Thursday, July 16, 2015 - 14:05 , Location: Skiles 006 , Shane Scott , Georgia Tech , email@example.com , Organizer: Shane Scott
This talk is an oral comprehensive exam in partial fulfillment of the requirements for a doctoral degree. To any topological surface we can assign a certain communtative algebra called a cluster algebra. A surface cluster algebra naturally records the geometry of the surface. The algebra is generated by arcs of the surface. Arcs carry a simplicial structure where the maximal simplices are triangulations. If you squint you can view a surface cluster algebra as a coordinate ring of decorated Teichmuller space with Penner's coordinate. Recent work from many authors has shown that the automorphisms of the surface cluster algebra which preserve triangulations arise from the mapping class group of the surface. But there are additional automorphisms that preserve meaningful structure of the cluster algebra. In this talk we will define surface cluster algebras and discuss future research toward understanding structure preserving automorphisms.
Wednesday, April 29, 2015 - 14:05 , Location: Skiles 006 , Robert Krone , Georgia Tech , firstname.lastname@example.org , Organizer: Robert Krone
For Prof. Wickelgren's Stable Homotopy Theory class
The Steenrod algebra consists of all natural transformations of cohomology over a prime field. I will present work of Milnor showing that the Steenrod algebra also has a natural coalgebra structure and giving an explicit description of the dual algebra.
Friday, April 24, 2015 - 14:00 , Location: Skiles 006 , Andrew McCullough , Georgia Institute of Technology , Organizer: Andrew McCullough
We will give a description of the Dehornoy order on the full braid group Bn, and if time permits mention a few facts about a bi-ordering associated to the pure braid group Pn.
Wednesday, April 22, 2015 - 14:05 , Location: Skiles 006 , Jonathan Paprocki , Georgia Tech , Organizer: Jonathan Paprocki
For Prof. Wickelgren's Stable Homotopy Theory class
Harer's homology stability theorem states that the homology of the mapping class group for oriented surfaces of genus g with n boundary components is independent of g for low degrees, increasing with g. Therefore the (co)homology of the mapping class group stabilizes. In this talk, we present Tillmann's result that the classifying space of the stable mapping class group is homotopic to an infinite loop space. The string category of a space X roughly consists of objects given by disjoint unions of loops in X, with morphisms given by cobordisms between collections of loops. Sending X to the loop space of the realization of the nerve of the string category of X is a homotopy functor from Top to the category of infinite loop spaces. Applying this construction for X=pt obtains the result. This result is an important component of the proof of Mumford's conjecture stating that the rational cohomology of the stable mapping class group is generated by certain tautological classes.
Monday, April 20, 2015 - 14:05 , Location: Skiles 006 , Shane Scott , Georgia Tech , Organizer: Shane Scott
Spin bundles give the geometric data necessary for the description of fermions in physical theories. Not all manifolds admit appropriate spin structures, and the study of spin-geometry interacts with K-theory. We will discuss spin bundles, their associated spectra, and Atiyah-Bott-Shapiro's K orientation of MSpin--the spectrum classifying spin-cobordism.
Wednesday, April 8, 2015 - 14:05 , Location: Skiles 006 , Xander Flood , Georgia Tech , email@example.com , Organizer: Alexander Flood
Complex-oriented cohomology theories are a class of generalized cohomology theories with special properties with respect to orientations of complex vector bundles. Examples include all ordinary cohomology theories, complex K-theory, and (our main theory of interest) complex cobordism.In two talks on these cohomology theories, we'll construct and discuss some examples and study their properties. Our ultimate goal will be to state and understand Quillen's theorem, which at first glance describes a close relationship between complex cobordism and formal group laws. Upon closer inspection, we'll see that this is really a relationship between C-oriented cohomology theories and algebraic geometry.
Wednesday, April 1, 2015 - 14:05 , Location: Skiles 006 , Benjamin Ide , Georgia Tech , Organizer: Benjamin Ide
In this talk, I prove that there is a bijection between [X, K(\pi, n)] and H^n(X; \pi). The proof is a good introduction to obstruction theory.
Wednesday, March 25, 2015 - 14:05 , Location: Skiles 006 , Jonathan Paprocki , Georgia Tech , Organizer: Jonathan Paprocki
Solutions to the Yang-Baxter equation are one source of representations of the braid group. Solutions are difficult to find in general, but one systematic method to find some of them is via the theory of quantum groups. In this talk, we will introduce the Yang-Baxter equation, braided bialgebras, and the quantum group U_q(sl_2). Then we will see how to obtain the Burau and Lawrence-Krammer representations of the braid group as summands of natural representations of U_q(sl_2).