Algebra Seminar Winter 2010
Ottawa-Carleton Institute of Mathematics and Statistics
Organizers: Benjamin Steinberg (Carleton) and Kirill Zainoulline (UOttawa)
Place and time:
UOttawa: Wed. 1:30 - 2:30p.m., KED B015
Carleton: Wed. 1:00 - 2:00p.m., MacPhail room (HP 4351)
(Carleton-uOttawa shuttle information)
13 Jan. (Carleton)
- Speaker: Benjamin Steinberg (Carleton)
- Title: Descent algebras and hyperplane face semigroups
- Abstract: Solomon associated to each finite Coxeter group W a subalgebra
of the group algebra, now called the descent algebra of W, that behaves as
a non-commutative version of the ring of class functions on W.
In his PhD thesis, inspired by ideas of Tits, Bidigare gave an alternative
approach to descent algebras via hyperplane face semigroups. There is a
natural semigroup structure on the faces of a hyperplane arrangement via
Tits projections. If W is a finite Coxeter group, then it acts by
automorphisms on the face semigroup of its reflection arrangement and
hence on its semigroup algebra. Solomon's descent algebra turns out to be
anti-isomorphic to the algebra of W-invariants. Saliola computed the
quivers of hyperplane face semigroups and used ideas and results from
these computations to compute the quiver of the descent algebra for type
Mantaci and Reutenauer had generalized Solomon's construction to create a
descent algebra for wreath products G wr S_n where G is a finite abelian
group (and it has since been generalized for non-abelian groups).
Hsiao introduced an analogue of the hyperplane face semigroup in this
context and showed how to construct the Mantaci-Reutenauer descent algebra
as an algebra of invariants. He asked whether Saliola's results can be
extended in this context. In joint work with Margolis we have computed the
quiver of Hsiao's algebra for an arbitrary group G and of related
20 Jan. (UOttawa)
Speaker: Kirill Zainoulline (UOttawa)
- Title: Degree formula for conective K-theory
- Abstract: We use the degree formula for connective $K$-theory
to study rational contractions of algebraic varieties.
As an application
we obtain a condition of rational incompressibility
of algebraic varieties and
a version of the index reduction formula.
Examples include complete intersection,
rationally connected varieties, twisted forms of abelian varieties
and Calabi-Yau varieties
27 Jan. (Carleton)
- Speaker: Lerna Pehlivan (Carleton)
- Title: Top to random shuffles and number of fixed points
- Abstract: Top to random shuffles have been studied by Diaconis,
Fill and Pitman. We
will discuss the distribution of the number of fixed points in a deck of
cards which is top to random shuffled m times. We will find closed form
expressions for the expectation and the variance of the number of fixed
points. Both calculations are proved using the irreducible representations
of symmetric groups.
24 Feb. (Carleton)
- Speaker: Paul Mezo (Carleton)
- Title: Endoscopy: ordinary and twisted
- Abstract: One may study representations of a group G in at least two
ways: on their own or through class functions on G. This duality is
commonly called harmonic analysis. If the group happens to be a real
(connected reductive) algebraic group then there is a third way, namely
the Local Langlands Correspondence. The Local Langlands Correspondence
hints that representations of G should be related to representations of
other, so-called, endoscopic groups of G. An endoscopic group is also a
real algebraic group which is typically "smaller" than G, and its
relationship to G is given through identities between characters or
class functions. After making these ideas more precise, we will
describe twisted endoscopy, in which an automorphism of G is introduced,
and conclude with present work on proving twisted endoscopic character
3 Mar. (UOttawa)
- Speaker: Gene Freudenburg (Western Michigan University, USA)
- Title: Locally nilpotent derivations of rings with roots adjoined
- Abstract: *.pdf
24 Mar. (UOttawa)
- Speaker: David Grimm (Konstanz, Germany)
- Title: Sums of squares in one variable function fields
- Abstract: Given a base field for which a certain arithmetic property is well understood,
what can be said about the same arithmetic property
of a one variable function field over that base field ?
One example for this sort of research is the result of Parimala and Suresh, that
the u-invariant of a one variable function field over a p-adic number field (p different from 2)
is exactly 8. Harbater, Hatmann and Krashen recently found an alternative proof for this result,
based on a local global principle for isotropy of quadratic forms over fields like the aforementioned.
We are interested in the question what are the possible values of the Pythagoras number
of a one variable function field over a base field
that is hereditarily pythagorean (or not).
(The Pythagoras number of a field F is the smallest natural number p(F) such that every sum of squares in F is a sum of n squares. A field is called hereditarily pythagorean,
if it is real and every finite extension field that is real has Pythagoras number 1).
We exploit the local global principle of Harbater, Hartmann and Krashen to prove
that in a one variable function field E over the laurant series field over the real numbers R((t)),
either p(E)=2 or p(E)=3 (j.w. K.J.Becher and J.Van Geel).
The field R((t)) is known to be hereditarily pythagorean.
Now assume K is not hereditarily pythagorean.
Does a one variable function field E/K exist such that E is real and p(E)=2?
We give partial negative answers and, as a special case, we will sketch a proof that p(E)
is at least 3 if the genus of E is zero,
generalizing a result of Becker who showed that
if E is the rational function field over K, then p(E) is at least 3.
31 Mar. (Carleton)
- Speaker: Rajender Adibhatla (Carleton)
- Title: The Herbrand-Ribet Theorem
- Abstract: The Herbrand-Ribet Theorem is a classical theorem in class field
theory which relates the p-divisibility of Bernoulli numbers to the
idempotents of the ideal class group of the cyclotomic field of pth roots
of unity of an odd prime p. Ribet's 1979 proof of this theorem was an
important milestone in modern number theory. His use of modular forms and
Galois representations paved the way for Mazur and Wiles' proof of the
Main Conjecture of Iwasawa Theory.
In this talk, I will outline some of the tools that Ribet used in his
proof. Time-permitting, I will also show how one might use companion forms
to simply part of his proof.
21 Apr. (UOttawa)
- Speaker: Aravind Asok (University of Southern California)
- Title: Connectedness, homotopy theory and birational invariants
- Abstract: The classical Luroth problem asks whether every subfield of a purely transcendental extension of a field k is itself purely transcendental. Much work was done in the 1970s to produce counterexamples to the Luroth problem. I will examine one approach (that of Artin and Mumford, as generalized by Colliot-Thelene and Ojanguren) that uses cohomological invariants to detect counterexamples. More precisely, I will discuss how techniques from the Morel-Voevodsky homotopy theory of algebraic varieties allow one to use
'higher' cohomological invariants to produce new counterexamples to the Luroth problem over the complex numbers.