This workshop focuses on certain kinds of discrete dynamical systems that are integrable and have interpretations in terms of cluster algebras. Some such systems, like the pentagram map and the octahedral recurrence, are motivated by concrete algebraic constructions (taking determinants) or geometric constructions based on specific configurations of points and lines in the projective plane. The systems of interest in this workshop have connections to Poisson and symplectic geometry, classical integrable PDE such as the KdV and Boussinesq equations and also to cluster algebras. The aim of the workshop is to explore geometric, algebraic, and computational facets of these systems, with a view towards uncovering new phenomena and unifying the work to date.

This workshop will focus on recent advances in combinatorial link homology theories (e.g., Heegaard-Floer homology and Khovanov homology), especially as they apply to questions about braids and, more generally, mapping class groups of surfaces. There will be short mini-courses on

1. Combinatorial knot Floer homology, with applications to contact geometry,
2. Braid foliations and the Jones conjecture,
3. Nielsen-Thurston theory, and
4. Garside theory and a linear order on the braid group,

along with a number of research and expository talks. The talks will emphasize the role that computation and experiment have thus far played in stating key conjectures and establishing key results. The workshop will culminate in a computational problem session, in which participants will discuss promising directions for future exploration and indicate which computations may be most useful in that exploration.

As the main goal of the workshop is to facilitate interaction... (more)

Over the past 25 years, experimental mathematics has developed as an important additional arrow in the mathematical quiver. Many mathematical scientists now use powerful symbolic, numeric and graphic (sometimes abbreviated "SNAG") computing environments in their research, in a remarkable departure from tradition. While these tools collectively are quite effective, challenges remain in numerous areas, including: (a) rapid, high-precision computation of special functions and their derivatives; (b) user-customizable symbolic computing; (c) graphical computing; (d) data-intensive computing; and (e) large-scale computing on parallel and GPU architectures (including algorithm and software design for such systems).

This workshop will convene mathematical and computer scientists who create or exploit these tools, together with computational tool developers and commercial vendors of mathematical software, to exchange approaches and extend the state of the art in the field, both in the design... (more)

Over the last two decades, algebraic and numerical techniques for nonlinear problems have begun a steady and relentless transition from mostly academic constructions, to widely used tools across the mathematical sciences, engineering and industrial applications. The workshop will bring together participants from many diverse fields including computer vision, cryptography, optimization and control, partial differential equations, robotics, and quantum computation, with the common interest in nonlinear algebraic computations. The main goal is to assess the state of the art, to stimulate further progress, and to accelerate developments by bringing together these diverse communities and have them share computational challenges and successes.

Most systems targeted by mathematical modeling in modern science and engineering are fundamentally multi-physical and multi-scale in nature. As such, they involve solving complex coupled, generally nonlinear, systems of partial differential equations (PDEs) built from subsystems of PDEs that mathematically model very different physical processes, often at very different scales.

Recent advances in high-performance computer hardware and advanced numerical algorithms have made it feasible to construct realistic mathematical models and to build corresponding numerical simulation software for these types of complex multi-physics/multi-scale problems. However, developing robust, efficient, and practical numerical algorithms for such simulation software that are capable of tackling these complex mathematical models is still extremely challenging in a number of fundamental ways. For example, we do not yet have robust methods that can handle strong coupling between different physics and/or... (more)

Mathematical models have been giving remarkable contributions in advancing knowledge and supporting decisions in several branches of medicine.

Some progress in applying predictive mathematical tools has been made, for example: surgical planning of the Total Cavopulmonary Connection in cardiac pediatrics is, in some hospitals, based on extensive numerical simulation. However, despite the significance, the impact of predictive modeling in the routine medical treatment falls behind.

