# 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:*

**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!

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*

**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

**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

**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

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 Zürich" - symposium with two laureates

The Institute for Mathematical Research (FIM) at ETH Zürich is hosting a one-day symposium on the 20th of January with talks given by Abel Laureates Sir Michael Atiyah and Endre Szemerédi and by John Rognes, chair of the Abel Prize Committee. All talks will take place in the main building of ETH Zürich, Rämistrasse 101, in lecture hall G 3.

(14.01.2016) More### The 3rd Heidelberg Laureate Forum opened

From August 23 to 28, the third Heidelberg Laureate Forum gathers 26 recipients of the Turing Award, the Fields Medal, the Abel Prize and the Nevanlinna Prize. Five Abel Laureates take part this year. They will meet with young researchers from over 50 nations. Kirsti Strřm Bull, President of the Norwegian Academy of Science and Letters, pointed out in her speech at opening ceremony that intellectual exchange between the laureates of mathematics and computer science and the young researchers is in the spirit of the Abel Prize that also supports activities for children and young people.

(25.08.2015) More### First step towards the Abel Prize 2016

The 15th of September 2015 is the deadline for nominating candidates for the Abel Prize 2016. The Abel Committee met for the first time at the Norwegian Academy of Science and Letters on the 26th and 27th of September to embark on the task of selecting a deserving candidate. The Academy's choice of Laureate is based on the Abel committee's recommendation. The Abel Committee is led by Professor John Rognes (left). To the right: Kristian Ranestad, chair of the Abel board.

(19.08.2015) More### Nash and Nirenberg received the Abel Prize from the King of Norway

John F. Nash Jr. and Louis Nirenberg received the 2015 Abel Prize from His Majesty King Harald V at the award ceremony in Oslo on 19 May. The two American mathematicians receive the prize "for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis." The laureates share the 6 mill NOK (about EUR 700,000 or USD 750,000) prize money.

(20.05.2015) More### Abel celebration in Oslo and Bergen

When the Abel laureates John F. Nash Jr. and Louis Nirenberg arrive in Oslo they can look forward to a week of mathematical celebrations in Oslo and Bergen. The highlight will be when they receive the Abel Prize from H.M. the King at the award ceremony on the 19th of May in the University Aula in Oslo. Earlier the same day they will be received in audience at the Royal Palace. The Abel Banquet at Akershus Castle will be hosted by Torbjřrn Rře Isaksen, Minister of Education and Research.

(11.05.2015) More