Viewing archives for CMI HIMR Summer School

Formalizing Class Field Theory

Class Field Theory in its cohomological form is one of the highlights of early 20th century mathematics, and is now understood as the abelian case of the Langlands Philosophy. Although it sounds like science fiction to many mathematicians, some computer scientists are arguing that AI methods are progressing so fast that soon computers will be helping humans to push back the boundaries of research in the Langlands Philosophy. However, there is currently no concrete evidence that this is happening. Furthermore, using a language model alone to do mathematics at this level is problematic, because language models are error-prone, and one error in a mathematical argument invalidates it.

This summer school does not have anything to do with AI, but it has a lot to do with class field theory. During the school, we will be teaching class field theory to the Lean theorem prover. You can imagine the school as a group of people collaborating on writing a Bourbaki-like document explaining class field theory. Or you can imagine it as a group of people turning class field theory into a bunch of levels of a puzzle game and then solving these levels. Or you can imagine it as a group of people creating training data for a theorem prover-backed AI which can then try and learn some of these interesting mathematical ideas.

Prerequisites: Students should have a good knowledge of Lean and a basic knowledge of cohomology groups, Cyclotomic fields, Kummer extensions and local fields. It is recommended that students work through some of the free online textbook Mathematics in Lean prior to the summer school. 

This event will be taking place in-person. Accommodation, lunch and subsistence will be provided; participants are expected to fund their own travel (limited funds may be available for those unable to secure travel costs from their home institution).

Registration will open in January 2025.

This summer school is jointly funded by the Clay Mathematics Institute, the Heilbronn Institute for Mathematical Research and the Mathematical Institute at the University of Oxford.

Details

Venue: Mathematical Institute, University of Oxford

CMI-HIMR Summer School on Symmetry and Randomness

This year’s summer school will focus on the mathematics of symmetry and randomness, where probability theory comes together with analysis, geometry and group theory to help understand highly symmetric structures. 

The mini-courses will present aspects of random walks on infinite graphs and groups in connection with geometric group theory; the mathematics of percolation theory especially on large transitive graphs; as well as spectral and mixing time estimates for finite Markov chains with an emphasis on the cut-off phenomenon, and much more. 

Students will have the opportunity to be introduced to these topics as well as to hear lectures by leading figures in the area.

Short Courses

Tom Hutchcroft (Caltech), Structure vs. expansion: probability on transitive graphs*

Justin Salez (Université Paris-Dauphine), The cutoff phenomenon for finite Markov chains*

Tianyi Zheng (UC San Diego), Random walks on self-similar groups: boundary behaviour and amenability*

*Provisional title

The summer school will also feature guest lectures from Itai Benjamini (Weizmann Institute of Science), Persi Diaconis (Stanford) and Geoffrey Grimmett (Cambridge).

Registration closes on Friday 15 March 2024

This event will be taking place in-person. Accommodation, lunch and subsistence will be provided, while participants are expected to fund their own travel (limited funds may be available for those unable to secure travel costs from their home institution).

Details

Venue: University of Bristol

CMI-HIMR Summer School on Probabilistic Group Theory

In recent years, probabilistic methods have been at the heart of many spectacular advances in group theory and related areas, finding a diverse range of applications. This summer school will introduce a wide audience of graduate students and early career researchers to some of the most exciting recent developments. The programme will feature four short courses from world-leading experts in the area, together with tutored problem sessions for participants.

Short Courses

Martin Liebeck (Imperial), Generation properties of finite and algebraic simple groups

Cheryl Praeger (Western Australia), Probabilistic group theory from an algorithmic perspective

Aner Shalev (HUJ), Words, groups and probability

Pham Huu Tiep (Rutgers), Character bounds for finite simple groups

The summer school will also feature guest lectures from Emmanuel Breuillard (Oxford), Ben Green (Oxford) and Colva Roney-Dougal (St Andrews).

