Programs & Events
Computational Challenges in the Theory of Lattices
Apr 23 - 27, 2018
This workshop will focus on the computational aspects of the theory of Euclidean lattices and on their applications to other areas in mathematics and computer science. It will put emphasis on computational challenges on lattice problems that have recently arisen from unexpected connections to other domains such as algebraic topology, automorphic forms, or cryptography.
A major goal of this workshop is to bring together researchers from different areas, working with Euclidean lattices, and to facilitate their interactions.
Topics will include the reduction theory of lattices and its applications, Voronoi algorithms and their use to compute the cohomology of arithmetic groups, the classification of lattice genera and the computation of spaces of modular forms, the algorithmic aspects of lattice based cryptography, in particular the relationship between the security of cryptographic primitives and the hardness of lattice problems.
Organizing Committee
- Christine Bachoc
- Henry Cohn
- Renaud Coulangeon
- Christopher Peikert
- Akshay Venkatesh
Public Lecture: Crowd Computing: Scientific discoveries by protein-folding game players
Apr 11, 2018
Can the brainpower of humans worldwide be brought to bear on critical problems posed in computational biology, such as the structure determination of proteins and designing novel enzymes? Yes! Citizen scientists—most of whom have little or no prior biochemistry experience—have uncovered knowledge that eluded scientists for years. Players of the online protein folding video game Foldit have contributed to several scientific discoveries through gameplay. Rather than solving problems with a purely computational approach, combining humans and computers can provide a means for solving problems neither could solve alone.
Computation and Optimization of Energy, Packing, and Covering
Apr 9 - 13, 2018
The packing and covering of equal geometric shapes, such as spheres or convex polyhedra, are classical geometric optimization problems. They have a long mathematical tradition and were for instance part of Hilbert's famous twenty-three problems for the 20th century. Nevertheless, seemingly simple packing and covering problems are still extremely hard to solve and generally, far from a solution. Likewise, minimal energy problems for pair potentials, of which best-packing is a special case, have many such unresolved questions.
However, in recent years several new developments with computer assisted approaches have led to previously unexpected breakthrough results. These involve massive computer searches and techniques from numerical optimization, as well as the creation and application of new optimization techniques, such as specific semi-definite programming bounds. New techniques for computer assisted certified proofs allow one to obtain results that would otherwise have been... (more)
Organizing Committee
- Achill Schürmann
- Salvatore Torquato
- Frank Vallentin
- Robert Womersley
Fast Algorithms for Generating Static and Dynamically Changing Point Configurations
Mar 12 - 16, 2018
This workshop focuses on fast algorithms for the generation of high quality point configurations and meshes such as hierarchical schemes combined with energy or geometrical optimization techniques. Energy methods utilizing appropriate potentials for a prescribed density on a given manifold have been effective in generating point configurations with good covering and packing properties. These methods rely on efficient energy, gradient, and potential computations which can be achieved by hierarchical algorithms that model a system in a recursively compressed (low-rank or low-dimensional) form where information is transmitted non-locally on a hierarchical tree structure. Different aspects of this technique can be found in the classical FFT, multigrid, and fast multipole method (FMM), as well as the recently developed fast direct solvers, multilevel models in statistics, and convolutional neural networks in deep learning.
Fast generation of point configurations and meshes for dynamically... (more)
Organizing Committee
- Natasha Flyer
- Adrianna Gillman
- Doug Hardin
- Jingfang Huang
- Edward Saff
Public Lecture: Sleeping Beauty and Other Probability Conundrums
Feb 28, 2018
Probability puzzles are notoriously tricky, and the best ones continue to intrigue the public and confound philosophers. We'll examine some of these questions and try to determine whether they uncover real problems with the foundations of probability theory, or just challenges to our flawed human intuition.
Optimal and Random Point Configurations
Feb 26 - Mar 2, 2018
This workshop will focus on probabilistic and physical aspects of systems of interacting points: their statistical mechanics, phase transitions, and ground states. Such systems include random point processes arising in probability and statistical physics, such as random matrices, determinantal processes, zeros of random polynomials, disordered ground states, and hyperuniform systems as well as configurations satisfying a geometric or analytic optimality constraint. Special cases also involve disordered and ordered sphere packings and covering problems.
