Research

Presentations

April 10, 2014 University of Kentucky Topology Seminar

Title: Spin Cobordism and Wedge Quasitoric Manifolds

Abstract: Quasitoric manifolds are smooth 2n-manifolds admitting a "nice" action of the compact n-torus so that the quotient of this action yields a (combinatorially) simple polytope. They are a generalization of smooth projective toric variaties and much is known about these manifolds in terms of complex cobordism theory. In fact they were used by Buchstaber and Panov to show that every cobordism complex class contains a (connected) quasitoric manifold.

Far less is known about spin quasitoric manifolds and spin cobordism which requires the calculation of KO-characteristic classes. We consider a procedure developed to investigate topological data for spin quasitoric manfolds which utilizes a wedge polytope operation on the quotient polytope. We'll discuss a list of results concerning these "wedge" quasitoric manifolds, including such topics as Bott manifolds, the connected sum, the Todd genus and lastly specific criteria in terms of combinatorial data allowing for the calculation of KO-characteristic classes of spin quasitoric manifolds.

October 31, 2013 University of Kentucky Topology Seminar

Title: Combinatorial Formulae for the \Chi_y Genus of Quasitoric Manifolds.

Abstract. We recall the definition of a quasitoric manifold as any smooth 2n-manifold admitting a nice action of the compact torus. We then consider an equivalent formulation in terms of combinatorial data and its related stably complex structure. Next we'll demonstrate Panov's proof for calculating the \Chi_y-genus of quasitoric manifolds in terms of this combinatorial description and elicit an explicit formula for the Todd genus. Lastly, we'll work through a couple of small dimensional examples and postulate some related conjectures concerning "wedge" quasitoric manifolds.

February 28, 2013 University of Kentucky Topology Seminar

Title: Wedge Quasitoric Manifolds

Abstract: Quasitoric manifolds (QTMs) are smooth compact manifolds admitting a well-behaved action of the compact torus so that the quotient of this action is diffeomorphic (as a manifold with corners) to a combinatorially simple polytope. We'll develop a procedure to attempt to view any QTM as a codimension 2 subquasitoric manifold of an "ambient" wedge QTM. We formulate these wedge QTMs on the level of polytopes from the wedge polytopal construction. The existence of such wedge QTMs in the general case is still unknown but we'll demonstrate a proof for the existence of such constructions for any Bott tower and discuss a similar conjecture concerning Bott manifolds and connected sums of the aforementioned. We will focus on small dimensional examples to view these constructions.

March 22, 2012 University of Kentucky Topology Seminar

Title: Quasitoric Manifolds and Generalized Bott ManifoldsAbstract: Based on the works of Choi, Masuda and Suh, we will discuss necessary and sufficient conditions for a quasitoric manifold (over a product of simplices) to be equivalent to a generalized Bott manifold. The argument is formulated around the bundle structure and cohomology rings of the manifolds.

April 14 and 28, 2012 University of Kentucky Topology Seminar

Title: The Ochanine Genus, Modular Forms, and the Brown-Kervaire Invariant Abstract: We consider a refinement of the universal elliptic genus, called the Ochanine or beta Genus. After a brief treatment of modular forms over graded rings, we examine certain modular forms over KO(lower star). We then show that the beta genus applied to a spin manifold is such a modular form. In a follow-up to this discussion, we will use these constructions to look at the Brown-Kervaire Invariant.

November 9 and 30, 2011 University of Kentucky Topology Seminar

Title: Toric Varieties and Quasitoric Manifolds in Cobordism

Abstract: Toric varieties are rich mathematical objects that connect several subjects: algebraic geometry, combinatorics, and topology to name a few. Quasitoric manifolds though similar in several aspects to smooth complete toric varieties, are preferable when dealing with cobordism theory. Indeed, in dimensions greater than 2, every complex cobordism class contains a quasitoric manifold. We will take a look at some of the properties that make quasitoric manifolds agreeable under cobordism, including the connected sum and a quick look at Hirzebruch's T-y genus.

February 17, 2011 University of Kentucky Topology Seminar

Title: Unit Tangent Vector Fields on Spheres

Abstract: We discuss a lower bound to the number of orthonormal tangent vector fields to the n-sphere. This will be achieved via orthonormal multiplications as they relate to Clifford Algebras. We will talk about some concrete examples of Clifford Algebras and specifically how their structures generate these vector fields on spheres. This talk should be accessible to undergraduate math majors. If you've had a topology or analysis course that would be a plus, but everyone is welcome!