The ultimate goal of this workshop is to foster collaboration between mathematicians and medical doctors on modeling cardiovascular system. The workshop is organized into two lines that reflect the special format of the workshop: (a) "Core topics" are up-to-date research areas in mathematics and scientific computing that still present several open exciting challenges, which can require developing new numerical models, computational approaches and validation techniques; (b) "New challenges" are a set of... (more)

Being central to gas dynamics, the Boltzmann equation describes gas flows at the microscopic level in regimes from free molecular to continuum. Its descriptive power makes it indispensable for predicting non-continuum phenomena in gases when experimental data is limited or not available. The Boltzmann equation is used in a wide range of applications, from external aerodynamics and thruster plume flows to vacuum facilities and microscale devices. Accurate solution of the Boltzmann equation for modeling gas flows arising in aerospace applications continues to be a challenge. Existing numerical capabilities fall short of capturing the complexities of engineering design. Reasons for this range from the absence of mathematical models that capture the physics properly to higher dimensionality of kinetic models and the resulting high cost of computations to the failure of mathematical theories to handle complex geometries of real life applications.

The goal of this workshop is to facilitate... (more)

In addition to advancing research and discovery in pure and applied mathematics, computation is pervasive across the sciences and now computational research results are more crucial than ever for public policy, risk management, and national security. Reproducibility of carefully documented experiments is a cornerstone of the scientific method, and yet is often lacking in computational mathematics, science, and engineering. Setting and achieving appropriate standards for reproducibility in computation poses a number of interesting technological and social challenges. The purpose of this workshop is to discuss aspects of reproducibility most relevant to the mathematical sciences among researchers from pure and applied mathematics from academics and other settings, together with interested parties from funding agencies, national laboratories, professional societies, and publishers. This will be a working workshop, with relatively few talks and dedicated time for breakout group discussions... (more)

Many important advances in material and biomedical science will come from controlling the chemical properties and nanoscale morphology of polymer mixtures. Predicting the longtime continuum-level properties of such complex systems poses a canonical computational challenge due to the disparate length and time scales separating the molecular description from the macroscopic behavior, particularly the evolution of morphology. This workshop focuses on four overlapping approaches to bridging this gap: Accelerated Molecular Methods, Coarse-Graining of Molecular Dynamics, Computational Approaches to Self-Consistent Mean Field, and Coupled Molecular and Continuum-Variational models. The goal is to spur the development of hybrid computational methods with the capacity to identify and characterize the rare events and the driving forces which steer the systems towards equilibrium, and connect the burgeoning growth in parallel-computation techniques for particle-based systems with recently... (more)

FEEC is a recent advance in the mathematics of finite element methods that employs differential complexes to construct stable numerical schemes for several important types of application problems. It has aroused great interest because it both presents interesting mathematical problems and shows great potential for application in computational science and engineering. The concentrated sequence of lectures in this program will provide participants with an understanding of the mathematical tools required to fully grasp the concepts in FEEC. ICERM is pleased to host this NSF-CBMS Regional Research Conference.

Douglas Arnold will deliver 10 lectures, one in the morning and one in the afternoon of each day of the workshop. Richard Falk and Anil Hirani will both give one presentation.

Limited funding is available to support participants.

The theme of this workshop is the computation, modeling, and mathematical analysis of heterostructured nanocrystalline materials. This includes quantum dots, nanowires, graphene, and grain boundaries. These various phenomena will be discussed in the context of modeling and computation on different scales ranging from density functional theory to continuum mechanics. The workshop will also address various techniques that allow one to combine models on different scales to yield efficient computational methods.

As concurrency in scientific computing pushes beyond a million threads and performance of individual threads becomes less reliable for hardware-related reasons, attention of mathematicians, computer scientists, and supercomputer users and suppliers inevitably focuses on reducing communication and synchronization bottlenecks. Though convenient for succinctness, reproducibility, and stability, instruction ordering in contemporary codes is commonly overspecified. This workshop attempts to outline evolution of simulation codes from today's infra-petascale to the ultra-exascale and to encourage importation of ideas from other areas of computer science into numerical algorithms, new invention, and programming model generalization.