While systems of interacting particles, their free energy and crystallization properties have been studied for a long time in the statistical physics community, there has also been much activity recently, both in the random matrix community and probability communities and in the complex analysis community to understand the microscopic laws of eigenvalues of random matrices and points in beta-ensembles, as well as understanding and... (more)
Organizing Committee
- Peter Grabner
- Doug Hardin
- Arno Kuijlaars
- Sylvia Serfaty
Point Configurations in Geometry, Physics and Computer Science
Feb 1 - May 4, 2018
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)
Organizing Committee
- Christine Bachoc
- Henry Cohn
- Peter Grabner
- Doug Hardin
- Edward Saff
- Achill Schürmann
- Sylvia Serfaty
- Salvatore Torquato
- Robert Womersley
Geometry and Topology of Data
Dec 11 - 13, 2017
The scale, dimensionality, and complexity of large data has given rise to new topological and geometric methods for understanding what features in a data set are robust under perturbations of the system. Tools from algebraic topology and coarse geometry have been brought fruitfully to bear in a number of contexts leading to a surge of interest in persistent homology, combintorial geometry, and discrete Morse theory.
Likewise, new frameworks have emerged from harmonic analysis to develop diffusion geometries for large data, enabling multi-scale analyses, and other dynamical approaches to understanding complex data sets. Tools for enabling visualization of each of these methods are in development and increasingly granting researchers the ability to understand their data in new ways.
This workshop will bring together a broad range of researchers for a short workshop to attempt to set directions for future research. This workshop is part of the
Organizing Committee
- Jeffrey Brock
- Lorin Crawford
- Ani Eloyan
- Bjorn Sandstede
Recent Advances in Seismic Modeling and Inversion: From Analysis to Applications
Nov 6 - 10, 2017
This workshop will bring together academic and industrial researchers with the goal of addressing some of the key challenges in the analysis of seismic inverse problems, with emphases on reconstruction, big data and fast algorithms. We aim to facilitate interactions among scientists addressing all aspects of these problems, from analysts addressing such questions as stability and uniqueness through geophysicists developing new acquisition systems and applying cutting-edge ideas to field data sets. The workshop will place particular emphasis on fast algorithms that address the unique big-data requirements of seismic imaging from the reservoir to whole-Earth scale.
Specific topics will include analysis of seismic inverse problems leading to reconstruction (iterative or direct) from finite data; emerging acquisition technologies; uncertainty quantification; big-data simulation; inversion and model reduction, including compressed sensing; fast solvers in frequency and time; anisotropy;... (more)
Organizing Committee
- Maarten de Hoop
- Vladimir Druskin
- Alison Malcolm
- Alexander Mamonov
- Lexing Ying
Advances in model reduction for large-scale forward and inverse scattering problems
Nov 3, 2017
Model order reduction is a wide topic in computational mathematics that is generally used to approximate the response of complex systems. A recent development in this field is concerned with using reduced order models for solving efficiently and accurately inverse problems for partial differential equations. Such reduced order models are called data driven, because they are constructed from data interpolation conditions. Moreover, they are designed to respect the physics of the problem, such as loss of resolution away from the surface of measurements in diffusive inverse problems and causality conditions in inverse problems for the wave equation. This workshop will be focused on this novel approach to inversion, with particular emphasis on applications to inverse scattering problems arising in seismic imaging. The workshop will also celebrate the work of Dr. Vladimir Druskin, who has been making outstanding contributions to this field.
Organizing Committee
- Liliana Borcea
- Alexander Mamonov
- Shari Moskow
- Mikhail Zaslavskiy
Research Cluster: Wave Propagation and Inversion in Seismic Applications
Oct 23 - Nov 21, 2017
Seismic inversion is the process of transforming seismic data generated by active or passive sources into a quantitative description of the subsurface properties of the earth. It addresses important problems related to our energy needs, to hazards such as earthquakes and volcanic eruptions, and to the general study of Earth's interior on a planetary scale. It draws extensively from the mathematical sciences by applying tools from signal processing, elastic and electromagnetic theory, partial differential equations, harmonic analysis, inverse-problem theory, numerical analysis, optimization, and statistics.
The sheer volume of seismic data also makes it arguably the oldest area with "big data." Theoretical and engineering developments have advanced this field tremendously in the past several decades; however, there remain many fundamental open questions, ranging from uniqueness and uncertainty through the nonlinear nature of these problems.
This cluster will bring together academic... (more)
Organizing Committee
- Vladimir Druskin
- Alison Malcolm
- Lexing Ying
Musical Geometry, Games, and Multimedia Art
Oct 18, 2017
Members of the community are invited to attend a public lecture that will explore the deep connection between elementary musical concepts (chord, scale, voice leading) and basic concepts of recent geometry (quotient space, orbifold, tangent vector).
In his talk, Dmitri Tymoczko will review the principles underlying this connection and then demonstrate some recent applications of these ideas, including a large-scale multimedia work for orchestra and live video, in which the images and sounds explore the same geometrical relationships, musical games in which users directly interact with abstract musical geometries, and new musical instruments based on these same ideas.