Friday, January 19th
Speaker: Paul Goerss, Northwestern
University
Title: Geometry, Duality, and Large
Scale Phenomena in the Homotopy Groups of Spheres
Abstract: 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
Abstract: 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
Speaker:
Jonathan
Campbell,
Vanderbilt
University
Title:
An
Introduction
To, and
Extension Of,
Algebraic
K-Theory
Abstract:
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
Speaker: Anna Marie Bohmann, Vanderbilt
University
Title:
K-theory and multiplication
Abstract:
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.