UW Algebra Seminar
Abstracts
Speaker:
Greg Landweber, University of Oregon
Title:
Off-shell supersymmetry and filtered Clifford supermodules
Date:
March 28
Abstract:
An off-shell representation of supersymmetry is a representation of the
super Poincare algebra on a dynamically unconstrained space of fields.
We describe such representations formally, in terms of the fields and
their spacetime derivatives, and we interpret the physical concept of
engineering dimension as an integral grading. We prove that formal
graded off-shell representations of one-dimensional N-extended
supersymmetry, i.e., the super Poincare algebra p^{1|N}, correspond to
filtered Clifford supermodules over Cl(N). We also prove that formal
graded off-shell representations of two-dimensional
(p,q)-supersymmetry, i.e., the super Poincare algebra p^{1,1|p,q},
correspond to bifiltered Clifford supermodules over Cl(p+q).
Our primary tools are the formal deformations of filtered superalgebras
and supermodules, which give a one-to-one correspondence between
filtered spaces and graded spaces with even degree-shifting injections.
This generalizes the machinery developed by Gerstenhaber to prove that
every filtered algebra is a deformation of its associated graded
algebra. Our treatment extends Gerstenhaber's discussion to the case of
filtrations which are compatible with a supersymmetric structure, as
well as to filtered modules in addition to filtered algebras. We also
describe the analogous constructions for bifiltrations and bigradings.
Speaker:
No seminar---Milliman Lectures
Title:
Date:
April 4
Abstract:
Speaker:
Arkady Berenstein, University of Oregon
Title:
Braided symmetric and exterior algebras
Date:
April 11
Abstract:
A classical problem of Representation Theory (coming back to Frobenius
and Schur)
reads: decompose symmetric and exterior powers of a simple G-module
(where G is a
complex reductive groups) into simple G-modules. The problem is
exceptionally hard
and the complete answer is unknown even for G=GL_2, the group of
invertible 2x2
matrices. One of the goals of my talk (based on a joint paper with
Sebastian
Zwicknagl and his follow-up research) is to propose a right setup in
which the
problem can be solved. This setup includes new quantum objects --
braided
symmetric and exterior algebras, which we associate to each
finite-dimensional
G-module and which share many properties of their classical
counterparts.
One of
the most fundamental properties of these braided algebras is the
associated
(super) Poisson structure. One can ask if this Poisson structure can
be lifted to
the ordinary symmetric and exterior algebra of a given G-module. I
will conclude
the talk with Sebastian's beautiful classification of those G-modules
for
which
the answer to the latter question is affirmative.
Speaker:
Jintai Ding, University of Cincinatti
Title:
The Zhuang-Zi algorithm :
A new algorithm for solving multivariate
polynomial equations over a finite field
Date:
Monday April 17 --- Unusual day!
Abstract:
The Zhuang-Zi algorithm is a new method of solving multivariate
polynomial equations over a finite field proposed recently. This work
grows out of recent development in the area of multivariate public key
cryptosystems, where one uses a set of multivariate polynomials over a
finite field as a public key. The theoretical foundation of this family
of cryptosystems is the proven theorem that solving a set of
multivariate polynomial equations over a finite field is, in general,
an
NP-hard problem. The basic idea of the Zhuang-Zi algorithm is very much
inspired by the ideas of Matsumoto, Imai, Patarin, Shamir, and etc in
multivariate public key cryptosystems, where one identifies a vector
space over a finite field as a large field.
In this talk, we will show how the Zhuang-Zi algorithm can be used to
solve a set of equations, and present some examples that could not
previously be solved with all other known algorithms but the Zhuang-Zi
algorithm could.
Speaker:
Title:
Date:
April 18
Abstract:
Speaker:
Eva Feichtner, University of Stuttgart
Title:
Tropical Discriminants
Date:
April 25
Abstract:
We use tropical geometry to take a fresh look at the theory of
A-discriminants of Gelfand, Kapranov and Zelevinsky. We show that
the tropical A-discriminant is the Minkowski sum of the row space
of A and the Bergman fan of the kernel of A. Moreover, the
tropical A-discriminant allows for an interpretation as a certain
set of regular polyhedral subdivisions of A.
We obtain a positive formula for the extreme monomials of any
A-discriminant, and we give a combinatorial characterization of
Delta-equivalence for regular triangulations of A.
This is joint work with Alicia Dickenstein and Bernd Sturmfels.
Speaker:
Matt Ballard, University of Washington
Title:
A Case of Homological Mirror Symmetry for (local) 3-folds
Date:
May 2
Abstract:
Kontsevich's Homological Mirror Symmetry conjecture relates two
seemingly
disparate constructs - coherent sheaves on algebraic varieties and
Lagrangian
submanifolds in symplectic manifolds. Each construct naturally
gives rise to a
differential-graded algebra (A_{\infty}-algebra if you prefer)
and the HMS
conjecture states, under suitable assumptions, that these
dg-algebras are
quasi-isomorphic. We will first investigate the Fano version of the
conjecture in
the case of the projective lane P^2 where many of the
technical issues can be
Speaker:
Matt Ballard, University of Washington
Title:
A Case of Homological Mirror Symmetry for (local) 3-folds:
Part II
Date:
May 9
Abstract:
I will outline Kontsevich's conjecture in the case of Calabi-Yau
manifolds. There are two ways to obtain a Calabi-Yau from
$\mathbb{P}^2$: taking an anti-canonical hypersurface or taking the
total space of the anti-canonical bundle. The first case
corresponds to compactifying the affine symplectic mirror of
$\mathbb{P}^2$ while the second corresponds to stabilizing W. After
outlining why the first case (in general) is difficult, we will
discuss how to verify homological mirror symmetry in the second
case.
Speaker:
Keir Lockridge, University of Washington
Title:
A derived-categorical characterization of global dimension
Date:
May 16
Abstract:
In 1966, Peter Freyd conjectured that the sphere generates (in
the sense of category theory) the finite part of the stable category of
spectra. This conjecture, known as the generating hypothesis, is a
central open question in stable homotopy theory. In this talk, we will
introduce GH in its original setting and discuss its generalization to
arbitrary triangulated categories. We will then focus our attention on
D(R) (the derived category of right R-modules) and show that the global
dimension of R is n if and only if a global analog of GH holds in D(R).
Speaker:
Izuru Mori , SUNY (Brockport)
Title:
Symmetry in the Vanishing of Ext-groups
Date:
Monday May 22---Unusual Day
Abstract:
In this talk, we will find a class of
rings R satisfying the following property: for every pair of
finitely generated right R-modules M and N, Ext ^i_R(M,
N)=0 for all i>>0 if and only if Ext _R^i(N, M)=0 for all
i>> 0. In particular, we will show that such a class of rings
includes a group algebra of a finite group and the exterior
algebra of odd degree.
Speaker:
Title:
Date:
May 23
Abstract:
Speaker:
Paul Hacking, Yale University and University of Washington
Title:
Moduli spaces of del Pezzo surfaces
Date:
May 30
Abstract:
A del Pezzo surface is a smooth surface such that the dual of
the canonical line bundle is ample. We describe nice
compactifications of
the moduli spaces of del Pezzo surfaces using ideas from both the
minimal
model program and tropical geometry. These spaces are exceptional
analogues of the Deligne--Mumford compactifications of moduli of
pointed
curves of genus zero associated to the exceptional root systems.
Joint work with Sean Keel and Jenia Tevelev.
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