A basic enzymatic mechanism

The aim of the course is to learn how tools from algebraic geometry (in particular, from computational and real algebraic geometry) can be used to analyze standard models in molecular biology. Particularly, these models are key ingredients in the development of Systems and Synthetic biology, two active research areas focusing on understanding, modifying, and implementing the design principles of living systems.

We will focus on the mathematical aspects of the methods, and exemplify and apply the theory to real networks, thereby introducing the participants to relevant problems and mechanisms in molecular biology. As a counterpart, however, the participants will also see how this field has in the past challenged current methods, mainly in the realm of real algebraic geometry, and has given rise to new general and purely theoretical results on polynomial equations. We will end our lectures with an overview of open questions in both fields.

This two week school will focus on spectral theory of periodic, almost-periodic, and random operators.  The main aim of this school is to teach the students who work in one of these areas, methods used in parallel problems, explain the similarities between all these areas and show them the `big picture'.

This school is associated with an upcoming research program at MSRI under the same title. The goal of the school is to equip students unfamiliar with these topics with the mathematical and theoretical computer science toolbox that forms the foundation of market and mechanism design.

Soap bubble: equilibrium solution of the rescaled mean curvature flow and constant curvature surface.

This graduate summer school will introduce students to two important and inter-related fields of differential geometry: geometric flows and minimal surfaces.

Geometric flows have had far reaching influences on numerous branches of mathematics and other scientific disciplines. An outstanding example is the completion of Hamilton’s Ricci flow program by Perelman, leading to the resolution of the Poincare conjecture and Thurston’s geometrization conjecture for 3-manifolds. In this part of the summer school, students will be guided through basic topics and ideas in the study of geometric flows.

Since Penrose used variations of volume to formulate and study black holes in general relativity (in his Nobel prize-winning work), the intriguing connections between minimal surfaces and general relativity have been a strong driving force for the modern developments of both research areas. This part of the summer school will introduce students to the basic theory of minimal submanifolds and its applications in Riemannian geometry and general relativity.

The curriculum of this program will be accessible and will have a broad appeal to graduate students from a variety of mathematical areas, introducing some of the latest developments in each area and the remaining open problems therein, while simultaneously emphasizing their synergy.

The overarching goal of this summer school is to expose the students both to modern forms of unsupervised learning — in the form of geometrical and topological data analysis — and to supervised learning — in the form of (deep) neural networks applied to regression/classification problems. The organizers have opted for a lighter exposure to a broader range of topics. Using the metaphor of a meal, we are offering 2 + 2 samplers — geometry and topology for data analysis + theoretical and practical deep learning — rather than 1 + 1 main dishes. The main goal, thus, is to inspire the students to learn more about one or several of the topics covered in the school.

The expected learning outcomes for students attending the school are the following:

1. An introduction to how concepts and tools from geometry and topology can be leveraged to perform data analysis in situations where the data are not labeled.

2. An introduction to recent and ongoing theoretical and methodological/practical developments in the use of neural networks for data analysis (deep learning).

Schur quartic x 4−xy3 = z 4−zu3 and several of the 64 lines that it contains

Derived algebraic geometry is an ‘update’ of algebraic geometry using ‘derived’ (roughly speaking, homological) techniques. This requires recasting the very foundations of the field: rings have to be replaced by differential graded algebras (or other forms of derived rings), categories by higher categories, and so on. The result is a powerful set of new tools, useful both within algebraic geometry and in related areas. The school serves as an introduction to these techniques, focusing on their applications.

The school is built around two related courses on geometric (‘derived spaces’) and categorical (‘derived categories’) aspects of the theory. Our goal is to explain the key ideas and concepts, while trying to keep technicalities to a minimum.

The goal of the summer school is for the students to first become familiar with the concept of concentration of measure in different settings (Euclidean, Riemannian and discrete), and the main open problems surrounding it. The students will later become familiar with the proof techniques that involve the different types of localization and obtain expertise on the ways to apply the localization techniques. After attending the graduate school, the students are expected to have the necessary background that would give them a chance to both conduct research around open problems in concentration of measure, find new applications to existing localization techniques and perhaps also develop new localization techniques.

This summer school will introduce graduate students to sketching-based approaches to computational linear and multi-linear algebra. Sketching here refers to a set of techniques for compressing a matrix, to one with fewer rows, or columns, or entries, usually via various kinds of random linear maps. We will discuss matrix computations, tensor algebras, and such sketching techniques, together with their applications and analysis.

Several executions of a 3-dimensional sumcheck protocol with a random order of directions (thanks to Dev Ojha for creating the diagram)

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.

This summer school will familiarize students with the basic problems of the mathematical theory of Euclidean quantum fields. The lectures will introduce some of its prominent models and analyze them via the so called “stochastic quantization” methods, involving recently developed stochastic and PDE techniques. This is an area which is highly interdisciplinary combining ideas ranging from the theory of partial differential equations, to stochastic analysis, to mathematical physics. Our goal is to bring together students which are maybe familiar with some but not all of these subjects and teach them how to integrate these different tools to solve cutting-edge problems of Euclidean quantum field theory.