Welcome to the *Algebra and its Applications* seminar
home page, which has been copied from Ettore's old webpage!

The seminar is organized by
Amod
Agashe. Please email
`agashe@math.fsu.edu` to contact me.

In 1976 Ribet published a proof of the converse of Herbrand's theorem. Though the result itself is an important result, the method of proof employed has far surpassed the result in terms of importance. We will review Ribet's proof and then see how it can be generalized to give results on Main Conjectures in Iwasawa Theory as well as provide lower bounds on Selmer groups attached to p-adic Galois representations.

We count the cardinality of the sets of points of twisted Shimura varieties over finite fields and then we compute the zeta function of the twisted Shimura varieties. This result is a generalization of some famous papers of Kottwitz, and it is a part of the Langlands program.

Let F be a finite extension of Q_p. About fifteen years ago, Barthel and Livne partially classified the irreducible mod p representations of GL_2(F). In particular, they found four mutually exclusive classes of such representations and classified three of them completely. In this talk, we will discuss what is known about the irreducible mod p representations of GL_n(F) and show that the problem of classifying them, analogously to the theory of complex representations of GL_n(F), reduces to that of classifying certain "supersingular" representations and that of classifying the constituents of parabolic inductions. All these notions will be explained in the talk. We will also sketch out the mod p local Langlands correspondence and mention the relevance of our results to understanding it.

We will give an alternative definition for the higher arithmetic Chow groups defined by Goncharov, which is suitable for quasi-projective arithmetic varieties over a field. These groups are the analog, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow group of Burgos and Gillet-Soul. Our new construction is shown to be functorial and is endowed with a product structure. This work is part of the speaker's thesis entitled "On higher arithmetic intersection theory", under the supervision of Dr. J. I Burgos Gil.

Click here for a pdf version of the abstract.

Four dimensional topology hosts phenomena not present in any other dimension. For example, if an $n$-manifold admits infinitely many different (non-diffeomorphic) smooth structures then $n=4$. The talk reviews the basic topological invariants in order to point out the clash between the smooth and topological categories. Extra attention is set on an open 4-manifold of particular physical interest and its peculiar behavior.

For first order difference equations, it is not hard to find a closed form solution in terms of Gamma functions. For difference equations of degree greater than 1, the question is how to decide if it has solutions that also satisfy a first order difference equation. Such solutions are called hypergeometric solutions. The goal of the talk is to find all hypergeometric solutions of linear difference equations with polynomial coefficients.

An overview of the main features/distinctions of exotic $mathbb{R}^4$'s.

The talk will give an introduction to Khovanov homology and its relation to the Jones polynomial.

A Fulton-MacPherson bivariant theory is a machinery that combines a (covariant) homology theory and a (contravariant) cohomology theory into one bivariant theory. The bivariant groups are associated to 'maps' and carry various structures (product, pull-back, push-forword). After defining bivariant theories, we consider the 'singular bivariant theory' for topological spaces which combines singular homology and singular cohomology. We show how certain standard constructions on singular (co)homology (e.g., cup/cap/slant products, Gysin maps, etc.) follow formally from the existence of the bivariant theory.

I will discuss analogues of regular representations of infinite-dimensional groups of upper triangular matrices, as studied by A. Kosyak. These representations generate von Neumann algebras, which are factors in certain cases. I will demonstrate that in the reducible case, the obtained factors are always of type III_1 (according to the classification of A. Connes).

The group SL(2,Z) admits a well known action on the complex upper half plane, as well as an action on a tree. Both of them can be pushed to the respective limit sets, which admit a Patterson-Sullivan measure. There is an equivariant map relating the two actions. Using groupoids we propose a tentative framework for comparing the noncommutative geometry of these actions.

These talks will present an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key novelty in this approach is that it is set up in a way that will make it possible to prove a new complexity result for this algorithm which is asymptotically sharp. We also introduce a practical improvement to prior algorithms which we call early termination. Our algorithm actually performs better than prior algorithms in many common classes of polynomials (including irreducibles).