Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                                Spring  2018

   Organizers:  Anna Marie Bohmann, Ioana Suvaina, Rares Rasdeaconu

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

                Friday, January 19th

                Speaker: Paul Goerss, Northwestern University

                Title:  Geometry, Duality, and Large Scale Phenomena in the Homotopy Groups of Spheres

The core project of homotopy theory is to calculate the homotopy classes of maps between finite simplicial complexes.
                However, even when the complexes are spheres, low dimensional calculations appear noisy and random. The key to developing a
                structure was to find large-scale periodic phenomena, a point of view initiated by Adams in the 1960s using K-theory. Forty years
                later we have gotten very good at this, especially after work of Hopkins and his coauthors connecting this study to arithmetic geometry,
                including the geometry of elliptic curves. In this talk, I hope to offer a window into this world, and explain some of the beautiful
                patterns and symmetries we have found. (Contact Person: Anna Marie Bohmann)

                Friday, March 2 - no meeting

Shanks Workshop: "Complex Differential Geometry", March 2-3, Vanderbilt University             

                Friday, March 16th

                Speaker: Yanli Song, Washington University at St Louis

                Title:  [Q,R]=0 and index theory

It is well-known that the presence of conserved quantities in a Hamiltonian dynamical system enables one to reduce the
                number of degrees of freedom of the system. This technique is nowadays known as symplectic reduction. Guillemin and Sternberg
                considered the problem: what is the quantum analogue of symplectic reduction? In other words, when one quantizes both a mechanical
                system with symmetries and its reduced system, what is the relationship between the two quantum-mechanical systems that one obtains?
                This is the so-called quantization commutes with reduction theorem. I will give an introduction to this theorem and then talk about its
                various generalization and its application in index theory. (Contact Person: Rudy Rodsphon)        

                Friday, March 23rd

Jonathan Campbell, Vanderbilt University

                Title:  An Introduction To, and Extension Of, Algebraic K-Theory

In this talk I'll introduce algebraic K-theory, and then explain how it can be extended to many non-algebraic situations. I will
                sketch applications for this extension --- for example, the rank filtration in algebraic K-theory due to Quillen seems closely related with
                the classical scissors congruence group. (Contact Person: Anna Marie Bohmann)      

                Friday, March 30th

Anna Marie Bohmann, Vanderbilt University

                Title:  K-theory and multiplication

Algebraic K-theory is a rich invariant of rings and of categories.  While it first arose in a purely algebraic context, seminal work of
                Quillen in the 70s showed that these invariants fundamentally come from topology: they are a sequence of abelian groups arising as the higher
                homotopy groups of a space. There are now a wide range of methods for building this space from an input category, each having different
                advantages.  In particular, many of these methods are "multiplicative," in that when the starting category has a sort of multiplication, the homotopy
                groups of the space produced form a graded ring.  In this talk, I will discuss what this means and why we care, and then discuss some recent work
                with Osorno in which we compare the multiplication structures produced by two different methods.

                Friday, April 13th

                Speaker: Jayadev Athreya
, University of Washington, Seattle

                Title:   Statistical regularities of self-intersections for geodesics on hyperbolic surfaces: local geometric properties

In joint work with S. Lalley, J. Sapir, and M. Wroten, we show that the tessellation of a compact, hyperbolic surface induced by a typical
                long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation --
                for instance, the fraction of triangles -- approach those of the limiting Poisson line process. (Contact Person: Spencer Dowdall)   

                April 14-15,
AMS Spring Southeastern Sectional Meeting, Vanderbilt University

                Friday, April 27th

                Speaker: Viatcheslav Kharlamov
, IRMA, Strasbourg, France

                Title:   About an invariant signed count of real lines on real projective hypersurfaces

                Abstract: As it was observed a few years ago, there exists a certain signed count of lines that, contrary to the honest "cardinal" count, is independent
                of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. In this invariant signed count
                the input of a line is given by its local contribution to the Euler number of a certain auxiliary universal vector bundle.

                The aim of the talk is to present other, in a sense more geometric, interpretations of the signs involved in the invariant count. In particular, this provides
                certain generalizations of Segre indices of real lines on cubic surfaces and Welschinger weights of real lines on quintic threefolds.

                This is a joint work with S.Finashin. (Contact Person: Rares Rasdeaconu)       


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