Programs & Events
Robust Discretization and Fast Solvers for Computable Multi-Physics Models
May 12 - 16, 2014
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)
Organizing Committee
- Franco Brezzi
- Jan Hesthaven
- Michael Holst
- Jinchao Xu
Eigenvectors in graph theory and related problems in numerical linear algebra
May 5 - 9, 2014
The analysis of problems modeled by large graphs is greatly hampered by a lack of efficient computational tools. The purpose of the workshop is to explore possibilities for designing appropriate computational methods that draw on recent advances in numerical methods and scientific computation. Specifically, the questions of how to form the matrices representing graph Laplacians, and how to compute the leading eigenvectors of such matrices will be addressed. It seems likely that these problems will be amenable to algorithms based on randomized projections that dramatically reduce the effective dimensionality of the underlying problems. Such techniques has recently proven highly effective for the related problems of how to find approximate lists of nearest neighbors for clouds of points in high dimensional spaces, and for constructing approximate low-rank factorizations of large matrices. In both cases, a key observation is that the problem of distortions of distances that is inherent to... (more)
Organizing Committee
- Anna Gilbert
- Peter Jones
- Gunnar Martinsson
- Van Vu
Research Cluster: Towards Efficient Algorithms Exploiting Graph Structure
Apr 24 - May 3, 2014
This working group will develop new theoretically grounded approaches to practical problems on graphs and networks using the arsenal of graph structure theory and algorithms (treewidth, minors, fixed-parameter tractability, approximation algorithms, etc.).
Our approach is to combine the expertise of a mix of junior and senior researchers from three disciplines: mathematics (graph theory), computer science (fixed-parameter and approximation algorithms), and applied network analysis (social networks, power grid, bioinformatics, etc.). During this research cluster, we will identify specific practically motivated problems, and tackle the key associated mathematical challenges, with a goal of ultimately encouraging broader adoption of graph-structure-based tools in the computational community. This goal is particularly important given the emergence of vast quantities of relational data (a.k.a. networks) and increased need for analysis via scalable algorithms.
We face several challenges in... (more)
Organizing Committee
- Erik Demaine
- Daniel Marx
- Blair Sullivan
Electrical Flows, Graph Laplacians, and Algorithms: Spectral Graph Theory and Beyond
Apr 7 - 11, 2014
Spectral graph theory, which studies how the eigenvalues and eigenvectors of the graph Laplacian (and other related matrices) interact with the combinatorial structure of a graph, is a classical tool in both the theory and practice of algorithm design. The success of this approach has been rooted in the efficiency with which eigenvalues and eigenvectors can be computed, and in the surprisingly large number of ways that a graph's properties are connected to the Laplacian's spectrum---particularly to the value of its second smallest eigenvalue, λ2.
However, while the eigenvalues and eigenvectors of the Laplacian capture a striking amount of the structure of the graph, they certainly do not capture all of it. Recent work in the field suggests that we have only scratched the surface of what can be done if we are willing to broaden our investigation to include more general linear-algebraic properties of the matrices we associate to graphs.
A particularly fruitful example of this... (more)
Organizing Committee
- Jonathan Kelner
- Ioannis Koutis
- Gary Miller
Stochastic Graph Models
Mar 17 - 21, 2014
Random graphs, stochastic processes on graphs and algorithms for computations on these structures continue to play a dominant role in algorithmic research and discrete mathematics, with recent applications ranging from web search and recommendation engines to social networks and system biology.
This workshop will be an opportunity for researchers from diverse fields to get together and share problems and techniques for handling and analyzing graphs structures. The connections---mathematical, computational, and practical---that arise between these seemingly-diverse problems and approaches will be emphasized.
Organizing Committee
- Susanne Albers
- Ravi Kumar
- Michael Mitzenmacher
- Eli Upfal
Research Cluster: Graphs with incomplete information
Feb 17 - Mar 15, 2014
How can we handle graph problems when the graph is only known imperfectly?
