# Colloquium Schedule - Spring 2013

## Thursday May 2, 2:30-4:0pm, Location DMSC 103

Allison Moore
Department of Mathematics
UT Austin

Mutation and knot Heegaard Floer homology
Abstract: Let K be a knot in S^3. In the 1970s, Conway defined
mutation, an operation which produces a new knot from K which can be
surprisingly difficult to distinguish from the original knot. In the
1980s, Ruberman generalized the notion of mutation to a topological
operation on a three manifold M containing an embedded surface F, in
which M is cut along F, and then reglued under the image of the
hyperelliptic involution \tau. Mutant pairs are commonly used to
determine how discriminating new invariants of knots or three manifolds
are. I am interested in how the knot Heegaard Floer complex behaves with
respect to such mutations. In particular, I'll describe an infinite family
of genus two mutant pairs with the same total dimension in knot Floer
homology, and also prove that in the case of pretzel knots, the existence
of an essential Conway sphere in the knot complement is evident from the
knot Floer complex. Parts of this work are joint with Starkston and
Lidman.

## Thursday April 25, 2:30-4:0pm, Location DMSC 104

Antonio DiCarlo
Department of Mathematics and Physics
Universita Roma Tre

Some exterior calculus (differential and discrete) for PDE people
Abstract: By their very name, PDEs are commonly understood in terms of partial derivatives. I prefer to look at them from a more structured point of view, bringing to the fore the underlying geometric structures and the corresponding calculus. To bridge the gap between differential and discrete calculus, I find it useful to introduce a new concept, namely, the boundary of a (multi-)vector field. To do so, I combine three scattered tools from exterior calculus: i) the definition of contravariant exterior derivative, mentioned in passing by Abraham, Marsden and Ratiu [1]; ii) the notion of k-measure, introduced by Fichera in a little-known paper [2]; iii) the intrinsic definition of the exterior derivative in terms of Lie derivatives, due to Palais [3]

## Thursday April 18, 2:30-4:0pm, Location DMSC 103

Bogdan D. Suceava
Department of Mathematics
California State University, Fullerton

Amalgamatic Curvature, Chen Invariants and New Inequalities for Curvature
Abstract: In the last two decades, there have been important advances in the study of new curvature invariants. In 2011, B.-Y. Chen published the monograph titled "Pseudo-Riemannian Geometry, Delta Invariants and Applications". Using as starting point this recent monograph, we focus our attention on several classes of inequalities in the geometry of submanifolds, starting with B.-Y. Chen's fundamental inequality for Riemannian submaniolds. Then we discuss Z. Lu's proof of the normal scalar curvature conjecture. Among the other recent results we will present, we will introduce a new curvature invariant called amalgamatic mean curvature; we explore its geometric meaning by proving an inequality relating it to the absolute mean curvature of the hypersurface. In our study, a new class of geometric object is obtained: the absolutely umbilical hypersurfaces.

## Thursday March 7, 2:30-4:0pm, Location DMSC 103

Scott Carter
Department of Mathematics and Statistics
University of South Alabama

Knotted Surfaces and Their Generalizations
Abstract: The study of knotted surfaces and their higher dimensional analogues was initiated in a 1925 paper by E. Artin. In the late 1950s and early 1960s a lot of progress was made in analyzing higher dimensional knots because homological techniques could be brought to the problems. Fox's Quick Trip" and Zeeman's subsequent definition of twist-spinning pointed to interesting distinctions between the classical case of loops knotted in 3-space and surfaces in 4-space.
Meanwhile, higher dimensional knottings can be understood by means of other homological methods. Diagrammatic techniques to study knotted surfaces were introduced among many Japanese scholars and brought to fruition by works of Giller and Roseman. The first section of this talk will discuss these historical developments.

In the second section of the talk, specific methods to define and calculate invariants will be outlined. These will be based upon the notion of a quandle and its associated quantities. Some other types of knottings in 4-dimensional space will also be discussed.

## Tuesday March 5, 2:30-4:0pm, DMSC 103

Benito M. Chen-Charpentier
Department of Mathematics
University of Texas at Arlington

The Method of Polynomial Chaos with Applications
Abstract: In mathematical modeling, especially in models of population growth, epidemics and other biological processes, there is a dependency on pa- rameters that are either measured directly or determined by curve fitting. Some of these parameters can have variability depending on experimental error, on differences in the actual population used and on many other factors. To deal with this variability, in this talk we consider that those parameters are random variables with given distributions and that there may be a correlation between some of them. We also consider that the unknown variables are stochastic processes. The method of polynomial chaos is a way to deal with random differential equations. We apply the method to some simple bacterial growth models and will solve the re- sulting equations numerically. We also present a variant of the method known as non-intrusive polynomial chaos that is easier to program and more efficient. We apply it to a model of virus propagation.

## Thurday February 28, 2:30-4:0pm, DMSC 104

Terry Loring
Department of Mathematics and Statistics
University of New Mexico

Antiunitary symmetries in operators algebras and topological insulators
Abstract: Topological insulators differ from ordinary insulators in that they have boundary states that allow for electron motion while the bulk remains insulating. There existence is explained using time-reversal symmetry, and other symmetries, defined by an antiunitary operator. Physicist classify topological insulators using real and complex K-theory, and the various K-theoretical invariants from physics can be studied using methods in numerical linear algebra and C*-algebras. I will discuss the challenges this brings to both subjects,

Specialized topics will arise, such as E-theory for real C*-algebras and symplectic eigensolvers. However most of the talk will stay in the more common language of linear algebra and operator theory.

## Tuesday February 12, 9:30-10:45am, EJCH 205

Matt Enright

Math in Video Games
Abstract: The rough topics I'm planning on discussing are design, graphics, and physics in games:

Design (weighted random numbers, planning, balance)
Involves: Random number generation, geometry, statistics
Graphics (lighting equation, projection in rasterization, procedural content)
Involves: Matrix multiplication, fractals.
Physics (collision tests, high order solvers)
Involves: Integrals, derivatives, solving systems of equations.

## Thursday January 24, 2:30-3:45pm, DMS 103

Dr. Monika Neda
Department of Mathematical Sciences