The University of Michigan Algebra Seminar
Fall 2010, Tuesday 3:10-4:00, 3096 East Hall
(organized by Harm Derksen)
University of New Brunswick
Cyclic polytopes and tilting theory for higher Auslander algebras
There is a natural bijection between basic tilting modules
for a linearly oriented quiver of type An
and triangulations of an (n+2)-gon. I will recall this
construction and then discuss a higher-dimensional generalization of it, in which the path algebra
is replaced by the d-th higher Auslander algebra of linearly oriented An
(in the sense of Iyama), and the polygon is replaced by a cyclic polytope of dimension
2d+2. The triangulations of the cyclic polytope then correspond bijectively to tilting
objects contained in the (d+1)-cluster tilting subcategory of the module category. The
structure of this collection of tilting modules (or equivalently of the collection of
triangulations) provides a higher-dimensional analogue of cluster combinatorics, which I
will be discussing further in the combinatorics seminar on September 24. The work I will
be presenting is joint with Steffen Oppermann, and is contained in
University of Florence
Defining equations for secant varieties of Segre Veronese varieties
Abstract: In work with Daniel Erman and Dustin Cartwright, we describe the
defining ideal of the rth secant variety of P2xPn
embedded by O(1,2), for arbitrary n and r ≤ 5. We also present the Schur module
decomposition of the space of generators of each such ideal. Our main
results are based a more general construction for producing explicit
matrix equations that vanish on secant varieties of products of
projective spaces. This extends previous work of Strassen and
Our matrix equations are simple to understand, and I will present them
at the level of basic linear algebra. Then I will show the invariant
presentation of the equations. I will explain why these equations are
always necessary defining equations of secant varieties. I will
describe our results which say when these equations are also
sufficient to define the ideal. In addition, I will give several
examples to illustrate the limits of these equations. Finally, I will
mention applications of these results to signal processing.
(joint with number theory seminar),
Singularities of ordinary deformation rings
Abstract: Modularity lifting theorems have played an important role in number
theory over the last 15 years. The first such theorem, proved by
Wiles, was deduced from a stronger result, an "R=T theorem."
Recently, Kisin proved a more general lifting theorem without
establishing R=T. I will outline how one can prove R=T in some
of these cases. The key technical step is establishing that the
relevant local deformation rings are Cohen-Macaulay. This is
acheived by producing an appropriate resolution of singularities.
In the first part of the talk, I will explain what a modularity
lifting theorem is, what an R=T theorem is and why they are
important. In the second part, I will explain the general ideas
of the proof of the new result.
A module-theoretic study of Kummer and Artin-Schreier theories
Abstract: In this talk I'll report on some recent developments in the
study of Galois module structures of Kummer and Artin-Schreier
theories. In particular we'll discuss some new lower bounds on
realization multiplicities for certain p-groups, and we'll discuss
some current work on the structure of completed p-power classes.
University of Michigan
Generalizing twisted homogeneous coordinate rings
Abstract:Sklyanin algebras play an important role in the study of
physical phenomenon. We will first review techniques of Artin-Tate-van den Bergh (ATV) that
describe the ring-theoretic and homological behavior of these structures.
In particular, we highlight the significance of twisted homogeneous coordinate rings.
The focus of the talk is to introduce a generalized twisted homogeneous coordinate ring P
associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising
geometry of these algebras yields an analogue to a result of ATV; namely that P is a factor
of the corresponding degenerate Sklyanin algebra.
Last modified: 11/22/10, maintained by
University of Michigan
Moduli of quiver representations and birational geometry
We use tilting theory and birational geometry to study moduli spaces
of quiver representations. From certain "representable" functor, we construct a birational
transformation from the moduli space of representations of one quiver to another new quiver
with one vertex less. The dimension vector and the stability for the new moduli are determined
functorially. We introduce several relative notion of stability to study such a birational
Some of its structure is understood but a more round picture may lead to an algorithm similar
to the minimal model program. Moreover, we deduce a formular comparing the induced ample
divisors for the two moduli. We will illustrate this theory by some surface examples.
If time permit, we may show some higher dimensional examples.