**July 17, 2017 at 09:00**

**Speaker(s):**
**David BRYDGES, Gordon SLADE, Ajay CHANDRA**

**Center(s): **University of British Columbia, Vancouver, Canada, University of Warwick, UK

Bilbao summer school 'Probabilistic approaches in Mathematical Physics'
**Monday 17 July - Saturday 22 July 2017**
**BCAM-Basque Center for Applied Mathematics**
Mazarredo 14, Bilbao, Basque Country, Spain

**Organisers:**
Stefan ADAMS (University of Warwick)

Jean-Bernard BRU (BCAM, University of Basque Country, Ikerbasque)

Walter de SIQUEIRA PEDRA (University of São Paulo)

**Sponsors:**
BCAM, University of São Paulo, FAPESP, CNPq, MASDOC (University of Warwick), International Association of Mathematical Physics (IAMP), MINECO

**Abstract:**
The recently developed theory of regularity structures by Martin Hairer has many relevant connections to the theory of renormalisation and stochastic PDE's. The renormalisation group (RG) goes back to Wilson (Nobel Prize 1982) and is a dominant calculational and conceptual tool of theoretical physics and probability. RG is a map that replaces a model by a simpler one having the same macroscopic predictions. By iteration to a fixed point one obtains a simple model having the macroscopic predictions of a whole class of more detailed models. The aim of the school is to give an elementary introduction to both RG and regularity structures, as well as their connection. The lecture series will be complemented by exercise/discussion sections in the afternoon. This will allow students to deepen their understanding by raising questions and doing exercises under guidance of PhD students and Postdocs working in that field. The second main objective is to highlight open problems in both fields and their connection to enable the student to gain insight in a currently important and active field in probability, applied probability, analysis, and mathematical physics.

**Lecturers:**
**David BRYDGES** is an internationally very well-known expert in RG and related subjects since the 1980s. He has developed a mathematical approach to RG which is specially designed for students with interest in probability and mathematical physics.

**Gordon SLADE** is famous for his fundamental contributions to the self-avoiding random walk and corresponding RG techniques. In landmark work in collaboration with David BRYDGES he obtained important results on the weakly self-avoiding random walk in dimension 4.

**Ajay CHANDRA** is currently a Postdoctoral researcher and lecturer at the University of Warwick working in the group of Martin HAIRER. He is working at the interface between regularity structures for stochastic PDEs and RG. He has developed a lecture series which will allow students to learn the basic techniques and to obtain an overview of current research and future open problems.

**Exercises:**
Organised by

**Ben WALLACE** (University of British Columbia,Vancouver) and

**Pavlos TSATSOULIS** (University of Warwick, UK)

**Students:**
We cordially invite students (PhD and/or MSc) who want to participate to register on the website and also by sending to

** pamp@bcamath.org** your CV with the name of your adviser and your research area. Indeed, the school will offer financial support during the week of the workshop for a limited number of students. In addition we only have very limited funds to support travel for some students. In case travel support is needed - please apply by email to the organiser.

**Postdocs:**
We also cordially invite young researchers (Postdocs or faculty). Even if our funds are rather limited, we are able to offer affordable accommodation. Depending on our budget we will be able to help younger researchers to participate.

**Poster session: **
We will organise a poster session where students can present a poster presenting their research and give a brief talk.

**Additional talks: **
There will be a couple additional talks by different speakers (including some PhD students or Postdoctoral researchers).

**Lecture series:**
**An Introduction to Regularity Structures**, by Ajay CHANDRA, University of Warwick

In this short course we will start by giving a short introduction and motivation for the study of singular parabolic SPDE. We will then investigate each of the components of Martin Hairer's theory of regularity structures and understand how they come together to prove local existence theorems for such equations.

See

**https://arxiv.org/abs/1508.03616**.

**Critical Phenomena and the Renormalisation Group**, by David BRYDGES and Gordon SLADE, University of British Columbia

The subject of phase transitions and critical phenomena in physics has had a major influence on mathematics for over half a century, especially in probability theory but also in other disciplines. Now the influence on mathematics is greater than ever before.

The subject is mainly focussed on the study of various specific models. This course will provide an introduction to the |ϕ|⁴ spin model. Emphasis will be placed on the universal critical behaviour of the |ϕ|⁴ model, characterised by universal critical exponents. Recent joint work with R. Bauerschmidt for the case of dimension d=4, based on a rigorous implementation of Wilson's renormalisation group method, will be introduced in the course.

The level of the course will be suitable for graduate students in probability, analysis, and mathematical physics; the course will provide the necessary background and will not assume specialised knowledge.

A brief overview of the main results that motivate the course can be found at:

**https://arxiv.org/abs/1602.04048**.

The overview reviews to several other papers, available at:

**https://www.math.ubc.ca/~slade/research.html**
We kindly ask you make your

**REGISTRATION**, (just below the info of this activity on the site of workshops on the web of BCAM) if you plan to participate.