Algebra Seminar,
Spring
2011, Wednesdays
(Contact: Andrei Rapinchuk, Weiqiang Wang)
Algebra
Seminar in the past (Past1, Past2)
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January 26
Postponed to next week due to
weather.
February 2
Title: On
division algebras having the same maximal subfields
Speaker:
Andrei Rapinchuk
February 9
Title: Introduction
to derived algebraic geometry, I
Speaker: Mike
Hill (UVA)
February 16
Title: Introduction
to derived algebraic geometry, II
Speaker: Mike
Hill (UVA)
February 23
Title: Anisotropic groups and split BN-pairs
Speaker: Matt Zaremsky (UVA)
Abstract: Reductive algebraic groups like GL_n(K)
admit a nice pair of generating subgroups (B,N) called a BN-pair that
tells us a lot about the structure of the group.
More generally if G is any
reductive k-group, then G(k) admits a canonical
(split spherical) BN-pair. If G is k-anisotropic however, this standard
BN-pair is trivial,
though it is not
immediately obvious that G(k) doesn't possess some "abstract"
non-trivial split spherical BN-pair. In this talk I will discuss some joint
work with
P. Abramenko
regarding a conjecture of Caprace and Marquis stating
that reductive anisotropic groups admit no non-trivial split spherical
BN-pairs.
March 2
No talk
March 9
No talk (Spring break)
March 16
Title: Linear representations of Chevalley
groups over rings: Two approaches
Speaker:
Igor Rapinchuk (Yale)
Abstract: I will report on some recent results for
representations of Chevalley groups over commutative rings. These results, in
particular,
confirm the
conjecture of Borel and Tits on abstract homomorphism
between the groups of rational points of algebraic groups for Chevalley groups
over fields of
characteristic zero. I will discuss two approaches to the problem, one using
the notion of algebraic rings, and the other relying on
an analysis of
the congruence subgroup problem.
March 18 (note the special
date, Friday 3:30, Kerchof 317)
Title:
Surface Geometry via Loop Groups
Speaker: Josef Dorfmeister (TU Munich)
Abstract:
The construction of surfaces of a particular type and/or shape has been of
interest since the beginnings of differential geometry.
A particularly well studied class of
surfaces is the class of minimal surfaces in R3, the surfaces in R3 with
vanishing mean curvature, H=0.
Examples for this class of surfaces
have been constructed by analytic methods, but also via the Weierstrass
representation formula.
In this talk we will discuss a
"generalized Weierstrass representation"
for all surfaces in R3 of (non-vanishing) constant mean curvature H.
We will begin by recalling (in
conformal coordinates) the classical partial differential equations of surface
theory.
This will lead naturally to consider
the moving frame of a CMC surface as a member of a loop group, a certain Banach Lie group.
A natural decomposition of elements
in such loop groups (Birkhoff decomposition) will
yield a holomorphic 2x2 ODE system of a specific type
and form.
We will explain how one can
construct, conversely, from holomorphic ODE systems
surfaces of constant mean curvature.
March 23
Title: Higher Dimensional
Thompson Groups
Speaker:
Johanna Hennig (UCSD)
Abstract:
The groups F \leq T \leq V
were defined by Richard Thompson in 1965 and used to construct finitely
presented groups with unsolvable word problems. T and V
were also the first examples of infinite, finitely presented
simple groups. Since then, these groups have been studied extensively using a
rich interplay of algebraic, topological,
and dynamical approaches. I will discuss recent work
regarding the higher dimensional analogues of Thompson groups, nV, including the fact that mV is
not isomorphic
to nV for n \neq
m, and that for every $n$ the group nV is finitely
presented and simple.
March 30
Title: Rank
Gradient of Finitely Generated Groups
Speaker:
Nathaniel Pappas (UVA)
Abstract: The rank gradient is a group invariant which
assign some real number greater then or equal to -1
to a finitely generated group. Rank
gradient was first
defined by M. Lackenby for the study of 3-manifold groups. Rank gradient has connections with other
group invariants from other fields of mathematics such as
cost and L2 Betti numbers. I
will discuss the relationship of rank gradient, cost, and L2 Betti numbers as well as the group theoretic properties of
rank gradient.
I will discuss the still open question of
what values can be achieved by rank gradient.
April 6
No talk
April 13
Title: Unipotent invariant
(complete) quadrics
Speaker: Mahir CAN (Tulane University)
Abstract: The
variety of complete quadrics, which is used by Schubert in his famous
computation of the number of space quadrics tangent to 9 quadrics in
general position, is a particular
(wonderful) compactification of the space of non-singular
quadric hypersurfaces in n dimensional complex
projective space.
In this talk, towards a theory of
Springer fibers for complete quadrics, I will describe our recent work on the unipotent invariant complete quadrics.
These results involve interesting combinatorics, and in particular, in the case of a regular unipotent element we obtain the Poincare polynomial of
a unipotent
fixed subvariety of quadrics as a rank generating
function of a poset. This is joint work with Michael
Joyce.
April 20
Title: Invariant theory of Weyl
groups and a spin analogue
Speaker: Constance Baltera (UVA)
April 27
No talk