Accommodation, lunch and subsistence will be provided, while participants are expected to fund their own travel (limited funds may be available for those unable to secure travel costs from their home institution).

Registration

Registration closes on 24th February 2023.

Details

Venue: University of Bristol

CMI-HIMR Summer School in Computational Number Theory

A postgraduate mathematics two-week summer school jointly funded by the Clay Mathematics Institute and the Heilbronn Institute for Mathematical Research. Student places are offered with accommodation, refreshments and subsistence allowance.

Eights speakers have been invited to deliver four hours of lectures each with discussion workshops scheduled for each topic: John Cremona (Warwick), Celine Maistret (Bristol), Adam Morgan (KCL), Steffen Mueller (Bonn), Rachel Newton (Reading), Samir Siksek (Warwick), Andrew Sutherland (MIT), John Voight (Dartmouth).

Image: Chrystal Cherniwchan

Details

Venue: University of Bristol

CMI-HIMR Integrable Probability Summer School

The domain of integrable probability seeks to understand universal phenomena through exactly solvable probabilistic systems of algebraic and representation theoretic origin. This school will feature five mini courses (and an accompanying research seminar run through July and August) on methods to find and analyze such solvable systems, as well as on more probabilistic approaches to expand the universality classes around them. While the target audience is graduate students or early career researchers who work in this area or nearby fields, all are welcome to attend these lectures.

As part of this Online Summer School, there will be five mini courses, each comprised of three lectures of an hour and fifteen minutes.

Mini Course Lecturers
Hugo Duminil-Copin (IHÉS), On the free energy of the six-vertex model
Talk 1, July 27, 2020
Talk 2, July 29, 2020
Talk 3, July 31, 2020

Rick Kenyon (Yale), Limit shapes and variational principles
Talk 1, July 29, 2020
Talk 2, July 30, 2020
Talk 3, July 31, 2020

Greta Panova (USC), Talk 1: Algebraic combinatorics basics; Talks 2 and 3: Algebraic combinatorics meets probability: statistical mechanics and asymptotics
Talk 1, July 28, 2020
Talk 2, July 30, 2020
Talk 3, July 31, 2020

Fabio Toninelli (TU Vienna), (2+1)-dimensional growth models and the AKPZ universality class
Talk 1, July 27, 2020
Talk 2, July 28, 2020
Talk 3, July 29, 2020

Michael Wheeler (Melbourne), An invitation to the q-Whittaker polynomials
Talk 1, July 28, 2020
Talk 2, July 29, 2020
Talk 3, July 30, 2020

In addition to the mini courses, weekly Friday seminars wil be held throughout July and August.  

Seminar Speakers
Amol Aggarwal (CMI and Columbia University), Limit shapes and local statistics for the stochastic six-vertex model
Jinho Baik (University of Michigan), Relaxation time limit of periodic TASEP
Evgeni Dimitrov (Columbia University), Two-point convergence of the stochastic six-vertex model to the Airy process
Yan Fyodorov (King’s College London), On eigenvector statistics in non-normal random matrices
Vadim Gorin (MIT and University of Wisconsin, Madison), Random matrices as seen from infinite beta, Part 1; Part 2
Alan Hammond (University of California, Berkeley), KPZ universality via Brownian Gibbs analysis
Alisa Knizel (Columbia University), Invariant measure for the open KPZ equation
Leonid Petrov (University of Virginia), Symmetric functions from vertex models
Jeremy Quastel (University of Toronto), From TASEP to the KPZ fixed point and KP
Tomohiro Sasamoto (Tokyo Institute of Technology), Spin current statistics for the quantum 1D XX spin chain and the Bessel kernel
Li-Cheng Tsai (Rutgers University), Moments and tails of stochastic PDEs
Jon Warren (University of Warwick), Reflecting Brownian motions and point-to-line last passage percolation
Nikos Zygouras (University of Warwick), Variants of RSK and polymers

With questions regarding the scientific program, please contact Ivan Corwin or Alexei Borodin.  With all other questions, please contact admin@claymath.org.
 

