Hjem

The Abel Lectures

Avi Wigderson, Professor at the School of Mathematics, Institute for Advanced Study, Princeton will be delivering this year's Science Lecture at the University of Oslo, on May 23rd. The lecture is called Randomness and pseudorandomness, and is one of four lectures to be held this day in connection with the Abel Prize week. The other speakers are Abel Laureate Endre Szemerédi, László Lovász and Timothy Gowers.

Avi Wigderson: Randomness and pseudorandomness:

Obligatorisk!

Abstract

Is the universe inherently deterministic or probabilistic? Perhaps more importantly - can we tell the difference between the two?

Humanity has pondered the meaning and utility of randomness for millennia.

There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable!

Obligatorisk!

Which of these applications survive if the universe had no randomness in it at all? Which of them survive if only poor quality randomness is available, e.g. that arises from "unpredictable" phenomena like the weather or the stock market?

Pseudorandomness is the study, by mathematicians and computer scientists, of deterministic structures which share some properties of random ones.

Understanding pseudorandom objects and constructing them efficiently leads to a surprisingly positive answers to the questions above, namely that much can be done with poor quality randomness, of even without any randomness at all. I plan to explain key aspects of this theory, and mention some of Endre Szemeredi's contributions to pseudorandomness.

The talk is aimed at a general audience, and no particular background will be assumed.

 

Endre Szemerédi: In every chaos there is an order

Obligatorisk!

Abstract:

The chaos and order will be defined relative to three problems.

1.    Arithmetic progressions

This part is connected to a problem of Erdős and Turán from the 1930's. Related to the van der Waerden theorem, they asked if the density version of that result also holds:

    Is it true that an infinite sequence of integers of positive (lower) density contains arbitrary long arithmetic progressions?

The first result in this direction was due to K. F. Roth, who proved that any sequence of integers of positive (lower) density contains a three-term arithmetic progression.

I will give a short history of the generalization of Roth's result and explain some ideas about the "easiest"" proof.

2.    Long arithmetic progression in subset sums

I will give exact bound for the size of longest arithmetic progression in subset sums. In addition, I shall describe the structure of the subset sums, and give applications in number theory and probability theory.

3.    Embedding sparse graphs into large graphs

 

László Lovász:  The many facets of the Regularity Lemm

Obligatorisk!

Abstract:

The Regularity Lemma of Szemeredi, first obtained in the context of his theorem on arithmetic progressions in dense seuqences, has become one of the most important and most powerful tools in graph theory. It is basic in
extremal graph theory and in the theory of property testing. Weaker versions with better bounds (Frieze and Kannan) and stronger versions (Alon, Fisher, Krivelevich and Szegedy) have been proved and used. However, the
significance of it goes way beyond graph theory: it can be viewed as statement in approximation theory, as a compactness result for the completion of the space of finite graphs, as a result about the dimensionality of a metric space associated with a graph, as a statement in information theory. It serves as the archetypal example of the dichotomy between structure and randomness as pointed out by Tao. Its extensions to hypergraphs, a difficult problem solved by Gowers and by Rodl, Skokan and Schacht, connects with higher order Fourier analysis.

 

Timothy Gowers:  The afterlife of Szemerédi’s theorem

Obligatorisk!

Abstract:

Szemerédi's theorem asserts that every set of integers of positive upper density contains arbitrarily long arithmetic progressions. This result has been extraordinarily influential, partly because of the tools that Szemerédi introduced in order to prove it, and partly because subsequent efforts to understand the result more fully have led to progress in many other areas of mathematics, including combinatorics, ergodic theory, harmonic analysis and number theory. I shall discuss some of these later developments, to which Szemerédi himself made several essential contributions.

The Abel Lectures

Program

Location: Georg Sverdrups hus, Aud. 1, University of Oslo

23 May 2012 at 10:00 - 23 May 2012 at 15:15

Coffee and tea from 9:30 outside Auditorium 1.


