The two topics, combinatorial theory of free resolutions and differential graded algebra techniques in homological algebra, each have a long and rich history in commutative algebra and its applications to algebraic geometry. Free resolutions are at the center of much of the study in the field and these two approaches give powerful tools for their study and their application to other problems. Neither of these topics is generally covered in graduate courses. Furthermore, recent developments have exhibited exciting interplay between the two subjects. The purpose of the school is to introduce the graduate students to these subjects and these new developments. The school will consist of two lectures each day and carefully planned problem sessions designed to reinforce the foundational material and to give them a chance to experiment with problems involving the interplay between the two subjects.Updated on Jul 26, 2019 03:43 PM PDT
[The image on this vase from Minoan Crete, dated on 1500-2000 BC, resembles an ancient solution to the Curve shortening flow - one of the most basic geometric flows. The vase is at Heraklion Archaeological Museum]
The purpose of the workshop is to introduce graduate students to some of the most important geometric evolution equations.
This is an area of geometric analysis that lies at the interface of differential geometry and partial differential equations. The lectures will begin with an introduction to nonlinear diffusion equations and continue with classical results on the Ricci Flow, the Mean curvature flow and other fully non-linear extrinsic flows such as the Gauss curvature flow. The lectures will also include geometric applications such as isoperimetric inequalities, topological applications such as the Poincaré onjecture, as well as recent important developments related to the study of singularities and ancient solutions.Updated on Aug 02, 2019 12:19 PM PDT
The purpose of the summer school will be to introduce graduate students to effective methods in algebraic theories of differential and difference equations with emphasis on their model-theoretic foundations and to demonstrate recent applications of these techniques to studying dynamic models arising in sciences. While these topics comprise a coherent and rich subject, they appear in graduate coursework in at best a piecemeal way, and then only as components of classes for other aims. With this Summer Graduate School, students will learn both the theoretical basis of differential and difference algebra and how to use these methods to solve practical problems. Beyond the lectures, the graduate students will meet daily in problem sessions and will participate in one-on-one mentoring sessions with the lecturers and organizers.Updated on Jul 26, 2019 03:42 PM PDT
Representation Theory has undergone a revolution in recent years, with the development of what is now known as higher representation theory. In particular, the notion of categorification has led to the resolution of many problems previously considered to be intractable.
The school will begin by providing students with a brief but thorough introduction to what could be termed the “bread and butter of modern representation theory”, i.e., compact Lie groups and their representation theory; character theory; structure theory of algebraic groups.
We will then continue on to a number of more specialized topics. The final mix will depend on discussions with the prospective lecturers, but we envisage such topics as:
• modular representation theory of finite groups (blocks, defect groups, Broué’s conjecture);
• perverse sheaves and the geometric Satake correspondence;
• the representation theory of real Lie groups.Updated on Aug 08, 2019 09:36 AM PDT
The topic of random graphs is at the forefront of applied probability, and it is one of the central topics in multidisciplinary science where mathematical ideas are used to model and understand the real world. At the same time, random graphs pose challenging mathematical problems that have attracted the attention from probabilists and combinatorialists since the 1960, with the pioneering work of Erdös and Rényi. Around the turn of the millennium, very large data sets started to become available, and several applied disciplines started to realize that many real-world networks, even though they are from various different origins, share many fascinating features. In particular, many of such networks are small worlds, meaning that graph distances in them are typically quite small, and they are scalefree, in the sense that there are enormous differences in the number of connections that their elements make. In particular, such networks are quite different from the classical random graph models, such as proposed by Erdös and Rényi.Updated on Jul 26, 2019 03:40 PM PDT
Probability theory, statistics as well as mathematical physics have increasingly been used in computer science. The goal of this school is to provide a unique opportunity for graduate students and young researchers to developed multi-disciplinary skills in a rapidly evolving area of mathematics.
The topics would include spin glasses, constraint satisfiability, randomized algorithms, Monte-Carlo Markov chains and high-dimensional statistics, sparse and random graphs, computational complexity, estimation and approximation algorithms. Those topics will fall into two main categories, on the one hand problems related to spin glasses and on the other hand random algorithms.
The part of the summer school dedicated to spin glasses will be split into three parts: an introductory course about traditional spin glasses followed by two more advanced courses where spin glasses meet computer science in addition to a talk on dynamics of spin glasses. The part of the summer school on random algorithms will consist of an introductory course on phase transitions in large random structures, followed by advanced courses on theoretical bounds for computational complexity in reconstruction and inference, and on understanding rare events in random graphs and models of statistical mechanics.
The two introductory courses on spin glasses and on random algorithms will be accompanied by three exercises sessions of one hour. A one hour exercises session will follow each of the three sessions of a course for both the introductory course on spin glasses and the introductory course on random algorithms. Exercises sessions will be led by an assistant, but will primarily focus on participation of the students.Updated on Sep 18, 2019 03:30 PM PDT
[Image: The simplest interesting case of linkage (liaison) of curves in projective 3-space. We see two quadric surfaces, one of which is a cone, meeting in the union of a line (vertical in the illustration) and a twisted cubic (snaking up from the bottom left to the upper right, tangent to the line at the origin.]
