effective models in a wide variety of sciences and applications, to represent the dynamics of particles in physics, cells in biology, people in urban mobility studies, but also, more abstractly in the context of mathematics, as sample particles in Monte Carlo simulations or parameters of neural networks in machine learning.

This workshop aims at bringing together researchers in analysis, computation, inference, control and applications, to facilitate cross-fertilization and collaborations.

Certain problems in mathematics, physics, and engineering are formulated as minimizing cost functions that take as input a set of points on a compact manifold. In applied and computational harmonic analysis one is usually interested in finding tight frames and equiangular tight frames, which are respectively minimizers of different cost functions. In quantum information theory, the study of SIC-POVMS is equivalent to the existence of a point configuration made of antipodal points on a complex sphere. There seems to be a phenomenon where highly symmetric configurations are optimizers and optimizers often exhibit (partial) symmetries. The theory of spherical designs in combinatorics and discrete geometry with applications in approximation theory in the form of cubature formulas is deeply related to point configurations and distributions. Training a neural network involves minimizing a cost function relating to the desired task; it was recently discovered that doing so often results in... (more)