# Calendar

## Next Week

## Three Weeks

### Special Colloquium, Manuel Rivera, University of Miami and CINVESTAV, MATH 175

Monday, Jan 7 3:30 pm - 4:30 pm

TBA

Title: On characteristic classes of manifold bundles -- Abstract: Around 1955, Chern conjectured that affine manifolds have vanishing Euler number. One way to think about affine structure on manifolds is to construct a flat torsion free connection on the tangent bundle. Benzecri and Milnor studied this question in dimension 2 and later in 80's Milnor studied how characteristic classes of vector bundles change when we have certain geometric structures on the bundle, in particular flat structure. -- In this direction, he formulated a conjecture in algebraic K-theory. To be more precise, 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 and Friedlander conjectured that this map induces an isomorphism on cohomology with finite coefficients. 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 and the moduli space of manifolds to show that similar to finite dimensional Lie groups, the map from $BG^{\delta}$ to $BG$ induces a split surjection on cohomology with integer coefficients in the stable range. I will also discuss applications of these results in foliation theory, in particular, characteristic classes of surface bundles.

### Special Colloquium, Patricia Alonso Ruiz, University of Connecticut, MATH 175

Thursday, Jan 10 3:30 pm - 4:30 pm

Title: Diffusion processes on rough spaces and random geometric models. -- Abstract: Mathematical models in physics, biology, engineering, material sciences and social media are usually developed to understand structures and phenomena of high complexity. The more realistic a model is, the more “roughness” we expect to find. Introducing randomness allows us to treat richer, more complicated situations. In this talk I will outline how my research in analysis and probability aims to gain a better understanding of these structures: On the one hand, studying the stochastic processes intrinsic to a certain space can also reveal geometric and analytic information about it. On the other hand, the geometric properties of random sets and the asymptotic behavior of associated functionals are key in the development of statistical tools to analyze spatial data.

### Special Colloquium, Nicholas Cook, University of California, Los Angeles, MATH 175

Friday, Jan 11 3:30 pm - 4:30 pm

TBA

## 2019

### Special Colloquium, Xiaomeng Xu, Massachusetts Institute of Technology, MATH 175

Monday, Jan 14 3:30 pm - 4:30 pm

TBA

### Special Colloquium, Haizho Yang, National University of Singapore, MATH 175

Thursday, Jan 17 3:30 pm - 4:30 pm

TBA

TBA

Title: Discontinuous Galerkin methods for waves and fluid flow -- Abstract: The discontinuous Galerkin (DG) methods are a class of numerical methods for solving partial differential equations. They combine features of the finite elements and finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising form a wide range of applications. In this talk, we present some recent work on discontinuous Galerkin (DG) methods for waves and fluid flow. Three topics will be covered, including (1) new energy-conserving DG methods for linear hyperbolic waves, and nonlinear dispersive waves, (2) globally divergence-free DG methods for incompressible flow, and (3) hybridizable DG methods for Darcy flow. Special attention will be paid to the key idea of the construction of each DG scheme. Ample numerical results will be shown to illustrate the performance of these methods.

### Special Colloquium, Rolando de Santiago, University of California, Los Angeles, MATH175

Wednesday, Jan 23 3:30 pm - 4:30 pm

Title: Classification and Rigidity in Group von Neumann Algebras. -- Abstract: The works of F. Murray and J. von Neumann outlined a natural method to associate a von Neumann algebra to a group. Since then, an active area of research seeks to investigate which structural aspects of the group extend to its von Neumann algebra. The difficulty of this problem is best illustrated by Conne's landmark result which states all ICC amenable groups give rise to isomorphic von Neumann algebras. In essence, standard group invariants are not typically detectable for the resulting von Neumann algebra. When the group is non-amenable, the situation may be strikingly different. This talk surveys advances made in this area, with an emphasis on the results stemming from Popa's deformation/rigidity theory. I present several instances where elementary group theoretic properties, such as direct products, can be recovered from the algebra. We will also discuss recent progress made by Ben Hayes, Dan Hoff, Thomas Sinclair and myself in the case where the underlying group has positive first $\ell^2 $-Betti number. We will explore the relationship between s-malleable deformations of von Neumann algebras and $\ell^2 $ co-cycles which lays the foundation for our work. -- Refreshments at 3:00 pm in MATH 311