Seminar in mathematics for Masters students

Term
Fall 2010
Institution
VU Amsterdam
Links

Contents

Overview

The aim of this compulsory seminar for Master students at the VU and at the UvA is twofold: firstly, to make you acquainted with working with original, classical papers and extracting the essence of (an aspect of) it that is suitable to be presented in a talk, and secondly, to practice giving good mathematical talks to a mathematical audience. The format will be as follows: students form small work groups (around 4 students in each group) with a common interest in a particular mathematical direction. They work together on one or more pieces of literature and present those, or pieces thereof, in a one-hour talk. Students are also required to give feedback to other students for their talks. Tilman and Eric are available for advice and help with the papers and the preparation of the talk.

Time Thursdays 14.00-17.00, Sept 9 – Oct 21 and Nov 4 – Dec 16
Place

Block 1: VU, W&N Gebouw. Block 2: UvA, Science Park

Sept 9: S-640
Sept 16: C-147
Sept 23 – Oct 14: P-656
Oct 21: F-654

Nov 4: G2.02
Nov 11: A1.04
Nov 18: A1.10
Nov 25: A1.04
Dec 2: BV-0H19 (Bellevue building, VU)
Dec 9: A1.10
Dec 16: A1.10

ECTS credits

6

Target group Master students (compulsory), other participants should inquire
Requirements Presence (for the full 3 hours), giving two talks, working with peers, giving feedback
Assessment Quality of the talks

Syllabus

September 9: First meeting, organization, formation of work groups.

Group Members
Dynamical Systems

Ondrej Budáč, Aleksander Czechowski, Robert Noest, Martino Pitruzzella

Algebraic Topology Milo Bogaard, Rik Danko, Joachim de Ronde, Wicher Malten, Shan Shah
Graph Theory Fadaei Faryaneh, Roel Niessen, Piet van Eegken, Evalien IJsendijk
Expander Graphs Apo Cihangir, Sarah Gaaf, Gabriel Molinari, Rachid Tahri
Probability Theory Ellis de Groote, Marijn Jansen, Jurriën Knibbe, Aris Meems
Queuing Theory Rens Bankras, Doortje de Wiljes, Kick Koning, Richard Kraaij, Dorthe van Waarden

September 16: Workshop on how to give a good mathematics seminar.

Sources:

September 23: Rens, Rik
September 30: Shan
October 7: Jurriën, Rashid
October 14: Ellis, Sarah
October 21: Milo, Roel
November 4: Dorthe, Wicher
November 11: Evalien, Faryaneh
November 18: Doortje, Robert
November 25: Martino, Piet
December 2: Apo, Richard
December 9: Gabriel, Kick
December 16: Joachim, Marijn

Papers

Algebraic Topology

Algebra

Dynamical Systems

Analytic continuation and fundamental solutions of linear partial differential operators

Let P be a nonnegative polynomial on Rn, and let f be a smooth compactly supported function on Rn. For any complex number s with Re(s)>0, the integral Is(f) of the complex function f Ps dx over Rn is convergent and depends holomorphically on s.

In 1972 Joseph Bernstein proved by algebraic methods(!) that such integrals always satisfy a functional equation in s, and as a result, admit a meromorphic extension to the whole complex plane. This resolved a famous conjecture by Gelfand.

As an application one obtains the classical result of Hormander-Lojasiewicz that all linear partial differential operators with constant coefficients admit a tempered fundamental solution. The method to derive such a fundamental solution from the analytic continuation of the above type integrals goes back to Marcel Riesz.

A beautiful account of the algebraic aspects of this story can be found in the introduction, Chapter 1, and the first section of Chapter 7 of:

The original papers of Bernstein:

The above "elementary" proof of Bernstein was not the first general proof, but the original proofs (in 1968, due to Bernstein and Gelfand, and to Atiyah independently) rely on a very deep theorem, the "resolution of singularities" theorem of Hironaka. On the other hand, this original proof reveals more information, namely the deep fact that the poles of Is(f) are all located in negative rational numbers which appear in finitely many arithmetic progressions (depending on the polynomial P). It is very interesting to study Atiyah’s paper as well:

To get a better idea of the nature of the fundamental solutions for constant coefficient differential operators it is useful to study the first 3 chapters of the classical paper by Marcel Riesz, in which he uses analytic continuation of Riemann-Liouville integrals to study the fundamental solutions of wave equations and the Huygens principle:

Ramanujan Graphs and expanders.

Expander graphs are graphs which are sparse but at the same time have a high connectivity. This is a highly desirable property for networks, but expander families of graphs are difficult to construct. One way to quantify the expander quality of a graph is by the "spectral gap", the difference between the largest and second largest eigenvalue of the combinatorial Lapace operator of the graph. The theoretical optimum is reached by so-called Ramanujan graphs, which were introduced and constructed by Lubotzky, Phillips and Sarnak, and by Margulis, in 1988, by a number theoretical methods.

We propose to study this subject starting with the following two classical papers:

The following nice book gives a good exposition on the topic:

See also