Mathematics

Pseudo-Anosov maps arising from Penner’s construction

Balazs Strenner (Wisconsin Math)

Abstract: By the Nielsen-Thurston classification theorem, a generic surface homeomorphism is a pseudo-Anosov map, which, roughly speaking, stretches the surface in one direction by a factor and shrinks it in another direction by the same factor. Other than their importance in studying mapping class groups, pseudo-Anosov maps also have rich connections with 3-manifolds and Teichmuller spaces. Unlike their simpler analogues on the torus, which can easily be classified using matrix actions on the plane, pseudo-Anosov maps on higher genus surfaces are much harder to construct. Penner gave a very general construction for pseudo-Anosov maps, and he conjectured that virtually all pseudo-Anosov maps arise this way. This conjecture was known to be true on some simple surfaces, including the torus. Recently, a new approach to the conjecture was suggested by Shin, by connecting this topological question to a linear algebra problem. We discuss progress on the conjecture following this approach. View at http://youtu.be/QQx6HxI9pHQ

Structure-dynamics relation in networks: stability, optimization, and sensitive dependence on network structure

Takashi Nishikawa (Northwestern University)

Abstract: Much of the recent research in complex networks has been focused on establishing relations between network structure and dynamics as well as on exploring these relations for optimizing network processes. As a simple yet intriguing example of collective dynamics in networks, I will first focus on synchronization, in which individual dynamical units keep in pace with each other in a decentralized fashion, and discuss how the stability of such motion is affected by the network structure. Using networks having best synchronization properties, I will show that “less can be more” in networks: negative interactions as well as link removals can be used to systematically improve and optimize synchronization properties. I will then step beyond synchronization and demonstrate a rather general phenomenon: optimization can lead to cusp-like sensitive dependence of dynamics on network structural parameters, such as the number of nodes and links, as well as on the magnitude of generic structural perturbations. This phenomenon will be shown to be observed for a wide range of network dynamics, including Turing instability in activator-inhibitor systems and generator dynamics in power grids.

Prime Geodesic Theorems (Part I)

Melinda Lanius (UIUC Math)

Abstract: The prime number theorem tells us that the number of primes less than or equal to a positive real number $x$ grows asymptotically like $\frac{x}{\log x}$. In the context of differential geometry, mathematicians have proven similar results. A prime geodesic on a hyperbolic surface is a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime because their lengths have an asymptotic distribution similar to the prime number theorem. In this series of talks, I'll discuss some of the components that go into proving such results. No knowledge of manifolds is assumed. There will be plenty of examples and pictures.

The Haagerup property for arbitrary von Neumann algebras

Martijn Caspers (West-Fälische Wilhelms Universität, Münster)

Abstract: We introduce a natural generalization of the Haagerup property of a finite von Neumann algebra to an arbitrary von Neumann algebra M equipped with a normal, semi-finite, faithful weight. We prove that our notion is weight independent and hence is an property of M itself. We shall discuss stability properties of the Haagerup property regarding crossed products, free products and graph products. We also discuss noncommutative counterparts of the existence of a proper, continuous, conditionally negative definite function on a group that has the Haagerup property. The talk as based on joint work with Adam Skalski and partly on a joint project with Pierre Fima.

Bounds for the Hilbert-Samuel Multiplicity - Part II

Javid Validashti (UIUC Math)

Abstract: A classical inequality due to Lech states that the Hilbert-Samuel multiplicity of a zero-dimensional ideal in a regular local ring is bounded above by the normalized colength of that ideal. Simple examples show that this inequality is sharp asymptotically, but it gives a very weak bound for the Hilbert-Samuel multiplicity in general. In this talk, I will discuss improvements of Lech's inequality, which also yield stronger inequalities on the Hilbert coefficients of an ideal. Joint work with Craig Huneke (University of Virginia) and Ananth Hariharan (Indian Institute of Technology).

The Laplacian determinant for periodic planar graphs

Richard Kenyon (Brown University)

Abstract: The Laplacian on a periodic planar graph has a rich algebraic and integrable structure, which we usually don't see when we do standard potential theory. We discuss these combinatorial, algebraic and integrable features, and in particular interpret combinatorially the points of the "spectral curve" of the laplacian in terms of probability measures on spanning trees.