62nd Algebra Day
Saturday, April 5, 2008
|10:00||Coffee (Department lounge - KED 104)|
|10:30||Nantel Bergeron (York University), Combinatorial Hopf algebras and dual graded graphs|
|With Li, I have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras ⊕n≥0 An can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono, constructed dual graded graphs from primitive elements in Hopf algebras. With Li and Lam, I apply the composition of these constructions to towers of algebras. We show that if a tower ⊕n≥0 An gives rise to graded dual Hopf algebras then we must have dim An=rnn! where r = dim A1. This shows that combinatorial Hopf algebras obtained by this procedure fall into a very rigid framework and can potentially be classified.|
|11:30-13:30||Lunch (Department lounge - KED 104)|
|13:30||Miklos Abert (University of Chicago), Profinite actions from an ergodic point of view|
We study asymptotic properties of chains of subgroups in residually finite
groups using the dynamics of boundary representations and the structure of
periodic invariant measures on Bernoulli actions.
This allows us to analyze when the Schreier graphs coming from a chain of subgroups can approximate another action of the group. For chains with property tau, we exhibit a strong rigidity result, while for amenable groups, we prove that every chain approximates every action.
As a byproduct, we show that covering towers of regular graphs admit a new kind of spectral restriction which is related to the independence ratio. This leads us to solve a problem of Lubotzky and Zuk.
In another direction, we relate the cost of a boundary representation to the growth of rank and the first L2 Betti number of the group. This allows us to relate the 'fixed price problem' of Gaboriau to the 'rank vs Heegaard genus' conjecture in 3-manifold theory and show that they contradict each other.
|14:30||Coffee (Department lounge - KED 104)|
|15:00||Olivier Schiffmann (ENS, France and IAS), Macdonald polynomials and elliptic curves|
|NOTE: This talk will take place in room FTX 232. The speaker will be presenting live via the internet.
We give a construction of Macdonald polynomials in terms of the moduli space of vector bundles on an elliptic curve (defined over a finite field). We give an interpretation of certain positivity conjectures of Garsia and Haiman in this context.
|16:00||Reception (Department lounge - KED 104)|
The first two talks will take place in room KED B005, located in the Department of Mathematics and Statistics at the University of Ottawa (click here for map). The last talk will take place in FTX 232 (for those who do not know where this is, meet in the lobby of the math department at 14:50 and we will walk over). Information on travel and accommodation can be found here. The Quality Hotel is located a few blocks from the department.
For further information please contact Alistair Savage.
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