Algebra-Topology Seminar
We meet on Wednesdays, 2:15 pm in HG G 19.1. If you have any questions send an email to one of the organizers.
Autumn Semester 2009
| Date |
Speaker |
Title |
| Sep. 23, 2009 |
Tim Burness
|
Base sizes for algebraic groups |
| Abstract: |
Let G be a permutation group on a set S. A base for G is a subset B of S such that the pointwise stabilizer of B in G is trivial. We write b(G) for the minimal size of a base for G.
Bases for finite permutation groups have been studied since the early days of group theory in the nineteenth century. More recently, various bounds on b(G) have been obtained in the case where G is a finite simple group, culminating in the recent proof, using probabilistic methods, of a conjecture of Cameron.
In this talk, I will report on some recent joint work with Bob Guralnick and Jan Saxl on base sizes for actions of algebraic groups. Let G be a simple algebraic group over an algebraic closed field and let S = G/H be a transitive G-variety, where H is a maximal closed subgroup of G. Our goal is to determine b(G) exactly, and to obtain similar results for some additional base-related measures which arise naturally in the algebraic group context. I will explain the key ideas and present some of the results we have obtained thus far. I will also describe some connections to the related finite groups of Lie type. |
| Speakers: |
Dr. Tim Burness
(School of Mathematics, University of Southampton)
|
|
| Sep. 30, 2009 |
Christine Riedtmann
|
Irreducible components of module varieties |
| Abstract: |
Let A be a finite dimensional associative algebra with unit over an algebraically closed field k, and fix a natural number d. The A-module structures on kd form an affine algebraic variety moddA on which the group GL(d,k) operates by changing basis. In general, moddA is reducible. For each irreducible component C, I will define an ideal IC in k[moddA] whose zero set contains C. In case A is hereditary or admits only finitely many indecomposable modules (up to isomorphism), C is the zero set of IC, but in most cases IC is too small, and no source for additional polynomials vanishing on C is known. I will show on a simple example how to obtain the "missing" equations. |
| Speakers: |
Prof. Christine Riedtmann
(Mathematisches Institut, Universität Bern)
E-Mail:
|
|
| Oct. 7, 2009 |
Matthias Künzer
|
Heller triangulated categories |
| Abstract: |
In the beginning of the sixties, Grothendieck got stuck when composing several derived functors, since the spectral sequence describing the behaviour of the composition of R^i F with R^j G is complicated and yields only an approximation. The simple formula RG o RF = R(G o F) was better. The price to pay was to establish the context of derived categories, in which this formula is valid.
The derived category of an abelian category is no longer abelian in general. So Puppe and Verdier developed the formalism of triangulated categories, where distinguished triangles act as a substitute for short exact sequences. Verdier's octahedral axiom then describes the behaviour of distinguished triangles with respect to composition.
However, Verdier's octahedra themselves are not as well-behaved as one would wish. For instance, they are not determined up to isomorphism by the commutative triangle they are built on.
Now derived categories carry a stronger exactness structure still, that of a closed Heller triangulated category. It gives rise not only to distinguished triangles, called 2-triangles, but also to distinguished octahedra, called 3-triangles, which are determined up to isomorphism by their commutative triangle. Moreover, there exist 4-triangles, 5- triangles, etc.; they form a simplicial set which can be used to define triangulated K-theory.
Heller triangulated categories are defined without reference to properties of n-triangles in the axioms. Such a reference would become cumbersome e.g. when considering exact functors. The price to pay is to work in the context of stable categories of n-pretriangles, so that one can compare two shift functors defined there. |
| Speakers: |
Dr. Matthias Künzer
(Lehrstuhl D für Mathematik, TH Aachen)
E-Mail:
|
|
| Oct. 14, 2009 |
David Cimasoni
|
Discrete Dirac operators and Kasteleyn matrices |
| Abstract: |
A dimer configuration (aka perfect matching) on a graph G is a choice of edges of G such that each vertex of G is adjacent to exactly one of these edges. The number of such dimer configurations on a graph G can be computed using Pfaffians of "Kasteleyn matrices": these are signed adjacency matrices determined by an orientation of the edges of G. For a graph embedded in a genus g surface S, one needs 2^{2g} of them to compute the number of dimer configurations.
In this talk, I will first recall these basic facts. Then, I will explain to which extend these 2^{2g} matrices can be understood as discrete analogs of the 2^{2g} Dirac operators on the surface S. |
| Speakers: |
Dr. David Cimasoni
(Department of Mathematics, ETH Zürich)
E-Mail:
|
|
| Oct. 21, 2009 |
Gabriella Kuhn
|
cancelled due to illness |
| Speakers: |
Prof. Gabriella Kuhn
( Home Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca)
E-Mail:
|
|
| Oct. 28, 2009 |
Nicole Lemire
|
Upper Bounds for the Essential Dimension of the Moduli Stack of SL_n-bundles over a Curve |
| Abstract: |
In joint work with Ajneet Dhillon, we find upper bounds for the essential dimension of various moduli stacks of SL_n-bundles over a curve. When n is a prime power, our calculation computes the essential dimension of the moduli stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case. |
| Speakers: |
Prof. Nicole Lemire
(Department of Mathematics, University of Western Ontario)
E-Mail:
|
|
| Nov. 4, 2009 |
|
no seminar |
|
|
| Nov. 11, 2009 |
|
no seminar |
|
|
| Nov. 18, 2009 |
Gregor Masbaum
|
How to approximate quantum representations of mapping class groups by finite groups |
| Abstract: |
The Witten-Reshetikhin-Turaev TQFT-invariants of 3-manifolds give rise to finite-dimensional representations of mapping class groups of surfaces. I will show how to approximate these representations by representations into finite groups, using the theory of Integral TQFT developed in joint work with P. Gilmer. |
| Speakers: |
Prof. Gregor Masbaum
(IMJ, Paris)
E-Mail:
|
|
| Nov. 25, 2009 |
Pedro Vaz
|
The diagrammatic Soergel category and sl(N)-foams |
| Abstract: |
For each N > 3, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams. |
| Speakers: |
Pedro Vaz
(IMJ, Paris)
E-Mail:
|
|
| Dec. 2, 2009 |
Lesya Bodnarchuk
|
t.b.a. |
| Speakers: |
Lesya Bodnarchuk
(FIM, EPDI)
|
|
| Dec. 9, 2009 |
Stephan Wehrli
|
t.b.a. |
| Speakers: |
Stephan Wehrli
(IMJ, Paris)
|
|
| Dec. 16, 2009 |
Norbert A'Campo
|
t.b.a. |
| Speakers: |
Prof. Norbert A'Campo
(Departement Mathematik, Universität Basel)
E-Mail:
|
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