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Friday, September 4, 2015 - 14:05 ,
Location: Skiles 006 ,
TBA ,
Georgia Tech ,
Organizer: Sudipta Kolay

Wednesday, September 2, 2015 - 14:05 ,
Location: Skiles 006 ,
Jonathan Paprocki ,
Georgia Tech ,
Organizer: Jonathan Paprocki

We review the basics of hyperbolic geometry in preparation for studying mapping class groups.

Thursday, July 16, 2015 - 14:05 ,
Location: Skiles 006 ,
Shane Scott ,
Georgia Tech ,
scottsha@gatech.edu ,
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 ,
krone@math.gatech.edu ,
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 15, 2015 - 14:05 ,
Location: Skiles 006 ,
Elizabeth Bolduc ,
Georgia Tech ,
Organizer:

Friday, April 10, 2015 - 14:05 ,
Location: Skiles 006 ,
Xander Flood ,
Georgia Tech ,
Organizer: Alexander Flood

Wednesday, April 8, 2015 - 14:05 ,
Location: Skiles 006 ,
Xander Flood ,
Georgia Tech ,
aflood3@math.gatech.edu ,
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.