10.00  Welome by Pro-Rector Inga Bostad, President of The Norwegian Academy of Science and Letters Nils Chr. Stenseth, and Chair of the Abel Committee Ragni Piene.

10.10  Professor Endre Szemerédi: "In Every Chaos There is an Order"

11.00  Coffee/tea

11.30  Professor László Lovász: The many facets of the Regularity Lemma"

12:30 Lunch (requires registration)

13.30 Professor Timothy Gowers: "The afterlife of Szemerédi's theorem"

14:15  Coffee/tea

14:30  Science Lecture: Avi Wigderson, "Randomness and Pseudorandomness"

15.15  Ending by Chair of the Abel Committee Ragni Piene

Abel in Paris, January 28th 2015

Lecture by Abel laureate Mikhail Gromov

The Institut Henri Poincaré in Paris will host the "Abel in Paris" event in the afternoon of Wednesday January 28th.  In connection with the final meeting of the current Abel committee, whose task it is to select the Abel prize laureate for 2015, there will be an afternoon of lectures, aimed at a broad mathematical audience, by Abel laureate Mikhail Gromov (IHES) and Abel committee members Éva Tardos (Cornell) and Rahul Pandharipande (ETH Zürich).

(22.12.2014) More

Abel Laureates meet young researchers in Heidelberg

The 2nd Heidelberg Laureate Forum brings together 200 gifted young researchers and winners of the most prestigious prizes in mathematics (Abel Prize and Fields Medal) and computer science (Nevanlinna Prize and Turing Award). The three Abel Laureates who are present here in Heidelberg are Sir Michael Atiyah, Srinivasa S. R. Varadhan and John Tate (picture). 

(23.09.2014) More

Abel Lecture by John Milnor at SEOUL ICM 2014

"Topology through Four Centuries" is the title of John Milnor's Abel Lecture. The Abel Lecture will take place on Friday the 15th of August from 6:00 -7:00 p.m. in 3F Hall D, Coex Convention Center and will be open to everybody. John Milnor received the Abel Prize in 2011 "for pioneering discoveries in topology, geometry and algebra".

 

(28.07.2014) More

Abel celebration in Oslo, Stavanger and Stockholm

When Abel Laureate Yakov G. Sinai arrives in Oslo on 18 May together with his wife Elena B. Vul, who is also a mathematician, they can look forward to a week of mathematics celebration in Oslo, Stavanger and Stockholm. The highlight will be when Yakov G. Sinai receives the Abel Prize from HRH The Crown Prince at the award ceremony on 20 May in the University Aula in Oslo. Earlier the same day Sinai will be received in audience at the Royal Palace. The Minister of Education and Research, Torbjřrn Rře Isaksen, will be among the many prominent guests at the award ceremony.

(13.05.2014) More

Mathematical billiards and chaos

Abel Lectures 2014 at the University of Oslo, Norway

The Abel Laureate Yakov Sinai will give his prize lecture at the University of Oslo on 21 May. This will be followed by two Abel Lectures by Gregory Margulis and Konstantin Khanin. Domokos Szász will give the popular science lecture titeled "Mathematical billiards and chaos". See lecture summaries below.

(08.05.2014) More
Det Norske Videnskaps-Akademi
Drammensveien 78
N-0271 Oslo
Telefon: +47 22 12 10 90
Telefaks: +47 22 12 10 99
E-post: abelprisen@dnva.no
 
Nettredaktřr: Anne-Marie Astad
Design og teknisk lřsning: Ravn Webveveriet AS
 
The Norwegian Academy of Science and Letters
Drammensveien 78
N-0271 Oslo, Norway
Telephone: + 47 22 12 10 90
Fax: + 47 22 12 10 99
E-mail: abelprisen@dnva.no
Web editor: Anne-Marie Astad
Design and technical solutions: Ravn Webveveriet AS