The theory of algebraic curves, arguably the oldest branch of algebraic geometry, has seen major developments in recent years, for example in the study of syzygies, and around questions about moduli spaces and Hilbert schemes of curves. The theory is rich in research activity and unsolved problems. There is an encyclopedic work by Arbarello, Cornalba, Griffiths and Harris, but there is no modern text that could be used as a textbook and that goes beyond the basics of the theory. We have embarked on a project to write a book at roughly the level of the wonderful book on complex algebraic surfaces by Arnaud Beauville. The intent can be seen from a list of some major topics it will treat:
- Linear series and Brill-Noether theory
- Personalities: curves in projective space with low genus and degree
- Overview of moduli and Jacobians
- Hilbert schemes
- Syzygies and linkage
The school will have two series of lectures, one by Harris and one by Eisenbud. Harris’ lectures will focus on the more geometric side of the theory, including Brill-Noether theory, families of curves and Jacobians; while Eisenbud’s lectures will focus on the more algebraic side of the theory, including properties of the homogeneous coordinate rings of curves (Cohen-Macaulay, Gorenstein, free resolutions, scrolls, ...) Both lecturers will rely on chapters from the forthcoming book, which should be finished in large part by the time of the school. In addition, some of Eisenbud’s lectures will treat the use of Macaulay2 to investigate the projective embeddings of curves.Updated on Aug 14, 2019 03:45 PM PDT
Proofs are at the foundations of mathematics. Viewed through the lens of theoretical computer science, verifying the correctness of a mathematical proof is a fundamental computational task. Indeed, the P versus NP problem, which deals precisely with the complexity of proof verification, is one of the most important open problems in all of mathematics.
The complexity-theoretic study of proof verification has led to exciting reenvisionings of mathematical proofs. For example, probabilistically checkable proofs (PCPs) admit local-to-global structure that allows verifying a proof by reading only a minuscule portion of it. As another example, interactive proofs allow for verification via a conversation between a prover and a verifier, instead of the traditional static sequence of logical statements. The study of such proof systems has drawn upon deep mathematical tools to derive numerous applications to the theory of computation and beyond.
In recent years, such probabilistic proofs received much attention due to a new motivation, delegation of computation, which is the emphasis of this summer school. This paradigm admits ultra-fast protocols that allow one party to check the correctness of the computation performed by another, untrusted, party. These protocols have even been realized within recently-deployed technology, for example, as part of cryptographic constructions known as succinct non-interactive arguments of knowledge (SNARKs).
This summer school will provide an introduction to the field of probabilistic proofs and the beautiful mathematics behind it, as well as prepare students for conducting cutting-edge research in this area.Updated on Sep 12, 2019 01:58 PM PDT
The purpose of the summer school is to introduce graduate students to key mainstream directions in the recent development of geometry, which sprang from Riemannian Geometry in an attempt to use its methods in various contexts of non-smooth geometry. This concerns recent developments in metric generalizations of the theory of nonpositively curved spaces and discretizations of methods in geometry, geometric measure theory and global analysis. The metric geometry perspective gave rise to new results and problems in Riemannian Geometry as well.
All these themes are intertwined and have developed either together or greatly influencing one another. The summer school will introduce some of the latest developments and the remaining open problems in these very modern areas, and will emphasize their synergy.Updated on Jul 31, 2019 11:07 AM PDT
The study of nonnegative polynomials and sums of squares is a classical area of real algebraic geometry dating back to Hilbert’s 17th problem. It also has rich connections to real analysis via duality and moment problems. In the last 15 years, sums of squares relaxations have found a wide array of applications from very applied areas (e.g., robotics, computer vision, and machine learning) to theoretical applications (e.g., extremal combinatorics, theoretical computer science). Also, an intimate connection between sums of squares and classical algebraic geometry has been found. Work in this area requires a blend of ideas and techniques from algebraic geometry, convex geometry and representation theory. After an introduction to nonnegative polynomials, sums of squares and semidefinite optimization, we will focus on the following three topics:
- Sums of squares on real varieties (sets defined by real polynomial equations) and connections with classical algebraic geometry.
- Sums of squares method for proving graph density inequalities in extremal combinatorics. Here addition and multiplication take place in the gluing algebra of partially labelled graphs.
- Sums of squares relaxations for convex hulls of real varieties and theta-bodies with applications in optimization.
The summer school will give a self-contained introduction aimed at beginning graduate students, and introduce participants to the latest developments. In addition to attending the lectures, students will meet in intensive problem and discussion sessions that will explore and extend the topics developed in the lectures.Updated on Jul 26, 2019 03:40 PM PDT
The purpose of this two weeks school is to introduce graduate students to the state of the art methods and results in the study of incompressible Euler’s equations in general, and water waves in particular. This is a research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade.
The goal is to present the main current research directions in water waves. We will begin with the physical derivation of the equations, and present some of the analytic tools needed in study. The final goal will be two-fold, namely (i) to understand the local solvability of the Cauchy problem for water waves, as well as (ii) to describe the long time behavior of solutions.
Through the lectures and associated problem sessions, students will learn about a number of new analysis tools which are not routinely taught in a graduate school curriculum. The goal is to help students acquire the knowledge needed in order to start research in water waves and Euler equations.Updated on Jul 26, 2019 03:40 PM PDT
Upcoming Summer Graduate Schools