This postgraduate mathematics summer school is jointly funded by the Clay Mathematics Institute, the Heilbronn Institute for Mathematical Research, the Mathematical Institute at University of Oxford, and the NSF Integrable Probability Focused Research Grant DMS:1664619, DMS:1664650.

Details

Venue: Online

Downloads

Mini Course Abstracts

CMI-HIMR Dynamics and Geometry Summer School

The Clay Mathematics Institute and Heilbronn Institute for Mathematical Research are organising a virtual two-week summer school on “Dynamics and Geometry”, hosted by the School of Mathematics at the University of Bristol. This will be the third in a series of UK postgraduate mathematics summer schools, jointly funded by the Clay Mathematics Institute and the Heilbronn Institute for Mathematical Research.

Dynamics and Geometry are two intertwined areas of mathematics that have seen revolutionary breakthroughs in recent years. In this summer school world-leading experts will speak about some of these developments, alongside problem sessions and other opportunities for discussion and interaction. 

Mini Course Lecturers
Francisco Arana-Herrera (Stanford), Counting simple closed hyperbolic geodesics on surfaces
Elise Goujard (Bordeaux), Masur-Veech volumes of strata of moduli spaces of differentials
Ursula Hamenstädt (Bonn), Geometric flows with uniform hyperbolic localizations and applications
Yair Minsky (Yale), Fibered 3-manifolds
Amir Mohammadi (UC San Diego), Random walks and rigidity
Barak Weiss (Tel Aviv), Horocyclic and horospherical dynamics on the moduli space of translation surfaces 

Distinguished Lecturer
Alex Eskin (Chicago), Actions on non-abelian groups and measure rigidity (video)

Details

Venue: Online

CMI-HIMR Summer School on Unlikely Intersections in Diophantine Geometry

Unlikely intersection problems emerged around 25 years ago, with several different motivations. Bombieri, Masser and Zannier, Zilber, and Pink independently arrived at conjectures now known as the Zilber-Pink conjectures, which are still wide open in general. The Pila-Wilkie Theorem has become a key tool in the subject, and related developments in functional transcendence are now having an impact in other parts of diophantine geometry.

Key recent breakthroughs include Habegger and Pila’s proof of the Zilber-Pink conjecture for curves in abelian varieties, Tsimerman’s proof of the André-Oort conjecture for Ag, and work by Dimitrov, Gao, and Habegger on uniformities in Faltings’ Theorem. There have also been significant developments by Binyamini around the Pila-Wilkie Theorem towards effectivity and improved bounds.

This summer school will feature three short courses, each comprised of five lectures.  The school is aimed at a wide audience of graduate students and early career researchers; the requisite background material from model theory and o-minimality will be covered at the start of the week.  Daily afternoon tutorial sessions will allow students time to work on problems and to ask questions.

Short Course Lecturers
Gal Binyamini (Weizmann), Counting rational points on transcendental sets
Laura Capuano (Roma Tre), Unlikely intersection problems
Gabriel Dill (Oxford), Functional transcendence in diophantine geometry

The summer school will also feature guest lectures from Paola D’Aquino (Caserta), Lars Kühne (Copenhagen), and Jonathan Pila (Oxford).

Registration: Please register your interest by completing the application form; your c.v. and a short supporting letter from your PhD supervisor will be required with your application. Closing date for registration is 31 May 2022. Successful applicants will be notified by 15 June 2022.  Student places are offered with accommodation and meals.

With questions regarding the scientific program, please contact Gareth Jones.  With all other questions, please contact admin@claymath.org.

This postgraduate mathematics summer school is jointly funded by the Clay Mathematics Institute, the Heilbronn Institute for Mathematical Research and the Mathematical Institute at the University of Oxford.

We are regularly reviewing the Covid-19 situation and our current intention is to run an in-person residential program. Should we find ourselves unable to run an in-person program, it would be our intention to move to an online program. 

Details

Venue: University of Oxford