## Algebra Seminar Winter 2010## Ottawa-Carleton Institute of Mathematics and StatisticsOrganizers: Benjamin Steinberg (Carleton) and Kirill Zainoulline (UOttawa) |

- UOttawa: Wed. 1:30 - 2:30p.m., KED B015
- Carleton: Wed. 1:00 - 2:00p.m., MacPhail room (HP 4351)

(Carleton-uOttawa shuttle information)

- 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 A. 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). Recently, 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 semigroup algebras.

- 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

- 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.

- 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 identities.

- Speaker: Gene Freudenburg (Western Michigan University, USA)
- Title: Locally nilpotent derivations of rings with roots adjoined
- Abstract: *.pdf

- 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.

- 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.

- 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.