Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                                Fall  2017


   Organizers:  Anna Marie Bohmann, Ioana Suvaina, Rares Rasdeaconu

   Fridays, 3:10-4:00pm in SC 1310 (unless otherwise noted)



              
                Friday, August 25th (joint with Subfactor Seminar, 3:10-4:00pm)

                Speaker: Christian Fleischhack, University of Paderborn

                Title:  Loop Quantization of  Geometry

                Abstract:
Loop quantum gravity aims at a mathematically rigorous quantization of general relativity. It relies on
                a reformulation of gravity as a gauge field theory with constraints, which is canonically quantized. The classical
                configuration space A is formed by connections in some principal fibre bundle and gets compactified
                during quantization. The resulting space can be seen as the spectrum of an appropriate C*-algebra of bounded
                functions on A or as a projective limit of powers of the structure group. Moreover, it exhibits a canonical
                measure induced by the Haar measure. In my talk, I am going to present the background and the basic structures of the
                theory. If time permits, I will also discuss applications to the quantization of geometric entities like area or how
                diffeomorphism invariance restricts the freedom in quantizing the full theory. (Contact Person: Ioana Suvaina and
                Dietmar Bisch)



                Friday, September 22nd

                Speaker: Bert Guillou, University of Kentucky

                Title: The slice filtration for certain RO(K_4)-graded suspensions of HF_2

 
                Abstract:  
A space X can be described by its Postnikov tower, whose stages have only the homotopy groups of X in a range.
                Equivariantly, there is an analogue of the Postnikov filtration called the slice filtration. After reviewing some previously known
                examples, I will describe the slice filtration for twisted Eilenberg-Mac Lane spectra when the group of equivariance is K_4, the
                Klein four group. This is joint work with C. Yarnall. (Contact Person: Anna Marie Bohmann)



                Friday, September 29th

                Speaker: Jesse Leo Kass, University of South Carolina

                Title:  How to count lines on a cubic surface arithmetically

                Abstract:
A celebrated 19th century result is that a smooth cubic surface over the complex numbers contains
                exactly 27 lines.  Over the real numbers, the count of lines depends on the surface, but Segre showed that a certain
                signed count is independent of the surface.  In my talk, I will explain how to extend Segre's to an arbitrary field.
                This result is an application of recent ideas about Euler classes in A1-homotopy theory.  All work is joint with Kirsten
                Wickelgren. (Contact Person: Rares Rasdeaconu)



                Friday, October 6th


                Speaker: Peter Bonventre, University of Kentucky

                Title:  Genuine Equivariant Operads

                Abstract:
 
A classic result by Elmendorf states that G-spaces and G-coefficient systems are Quillen equivalent. Moreover,
                this result remains true if one replaces spaces with other well-behaved categories (including simplicial sets and categories).
                However, when considering G-operads, this equivalence fails to capture certain desired subtleties. In this talk, I will introduce
                (G-)operads and review this failure. I will then define a new algebraic gadget called genuine G-operads (playing the role of
                coefficient systems), and state an Elmendorf-type result in this context. This is joint work with L. Pereira. (Contact Person:
                Anna Marie Bohmann)


               
                Friday, Oct 13th - no meeting, Fall Break



                Friday, October 20th

                Speaker: Ryan Grady, Montana State University

                Title:  Manifold Invariants via Perturbative QFT

                Abstract: 
Assuming only elementary differential topology, I will introduce the Batalin-Vilkovisky formalism for quantum
                field theory (QFT). I will then give some elementary/foundational examples. Finally, I will introduce a family of QFTs
                (sigma models) and illustrate how they can be used to study the topology and geometry of smooth manifolds. (Contact Person:
                Anna Marie Bohmann)



                Friday, November 3rd

                Speaker:
Matthew Gentry Durham, Yale University

                Title:  Geometrical finiteness and Veech subgroups of mapping class groups

                Abstract: 
I will discuss work in progress with Dowdall, Leininger, and Sisto, in which we aim to develop a notion of
                geometrical finiteness for subgroups of mapping class groups.  Motivated by the theory of convex cocompact subgroups,
                which are precisely those which determine hyperbolic surface group extensions, I will describe some hyperbolic properties
                of the surface group extensions coming from lattice Veech subgroups. (Contact Person: Spencer Dowdall)




                Friday, November 10th

                Speaker: 
Joseph Migler, Ohio State University

                Title:  Torsion invariants in operator algebras and K-theory

                Abstract:
Determinant-type invariants known as joint torsion may be associated to collections of operators that commute
                modulo trace ideals.  These have been applied to problems in operator theory, asymptotics of determinants, and the classification
                of almost normal operators up to almost unitary equivalence.  We will review the construction of these invariants and discuss
                recent work on their applications. (Contact Person: Rudy Rodsphon)




                Wednesday, November 15th (3:10-4pm, in SC 1420)

                Speaker: 
Samuel Guerin, Universite Lyon 1, France

                Title: Real K-theory 

                Abstract:
It was realized by Atiyah and Singer in the sixties that the index theory of complex elliptic operators can be well
                understand using K-theory. This led them to a new proof of their famous theorem. They also show how these idea can be adapted
                for real elliptic operators. This led to the KO/KR theory. This new theory exhibits new features as an 8 fold periodicity and to new
                topological invariants, namely mod 2 invariants. Following Atiyah and Singer, we will speak about the Atiyah-Singer index theorem
                for families of real elliptic operators and how this gives the mod 2 index theorem for skew-adjoint elliptic operators.                          
             


                Friday, Nov 24th - no meeting, Thanksgiving Break
               
              






 

 Old Seminar Web-Pages: Fall 2009, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Fall 2015, Spring 2016, Fall 2016, Spring 2017