# Fall 2013

### Weak Galerkin finite element methods for flow in porous media

**James Liu**

#### Colorado State University

#### Thursday, Dec. 6, 2013 at 2:30 p.m. in DMS 104

**Abstract: **This talk presents a family of weak Galerkin finite element methods (WGFEMs) for the Darcy's law in single-phase flow. WGFEMs are numerical methods that rely on the novel concepts of discrete weak gradients. With the introduction of pressure unknowns both on the skeleton and inside the elements of a given mesh, WGFEMs offer accurate numerical solutions and easy implementation. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, there is no need for tuning problem-dependent penalty factors, and the resulted discrete linear systems are definite. We test WGFEMs on benchmark problems to demonstrate the strong potential of the new methods in handling strong anisotropy and heterogeneity in Darcy flow. Comparison of WGFEMs with the discontinuous Galerkin and mixed finite element methods will also be presented. If time permits, we shall also discuss applications of WGFEMs to two-phase flow.

**Constructing hyperbolic lattices**

#### Julien Paupert

#### Arizona State University

#### Dec 5, 2013 at 2:30pm in DMS 104

** Abstract: **A lattice in a Lie group is a discrete subgroup such that the quotient has finite volume (for example, it may be compact). Classical examples are Z^n in R^n and SL(2,Z) in SL(2,R). The latter is the prototype of a hyperbolic lattice, which has many rich geometric and algebraic properties. Starting from this example, we will see how to generalize it in several directions, leading for example to arithmetic groups and reflection groups. We will then survey some of the main results and open questions in this area.

The talk is intended to be accessible to a wide audience, including advanced undergraduates.

### Left-Orderability and Three-Manifolds

#### Tye Lidman

#### University of Texas, Austin

#### Thursday, November 14 at 2:30pm in DMS 104

**Abstract:** A group is called left-orderable if it can be given a left-invariant total order. After studying some basic properties, we will discuss the question of when the fundamental group of a three-manifold is left-orderable. In many cases, there is surprisingly a relationship between the topology of the manifold and this algebraic condition.

### The Brouwer Fixed-Point Theorem via Banach Algebras

#### Bruce Blackadar

#### University of Nevada, Reno

#### Thursday, Oct 31, 2013 at 2:30pm in PE 205

**Abstract: **The Brouwer Fixed-Point Theorem is one of the most fundamental results about the topology of Euclidean space. There are at least three important equivalent formulations (Bn denotes the closed unit ball in Rn and Sn1 its boundary sphere): Brouwer Fixed-Point Theorem: Every continuous function f : Bn ! Bn has a _xed point, i.e. there is a point x 2 Bn with f(x) = x. No-Retraction Theorem: There is no retraction from Bn to Sn1.

No-Contraction Theorem: The sphere Sn1 is not contractible, i.e. the identity map on Sn1 is not homotopic to a constant map. If n _ 3, there are no known simple proofs. There are several proofs, all of which use substantial machinery. The most straightforward proof uses a combinatorial theorem (Sperner's Lemma). We will describe an approach to the theorem using Banach algebras and K-theory. Although this approach is not elementary (it is rather like using a nuclear bomb to kill a y), it is appealing and puts the theorem in a larger context. I will not come close to setting up all the machinery needed for the proof in this talk, but I hope to show that the Fixed-Point Theorem is an integral part of, and a necessary consequence of, a circle of ideas from elementary topology, linear algebra and abstract algebra. I also hope to show the utility of the Banach

algebra approach in some parts of topology. This will be an in-between" talk, not research-level mathematics but also too advanced to be aimed at undergraduates. I will try to make most of the talk accessible to graduate students and advanced undergraduates.

### Generators and Relations for a Category of Spin Surfaces

#### Tarek Sami Fadali

Indiana University

#### "Thursday, October 24 at 2:30pm in WRB 2006

**Abstract: ** I will describe some of the work I did on my dissertation at Indiana University, involving spin surfaces. These can be pictured as surfaces, labeled by 0s and 1s, with a certain formula for how the labels change when the surfaces are glued together.One of my main theorems states that any spin surfaces can be built out of a short list of simple spin surfaces and provides a set of moves to get between any two such ways of building the same surface.This has some implications for 2-dimensional spin TQFT's (topological quantum field theories), providing a complete algebraic characterization for them. I will attempt to describe some of what goes into these results, with an emphasis on the general background material in topology that underlies them.

### Lie algebras up to homotopy and the geometry of closed differential forms

#### Chris Rogers

#### Mathematics Institute, University of G ̈ottingen

#### Tuesday, August 27 at 2:30 p.m. in DMS 105

**Abstract:** In topology, one often encounters weakened analogs of familiar algebraic laws in which strict equalities are replaced by homotopies. For example, concatenating based loops in a space is a kind of multiplication which is only associative up to homotopy. Similar "weakening" also arises in other homotopical contexts, such as chain complexes. In the first half of my talk, I will introduce objects called L∞-algebras or "Lie algebras up to homotopy". Invented by Schlessinger and Stasheff, these are complexes equipped with a Lie-like bracket which only satisfies the Jacobi identity up to "coherent" chain homotopy.

These structures play important roles in rational homotopy theory, deformation theory, and mathematical physics. In the second half, I'll explain how to canonically associate a L∞-algebra to any manifold equipped with a closed differential form of degree > 1. This is a generalization of the familiar Poisson algebra associated to a symplectic manifold. I will conclude by describing the role this L∞-algebra plays in the theory of gerbes and also in string geometry (i.e., the study of "spin structures" on loop spaces).

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