# 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 Prize celebrations in Oslo

#### Karen Uhlenbeck received the Abel Prize from H.M. King Harald V

His Majesty King Harald V presented the 2019 Abel Prize to Karen Uhlenbeck at an award ceremony in the University Aula in Oslo on the 21st of May. Uhlenbeck is "Professor Emerita of Mathematics and Sid W. Richardson Regents Chair at the University of Texas at Austin" and "Visitor in the School of Mathematics at the Institute for Advanced Study".

(17.05.2019) More### Abel lectures at the University of Oslo

#### The lectures will be streamed

Karen Uhlenbeck gave her Abel Prize lecture on the 22nd of May at the University of Oslo. Chuu-LianTerng and Robert Bryant gave lectures related to Uhlenbeck's work. The popular science lecture was given by stand-up mathematician Matt Parker.

(14.05.2019) More### Karen Uhlenbeck first woman to win the Abel Prize

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck of the University of Texas at Austin, USA “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.”

His Majesty King Harald V will present the Abel Prize to the laureate at the award ceremony in Oslo on the 21^{st} of May.

### Congratulations to Karen Uhlenbeck from University of Texas at Austin

"At the University of Texas at Austin and the Department of Mathematics, we are delighted and tremendously proud of Karen Uhlenbeck, recipient of the 2019 Abel Prize" - Thomas Chen, Chair of the University of Texas at Austin Math Department

(19.03.2019) More### Congratulations to Karen Uhlenbeck from AMS President

"On behalf of the American Mathematical Society, it is my great pleasure to congratulate Professor Karen Uhlenbeck, recipient of the 2019 Abel Prize. Professor Uhlenbeck has made legendary advances in several fields of mathematics. Her early groundbreaking work on harmonic maps gave rise to a new field, geometric analysis. Her analysis via gauge theory of solutions of Yang-Mills equations, had and will continue to have a profound influence on all future work in this field. She transformed the fields of geometry and analysis, crossing boundaries and making deep discoveries at the interfaces." AMS President Jill Pipher

(19.03.2019) More