Speaker: Kenshi Miyabe (Meiji University)

Title: Continuity of limit computable functions

Abstract:

The Darboux-Froda theorem says that, for every non-decreasing function,

the set of the points of non-continuity is at most countable.

This is probably the simplest case saying that every function

well-behaves at almost every point.

I introduce some similar theorems, and their computable versions.

Then, we discuss the relation with randomized algorithm.

Speaker: Andrew Sale (Cornell U.)

Title: On the outer automorphism groups of right-angled Artin and Coxeter groups

Abstract: In geometric group theory, a fundamental, and broad, question to answer is that of understanding the world of finitely presented groups. Two of the simplest examples are free groups Fn and free abelian groups Z^n. With Fn and Z^n being their extreme examples, right-angled Artin groups (RAAGs) give us some idea of what happens “between” these groups. RAAGs are an important class of groups which appear in diverse situations, perhaps most significantly in Agol’s proof of the virtual Haken conjecture.

In studying their outer automorphism groups, we are looking at a class of groups that again interpolates between two classically important families of groups: Out(Fn), the outer automorphism group of Fn, and GL(n,Z). While there are numerous similarities between these families, they also differ in some important contexts. One such context concerns the nature of quotients that they have, and I will describe a couple of properties that make rigorous the notion of “having many quotients”. I will explain what happens for outer automorphism groups of RAAGs, and also the closely related family of right-angled Coxeter groups, and the consequences this has for Kazhdan’s Property (T).

Speaker: Sam Nariman (Northwestern U.)

Title: On the homology of diffeomorphism groups made discrete.

Abstract: Let $G$ be a finite dimensional Lie group and $G^{delta}$ be the same group with the discrete topology. The classifying space $BG$ classifies principal $G$-bundles and the classifying space $BG^{delta}$ classifies flat principal $G$-bundles (i.e. those bundles that admit a connection whose curvature vanishes). The natural homomorphism from $G^{delta}$ to $G$ induces a continuous map from $BG^{delta}$ to $BG$. Milnor conjectured that this map induces an equivalence after the profinite completion. In this talk, we discuss the same map for infinite dimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms. In these cases, we use techniques from homotopy theory to show that the map from $BG^{delta}$ to $BG$ induces a split surjection on cohomology with finite coefficients in the stable range. If time permits, I will discuss applications of these results in foliation theory, in particular, characteristic classes of flat surface bundles.

Speaker: Chris Marks (Cal State Chico)

Title: Vector-valued modular forms and the bounded denominator conjecture

Abstract: This talk will primarily serve as an introduction to modular

forms, both scalar and vector-valued. After working through definitions

and the group theoretic background, I’ll discuss the so-called Bounded

Denominator Conjecture concerning Fourier coefficients of modular forms.

Speaker: Andreas Weinmann (Hochschule Darmstadt and Helmholtz Center Munich)

Title: Variational methods for the restauration of manifold-valued images and data

Abstract: Nonlinear manifolds appear as data spaces in various applications. One example in image processing is diffusion tensor imaging, where the data sitting in every voxel is a positive matrix representing the diffusibility of water molecules measured at the corresponding spatial location. Another example is color image processing, where instead of the RGB representation often other formats such as HSI or HSV are used which employ a circle to represent the hue of a color. A third example are registration problems (e.g., between a camera and an ultrasound devise) where time series of euclidean motions appear. Since the measured data is often noisy, regularization of these nonlinear data is necessary. In this talk, we propose algorithms for the variational regularization of manifold-valued data using non-smooth functionals. In particular, we deal with algorithms for TV regularization and with higher order methods including the TGV denoising of manifold-valued data. We present concrete applications in medical imaging tasks.