# Summer Graduate School

Parent Program: |
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Location: |
Courant Institute, New York |

## Show List of Lecturers

- Antonio Auffinger (Northwestern University)
- Wei-Kuo Chen (University of Minnesota)
- Eliran Subag (Weizmann Institute of Science)

## Show List of Teaching Assistants

- Ratul Biswas (University of Minnesota)
- Yuxin Zhou (University of Chicago)

While their original aim was to explain the strange behavior of certain magnetic alloys, the study of spin glass models has led to a far-reaching and beautiful physical theory whose techniques have been applied to a myriad of problems in theoretical computer science, statistics, optimization and biology. As many of the physical predictions can be formulated as purely mathematical questions, often extremely hard, about large random systems in high dimensions, in recent decades a new area of research has emerged in probability theory around these problems.

Mathematically, a mean-field spin glass model is a Gaussian process (random function) on the discrete hypercube or the sphere in high dimensions. A fundamental challenge in their analysis is, roughly speaking, to understand the size and structure of their super-level sets as the dimension tends to infinity, which are often studied through smooth objects like the free energy and Gibbs measure whose origin is in statistical physics. The aim of the summer school is to introduce students to landmark results on the latter while emphasizing the techniques and ideas that were developed to obtain them, as well as exposing the students to some recent research topics.

#### School Structure

The schedule will consist of two lectures and two problem sessions each day.

#### Prerequisites

The knowledge that is expected from the students is the material covered in first-year courses in probability theory and analysis. More advanced material in probability is helpful but not required as we will review what we need in class. As such, the school is well suited for any graduate student in mathematics with interest in probability or mathematical physics.

Students are expected to be familiar with concepts covered in first-year graduate courses in analysis and probability. A good preparation for this summer school includes Chapter 1 (Measure Theory), Chapter 2 (Laws of Large Numbers) and 3 (Central Limit Theory) in Probability: Theory and Examples, 5th ed, by R. Durrett (2019)

Exercises in these chapters are recommended.

#### Application Procedure

SLMath is only able to support a limited number of students to attend this school. Therefore, it is likely that only one student per institution will be funded by SLMath.

For **eligibility** and **how to apply**, see the **Summer Graduate Schools homepage.**

#### Venue

The summer school will be held at the Courant Institute in New York City. Students will be housed on the campus of NYU.

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

spin glasses

Gaussian fields

Free energy

Gibbs measure

Parisi formula

Replica symmetry breaking

Ultrametricity

complexity

Kac-Rice formula

critical points

TAP approach

approximate message passing

inference problems

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**