Analysis Seminar

Speaker: Sudeshna Basu (George Washington University)

"Non compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces"

Time: 14:00

Room: MC 107

We introduce Schatten-class operators in p-adic Hilbert spaces and study their properties and give several examples. We also show that the Trace class operators in p-adic Hilbert spaces strictly contains the class of completely continuous operators.This gives a totally new perspective to the space of compact operators in p-adic Hilbert spaces. Apart from its intrinsic interest as discussed in this paper, p-adic functional analysis is a fast growing research area and has numerous applications in differential equations, statistics, quantum physics, dynamical systems, cognitive sciences, psychology and sociology, to name a few.

Geometry and Topology

Speaker: Ajneet Dhillon (Western)

"The Fundamental Group Scheme-An Overview "

Time: 15:30

Room: MC 107

Analysis Seminar

Speaker: Seyed Mohammad Hadi Seyedinejad (Western)

"Testing local regularity of complex analytic mappings by fibred powers"

Time: 14:40

Room: MC 107

This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"A Topological proof of Abel-Ruffini theorem"

Time: 16:00

Room: MC 107

Galois referred to his theory as the ``Theory of Ambiguities". In his last letter he writes: “Mes principales m'editations depuis quelque temps etaient dirigees sur l’application a l’analyse transcendante de la theorie de l’ambiguite. Il s’agissait de voir a priori dans une relation entre quantites ou fonctions transcendantes quels echanges on pouvait faire, quelles quantites on pouvait substituer aux quantites donnees sans que la relation put cesser d’avoir lieu. Cela fait reconnaitre tout de suite l’impossibilite de beaucoup d’expressions que l’on pourrait chercher. Mais je n’ai pas le temps et mes id´ees ne sont pas encore bien d´eveloppees sur ce terrain qui est immense...”

This idea of Galois is indeed so rich and the territory is so vast that even after 200 years of mathematics we are still not sure it has fully delivered all its potential. I shall introduce the idea of analytic continuation and use it to define the monodromy of an algebraic function, as an instance of the application of ambiguities in analysis and algebra. I shall then indicate how this quickly leads to a proof of impossibility of solving general quintics by radicals, the Abel-Ruffini theorem.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Gauss-Bonnet Formula for Hypersurfaces"

Time: 10:30

Room: MC 108

Gauss–Bonnet theorem or Gauss–Bonnet formula is one of the star attractions of modern differential geometry. It states that for a compact oriented manifold M, the "curvatura integra" over M is equal to a multiple of Euler Characteristic of M.

We shall give an "extrinsic" proof for M as an embedded submanifold (actually an even dimensional hyper surface)of a Euclidean space. The proof is heavily based on the Poincare-Hopf index theorem which states that the sum of indexes of a smooth vector field over M is equal to the Euler characteristic of M.

Algebra Seminar

Speaker: Mohab Safey El Din (Universite Pierre et Marie Curie (Paris 6), INRIA Paris-Rocquencourt Research Centre)

"Polynomial System Solving over the Reals: Algorithms, Complexity, Implementations and Applications "

Time: 14:30

Room: MC 320

Solving non-linear algebraic problems is one of the major challenges in scientific computing. In several areas of engineering sciences, algebraic problems encode geometric conditions on variables taking their values over the reals. Thus, most of the time, one aims to obtain some informations on the real solution set of polynomial systems. The resolution of these problems often has a complexity which is exponential in the number of variables.

In this talk, I will review some geometric and algebraic techniques which enable to obtain fast practical algorithms meeting the best known complexity bounds. These algorithms are implemented in the maple package (RAGlib: The Real Algebraic Geometry library) which has the feature to provide algorithms of asymptotically optimal algorithms in real geometry. Its practical performances will be discussed and some applications will be presented.

This talk is based on joint work with J.C. Faugere, A. Greuet, E. Schost, PJ Spaenlehauer and L. Zhi.