COMS

The geometry of moduli spaces has been widely investigated and it is certainly one of the main subjects in Algebraic Geometry. In particular, moduli spaces of curves gave a major boost to the theory due to their many connections with Theoretical Physics and counting invariants. However, to count invariants it is very useful to apply intersection theory techniques that require the space to be proper, and moduli spaces are usually not proper. The problem of compactifying them is not trivial. Geometric Invariant Theory (GIT) may be a way to tackle the problem, but the limit points can be hard to describe geometrically.

Nowadays, new notions of stability and new techniques related to the minimal model program have been combined with GIT to provide new points of view to attack the problem and, more generally, to study the birational geometry of compactified moduli spaces. This has brought back many important insights again about counting invariants and mirror symmetry and has also been used to study other moduli problems, like the moduli of coherent sheaves or the moduli spaces of higher dimensional varieties.

The lecturers have been chosen among the leading experts of the area. Following the usual policy of the Advanced Schools at CRM, informal notes will be delivered by the speakers to the participants at the beginning of the school.

The goals of the Summer School are to encourage more developing scientists to become interested in this field and to encourage communication between mathematical modellers and public health scientists and epidemiologists who have been historically unaware of the uses of mathematical modeling.

In its simplest form, Hodge theory is the study of periods - integrals of algebraic differential forms which arise in the study of complex geometry, number theory and physics. Its difficulty and richness arises in part from the non-algebraicity of these integrals. According to the beautiful conjectures of Hodge, Bloch and Beilinson, what algebraic structure they have should be explained by (generalized) algebraic cycles. There has been much recent progress on these conjectures and on classifying spaces for periods, as well as their asymptotic and arithmetic; this conference will bring together leading scholars and students of these topics.

Given that our desire is to create a number of new collaborations, we would aim at a small number of high-quality talks per day with ample breaks between each. In addition to bringing together leading scholars on the different aspects of the asymptotic, symmetries, and arithmetic of periods together for a research conference, the organizers will also arrange for a prior workshop for graduate students and recent Ph.D.'s on major themes of the conference. We expect this expository part to run approximately 4 days, with lectures by the organizers and some of the invited speakers.

This workshop will bring together researchers from these different fields. The goal is to explore connections, learn and share techniques, and build bridges.

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.

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 has been the study of Laplacian linear systems, where the interplay between linear algebra and graph theory has led to progress in both fields. On the one hand, researchers have used the combinatorial structure of the corresponding graphs to facilitate the solution of these linear systems, resulting in solvers that run in nearly-linear time. On the other hand, one can use these linear systems to describe the behavior of electrical flows on a graph, which has provided a powerful new primitive for algorithmic graph theory. This interaction has already led to improved algorithmic results for many of the basic problems in algorithmic graph theory, including finding maximum flows and minimum cuts, solving traveling salesman problems, sampling random trees, sparsifying graphs, computing multicommodity flows, and approximately solving a wide range of general clustering and partitioning problems. In addition, researchers have recently shown how to exploit a wide range of other algebraic properties of matrices associated to graphs, such as the threshold rank, cut norm, sensitivity to perturbation, or hypercontractivity of the eigenspaces, to achieve impressive algorithmic results.

In this workshop, we will bring researchers together to study and advance this new emerging frontier in algorithmic graph theory.