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 , firstname.lastname@example.org , 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).
Friday, March 13, 2015 - 13:00 , Location: Skiles 006 , John Etnyre , Georgia Tech , Organizer: John Etnyre
In this talk I will finish the proof that braid groups are linear using the Lawrence-Krammer representation.
Wednesday, March 11, 2015 - 14:00 , Location: Skiles 006 , John Etnyre , Georgia Tech , Organizer: John Etnyre
In this talk I will discuss the Lawrence-Krammer representation of the Braid Group and begin to sketch the the proof that braid groups are linear.