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)

Limits of discrete random structures appear in different areas of probability, combinatorics, and machine learning. In statistical mechanics, probabilistic and combinatorial techniques are applied to rigorously describe the scaling limits of such random graphical models, which are closely related to phase transitions. In the vicinity of a phase transition, even a tiny change in some local parameter can result in dramatic changes in the macroscopic properties of the entire system. Random discrete structures are also useful mathematical models of large networks, which play a central role in our social and economic lives as the fabric over which we interact, form social connections, conduct economic transactions, transmit information, propagate disease, and much more.

The goal of this workshop is to integrate the algebraic combinatorics, probability, and machine learning paradigms of statistical mechanical models and to bring together researchers in related fields to discuss recent... (more)

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.

Estimators of various causal or statistical quantities are usually constructed with a particular target population in mind, that is, the population about which the investigators intend to draw inferences (e.g., decide on the implementation of a treatment strategy or use algorithm-derived predictions). Typically, however, the data used for estimation comes from a population that differs from the target population. How to ensure or evaluate whether the estimates generalize to the target population is a question that has received substantial attention in many scientific disciplines, but with the fields not always connecting with one another on overlapping challenges and solutions. This workshop will bring together experts from different disciplines to present state-of-the-science methods to address generalizability and discuss key challenges, and open problems.

Single-cell assays provide a tool for investigating cellular heterogeneity and have led to new insights into a variety of biological processes that were not accessible with bulk sequencing technologies. Assays generate observations of many different molecular types and a grand mathematical challenge is to devise meaningful strategies to integrate data gathered across a variety of different sequencing modalities. The first-order approach to do this is to analyze the projected data by clustering. Keeping more refined shape information about the data enables more meaningful and accurate analysis. Geometric methods include (i) Manifold learning: Whereas classical approaches (PCA, metric MDS) assume projection to a low-dimensional Euclidean subspace, manifold learning finds coordinates that lie on a not necessarily flat or contractible manifold. (ii) Topological data analysis: Algebraic topology provides qualitative descriptors of global shape. Integrating these descriptors across feature... (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 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 its 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.