Combinatorial algebraic geometry comprises the parts of algebraic geometry where basic geometric phenomena can be described with combinatorial data, and where combinatorial methods are essential for further progress.

Research in combinatorial algebraic geometry utilizes combinatorial techniques to answer questions about geometry. Typical examples include predictions about singularities, construction of degenerations, and computation of geometric invariants such as Gromov-Witten invariants, Euler characteristics, the number of points in intersections, multiplicities, genera, and many more. The study of positivity properties of geometric invariants is one of the driving forces behind the interplay between geometry and combinatorics. Flag manifolds and Schubert calculus are particularly rich sources of invariants with positivity properties.

In the opposite direction, geometric methods provide powerful tools for studying combinatorial objects. For example, many deep properties of... (more)

The Nobel-Prize-winning detection of gravitational waves from binary black hole systems in 2015 by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the LIGO Scientific Collaboration has opened a new window on the universe. In addition, the 2017 observation of both gravitational and electromagnetic waves emitted by a binary neutron star system marked a new era of multi-messenger astronomy. While these successes are a remarkable experimental feat, they also constitute a significant computational achievement due to the crucial role played by accurate numerical models of the astrophysical sources in gravitational-wave data analysis. As current detectors are upgraded and new detectors come online within an international network of observatories, accurate, efficient, and advanced computational methods will be indispensable for interpreting the diversity of gravitational wave signals. This semester program at ICERM will emphasize the fundamental mathematical and... (more)

Today's computational and experimental paradigms feature complex models along with disparate and, frequently, enormous data sets. This necessitates the development of theoretical and computational strategies for efficient and robust numerical algorithms that effectively resolve the important features and characteristics of these complex computational models. The desiderata for resolving the underlying model features is often application-specific and combines mathematical tasks like approximation, prediction, calibration, design, and optimization. Running simulations that fully account for the variability of the complexities of modern scientific models can be infeasible due to the curse of dimensionality, chaotic behavior or dynamics, and/or overwhelming streams of informative data.

This semester program focuses on both theoretical investigation and practical algorithm development for reduction in the complexity - the dimension, the degrees of freedom, the data - arising in these... (more)

The Illustrating Mathematics program brings together mathematicians, makers, and artists who share a common interest in illustrating mathematical ideas via computational tools.

The goals of the program are to:

• introduce mathematicians to new computational illustration tools to guide and inform their research;
• spark collaborations among and between mathematicians, makers and artists;
• find ways to communicate research mathematics to as wide an audience as possible.

The program includes week-long workshops in Geometry and Topology, Algebra and Number Theory, and Dynamics and Probability, as well as master courses, seminars, and an art exhibition.

Mathematical topics include: moduli spaces of geometric structures, hyperbolic geometry, configuration spaces, sphere eversions, apollonian packings, kleinian groups, sandpiles and tropical geometry, analytic number theory, supercharacters, complex dynamics, billiards, random walks, and Schramm–Loewner... (more)

Computer vision is an inter-disciplinary topic crossing boundaries between computer science, statistics, mathematics, engineering and cognitive science.

Research in computer vision involves the development and evaluation of computational methods for image analysis. This includes the design of new theoretical models and algorithms, and practical implementation of these algorithms using a variety of computer architectures and programming languages. The methods under consideration are often motivated by generative mathematical models of the world and the imaging process. Recent approaches also rely heavily on machine learning techniques and discriminative models such as deep neural networks.

Problems that will be considered in the program include image restoration, image segmentation, object recognition and 3D reconstruction. Current approaches to address these problems draw on a variety of mathematical and computational topics such as stochastic models, statistical methods,... (more)

The theory, algorithms, and software of linear algebra are familiar tools across mathematics, the applied sciences, and engineering. This ubiquity of linear algebra masks a fairly recent growth of nonlinear algebra in mathematics and its applications to other disciplines. The proliferation of nonlinear algebra has been fueled by recent theoretical advances, efficient implementations of core algorithms, and an increased awareness of these tools.

The benefits of this nonlinear theory and its tools are manifold. Pushing computational boundaries has led to the development of new mathematical theories, such as homotopy methods for numerical algebraic geometry, tropical geometry and toric deformations, and sums of squares methods for polynomial optimization. This uncovered many concrete nonlinear mathematical objects and questions, many of which are ripe for computer experimentation. In turn, resulting mathematical breakthroughs often lead to more powerful and efficient algorithms... (more)

The arrangement of point configurations in metric spaces, whether deterministic or random, is a truly interdisciplinary topic of great interest in mathematics, physics and computer science. Mathematical aspects involve optimization, discretization of manifolds, best packing and cubature, among others. For physics, such configurations arise in the study of crystallization, point processes connected with random matrices, self-assembling materials, jammed states, hyperuniformity and phase transitions. For computer science, extremal point configurations play a fundamental role in coding and information theory, and lattice-based protocols in cryptography and related computational complexity issues are of growing importance. Furthermore, there has been recent and substantial progress on related age-old problems (such as the Kepler conjecture).

The investigation of the above topics often evolves from the development of efficient computational methods that enable extensive numerical... (more)

Inversion and imaging with waves is of fundamental importance in both radar and seismic reconstruction. Mathematics provides the key technology in both areas and, despite differing in many important respects, they have much in common in their underlying mathematical frameworks, approaches, and challenges. This semester program will focus on advancing their common mathematical and computational methodologies, as well as selected subjects distinct to each area, in the context of new challenges and opportunities that have arisen in recent years. Both theory and applications will be of interest. Participants will be drawn from academia, industry, and governmental laboratories in order to broadly address theory, applications, and their synergy.

The program will be influenced by recent developments in wave propagation and imaging, data acquisition and analysis, and high-performance computing. Driven by the ongoing need for more realistic mathematical models and simulations, recent advances... (more)

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)

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)

Since its introduction by Felix Hausdorff in 1919, the concept of the Hausdorff dimension of sets and measures has been a versatile and powerful tool in classical analysis, geometry and geometric measure theory, mathematical physics and their numerous applications. However, there has been a particularly important symbiosis between dynamical systems and dimension theory. This connection arises both from application of dimension theory to the classification and geometric analysis of dynamical systems (and their invariant sets and measures), and the fact that many classical objects of study in mathematics arise from (sometimes implicit) dynamical systems, which often play a role in the dimension theory of said objects.

Recently, there has been substantial progress on a number of central problems in dimension theory, and while many old problems remain, many new ones have also presented themselves. These include a deeper understanding of the relationship between dimension, entropy and... (more)

In the late 1960s, Robert Langlands discovered a unifying principle in number theory providing a vast generalization of class field theory to include nonabelian extensions of number fields. This principle gives rise to a web of conjectures called the Langlands program which continues to guide research in number theory to the present day. For example, an important first instance of the Langlands program is the modularity theorem for elliptic curves over the rational numbers, an essential ingredient in the proof of Fermat's last theorem.

Despite its many successes, the Langlands program remains vague in many of its predictions, due in part to an absence of data to guide a precise formulation away from a few special cases. In this thematic program, we will experiment with and articulate refined conjectures relating arithmetic-geometric objects to automorphic forms, improve the computational infrastructure underpinning the Langlands program, and assemble additional supporting... (more)