# 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 Oxford" with lecture by Sir Andrew Wiles

It has become a tradition that the Abel Committee after its final meeting hosts an Abel seminar. This year the committee meeting at London Mathematical Society was followed by "Abel in Oxford" with Abel Laureate Andrew Wiles as the main speaker. Martin Bridson, head of the Mathematical Institute, University of Oxford and Camilla Serck-Hanssen, Vice President of the Norwegian Academy of Science and Letters gave introductory talks.

(02.02.2018) More### Yves Meyer received the Abel Prize from H.M. King Harald

H.M. King Harald presented the Abel Prize to Yves Meyer of the École normale supérieure Paris-Saclay, France at an award ceremony in Oslo on 23 May. He receives the prize for his pivotal role in the development of the mathematical theory of wavelets, says John Rognes, chair of the Abel committee. Among the prominent guests attending the award ceremony was the French ambassador to Norway, Jean-François Dobelle and the Norwegian Minister of Education and Research, Torbjřrn Rře Isaksen.

(26.05.2017) More### Three days of celebration for Abel Laureate Yves Meyer

His Majesty King Harald will present the Abel Prize to Yves Meyer at an award ceremony in Oslo on 23 May. He receives the prize "for his pivotal role in the development of the mathematical theory of wavelets", to quote the Abel committee. Yves Meyer, of the École normale supérieure Paris-Saclay, was the visionary leader in the modern development of this theory, at the intersection of mathematics, information technology and computational science.

(12.05.2017) More### The Abel lectures 2017

Abel Laureate Yves Meyer gave his prize lecture at the University of Oslo on the 24th of May, with following Abel lectures by Stéphane Mallat, Ingrid Daubechies og Emmanuel Jean Candčs. Watch the lectures here.

(08.05.2017) More### Congratulations from AMS President

"On behalf of the American Mathematical Society, it is my great pleasure to congratulate Professor Yves Meyer, recipient of the 2017 Abel Prize. Professor Meyer has been a visionary in a broad range of fields, including number theory and differential equations. His fundamental work in the theory of wavelets has transformed the world of signal processing and has led to a myriad of practical applications." -- AMS President Kenneth A. Ribet (University of California, Berkeley)

Photo: Jim Brook

(02.04.2017) More