Date: Thursday 23. January 2025
Time: 09:30 – 16:20
Venue: Lipschitz-Saal (Room 1.016)
Mathematikzentrum der Universität Bonn, Endenicher Allee 60

Registration: https://www.mathematics.uni-bonn.de/en/events/registration-abel/

The event is free, but registration is required.

A light lunch will be served.

Programme

09:30 – Welcome by Annelin Eriksen, president the Norwegian Academy of
Science and Letters and Helge Holden, chair of the Abel Committee,
moderator
09:40 – László Lovász, Abel laureate 2021
10:40 – Break
10:50 – Hee Oh, Yale University, USA, The Abel Committee
11:50 – Break, coffee, tea, water
12:00 – Peter Scholze, University of Bonn, Germany, Fields medalist 2018
13:00 – Light lunch
14:00 – Maryna Viazovska, Institute of Mathematics of the École
Polytechnique Fédérale de Lausanne, Switzerland, Fields medalist 2022
15:00 – Break
15:10 – Avi Wigderson, Abel laureate 2021
16:10 – Conclusion of symposium and welcome to reception Michael Hoch,
Rector University of Bonn
16:20 – Informal reception

Lectures

László Lovász: Electrical capacity and cut capacity: A tale of two communities

Abstract:
Discrete structures behave quite differently from continuous ones. The great
success story of mathematics in the 18-th and 19-th centuries was the
development of analysis, with extremely powerful tools such as differential
equations or Fourier series, and with by now very standard methods like the
famous (infamous?) epsilon-delta technique. Discrete mathematics had a later
start, but for importance of its applications it is catching up. Its proof
techniques are different, such as enumeration or induction. In the continuous
world, algorithms are mostly computations, with numerical analysis at the
center. In the discrete world, algorithmic ideas are more diverse, including
searching, recurrence, and (yes!) pulling in methods from continuous
mathematics.

I will argue that these worlds may be far apart in the social sense, but not in
the mathematical sense. Submodular setfunctions are fundamental in at least two
areas: matroid theory (initiated by Whitney in the 1930's), and in the theory
of electrical capacities (where Choquet introduced them in the 1950's). Both
areas have a lot of important research, but I have not found any collaboration
of even cross-references. I will illustrate that connections between
submodularity in the discrete and continuous worlds are deep and lead to
fruitful questions and results.

Hee Oh: A Euclidean Traveller's Journey in a Hyperbolic World

Abstract:
This lecture explores the journey of an imaginary traveler who moves along a Euclidean line. In a Euclidean torus world, she enjoys the sights of a subtorus, as established by Kronecker's classical theorem of 1884, which states that the closure of any line in the Euclidean torus is always a subtorus. But what happens when she ventures into a closed hyperbolic manifold world or even a hyperbolic manifold of infinite volume? We will describe her views (the closure of the Euclidean line she travels on) and delve into the mathematics that explain the various closures, accompanied by numerous illustrations.

Peter Scholze: Holomorphic functions on sectors

Abstract:
A standard phenomenon in complex geometry is that the asymptotic behaviour of functions like $\mathrm{exp}(1/x)$ depends on the direction from which the singularity $x=0$ is approached. This leads to the study of holomorphic functions on sectors, constraining $\mathrm{arg}(x)$ to an interval. I will explain a highly structured version of this story, including an accompanying "holomorphic functional calculus on sectors" that applies even to $\mathbb C((x))$, thereby showing that the analytic spectrum of the Laurent series field $\mathbb C((x))$ is not a point, but rather has a nontrivial map to the circle.

Maryana Viazovska:

Avi Wigderson: The Value of Errors in Proofs - the fascinating journey from Turing's 1936 R != RE to the 2020 breakthrough of MIP* = RE

Abstract:
In 2020, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title "MIP* = RE", impacting and surprising not only computational complexity theory but also some areas of math and physics. Specifically, it resolved, in the negative, the "Connes' embedding conjecture" in the area of von-Neumann algebras, and the "Tsirelson problem" in quantum information theory.
Read paper

As it happens, both acronyms MIP* and RE represent proof systems, of a very different nature. To explain them, we'll take a meandering journey through the classical and modern definitions of proof. I hope to explain how the methodology of computational complexity theory, especially modeling and classification (both problems and proofs) by algorithmic efficiency, naturally leads to the generation of new such notions and results (and more acronyms, like NP). A special focus will be on notions of proof which allow interaction, randomness, and errors, and their surprising power and magical properties. 

The talk will be non-technical, and requires no special background.