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)

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.

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.

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.

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.

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)

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)

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.

The mathematical focus of this workshop will include all aspects of the topology and geometry of low-dimensional manifolds and geometric group theory. It has been understood for over a century that these subjects are tightly connected, but the connections have become even deeper as the subjects have matured. Recent advances have given dramatic evidence of this. The workshop aims to further extend the interplay between these subjects.

Algorithms have been an important and consistent feature of all of these mathematical areas from the beginning. This includes both questions about the existence of algorithms and the development of practical algorithms for computing natural invariants. More recently, computer experiments and rigorous computer-assisted proofs have had a significant impact. It is natural to expect experimental and computational methods to play an expanding role in the theory of low dimensional spaces. Additional goals of the workshop are to explore the development of new... (more)

As part of the Mathematical Sciences Collaborative Diversity Initiatives, nine mathematics institutes (including ICERM) are pleased to host their annual pre-conference event, the 2013 Modern Math Workshop. This event precedes the SACNAS National Conference.

The Modern Math Workshop is intended to re-invigorate the focus of mathematics students and faculty at minority-serving institutions and the research careers of minority mathematicians.