In one setting, the input is a noisy version of some unknown ground truth graph, to which random edges have been added, destroying the structure : planarity, clustering, distances for example. In another setting, the graph itself can only be accessed via queries such as shortest path queries, distance queries, or cut queries, and must be inferred from the result to well-chosen queries ; this comes up in internet tomography. In a third setting, the graph evolves dynamically over time and solutions must adapt to edge additions and removals.
The cluster will gather researchers around a bi-weekly working group drawing on the skills of the participants in random graphs and discrete probability, optimization and linear, semi-definite or convex programming methods, structural graph properties, and randomized dynamic data structures.
Organizing Committee
- Claire Mathieu
Semidefinite Programming and Graph Algorithms
Feb 10 - 14, 2014
Semidefinite programming is playing an ever increasing role in many areas of computer science and mathematics, including complexity theory, approximation algorithms for hard graph problems, discrete geometry, machine learning, and extremal combinatorics.
This workshop will bring together researchers from these different fields. The goal is to explore connections, learn and share techniques, and build bridges.
Organizing Committee
- Monique Laurent
- David Phillips
- David Steurer
- Kilian Weinberger
Network Science and Graph Algorithms
Feb 3 - May 9, 2014
The study of computational problems on graphs has long been a central area of research in computer science. However, recent years have seen qualitative changes in both the problems to be solved and the tools available to do so. Application areas such as computational biology, the web, social networks, and machine learning give rise to large graphs and complex statistical questions that demand new algorithmic ideas and computational models. A wide variety of techniques are emerging for addressing these challenges: from semidefinite programming and combinatorial preconditioners.
In addition to three international conferences, the program will support several research clusters, concentrated periods of activity organized around a specific and timely approach to graph algorithms.
Organizing Committee
- Andrea Bertozzi
- Jonathan Kelner
- Philip Klein
- Claire Mathieu
- David Shmoys
- Eli Upfal
Research Cluster: Geometric analysis methods for graph algorithms
Feb 3 - 28, 2014
This working group will develop new mathematics at the interface between graph structures and high dimensional data and geometric analysis. In the last ten years we have seen an explosion of work in both (a) compressive sensing (sparsity, L1-based methods) and in (b) machine learning involve graphical structures for large scale and high dimensional data. The focus is on both analysis and algorithm development. In the case of new algorithms - codes will be tested against state of art machine learning algorithms. In the case of analytical results - we will draw on expertise in diverse areas of mathematics including differential geometry, nonlinear PDE, optimization, and spectral analysis of graphs. Application areas represented include machine learning, social network data, modularity optimization, L1-compressive sensing methods, and image processing.
One area of focus is community detection in large networks. A current approach for community detection consists in minimizing the... (more)
Organizing Committee
- Andrea Bertozzi
- Thomas Laurent
From the Clinic to Partial Differential Equations and Back: Emerging challenges for Cardiovascular Mathematics
Jan 20 - 24, 2014
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)
Organizing Committee
- Pablo Blanco
- Leopold Grinberg
- John Oshinski
- Anne Robertson
- W. Robert Taylor
- Alessandro VENEZIANI
Public Lecture: "Toy Models"
Nov 21, 2013
One of Tadashi Tokieda's lines of activity is inventing, collecting, and studying toys -- objects from daily life that can be found or made in minutes, yet which, if played with imaginatively, reveal behaviors so surprising that they intrigue scientists for weeks.
During this unique talk, Tokieda will display, demonstrate and discuss several toys. Some of the toys will be known but revisited, some will be original, and all will be surprising to mathematicians/physicists and amusing to everyone else.
Geometric Structures in Low-Dimensional Dynamics
Nov 18 - 22, 2013
This workshop will present topics in low-dimensional dynamics such as billiards, flows on flat surfaces, dynamics on moduli spaces, and piecewise isometric maps. One theme in the workshop will be the appearance of geometric structures such as hyperbolic space and Teichmüller space in connection with dynamical systems which are basically defined in terms of the Euclidean plane. Computer experiments are common in these areas, and will be discussed, but the emphasis will be on the mathematics that comes out of the experiments.
Organizing Committee
- Moon Duchin
- Pascal Hubert
- Howard Masur
- Richard Schwartz
- Anton Zorich