Microlocal analysis originated in the study of linear partial differential equations (PDEs) in the high-frequency regime, through a combination of ideas from Fourier analysis and classical Hamiltonian mechanics. In parallel, similar ideas and methods had been developed since the early times of quantum mechanics, the smallness of Planck’s constant allowing to use semiclassical methods. The junction between these two points of view (microlocal and semiclassical) only emerged in 1970s, and has taken its full place in the PDE community in the last 20 years. This methodology resulted in major advances in the understanding of linear and nonlinear PDEs in the last 50 years. Moreover, microlocal methods continue to find new applications in diverse areas of mathematical analysis, such as the spectral theory of nonselfadjoint operators, scattering theory, and inverse problems.Updated on Dec 01, 2020 02:04 PM PST
The courses in this summer school focus on mathematical models of group dynamics, how to describe their dynamics and their scaling limits, and the connection to discrete and continuous optimization problems.
The phrase "group dynamics" is used loosely here -- it may refer to species migration, the spread of a virus, or the propagation of electrons through an inhomogeneous medium, to name a few examples. Very commonly, such systems can be described via stochastic processes which approximately behave like the solution of an appropriate partial differential equation in the large-population limit.Updated on Jan 04, 2021 11:52 AM PST
The theory of Diophantine equations is understood today as the study of algebraic points in algebraic varieties, and it is often the case that algebraic points of arithmetic relevance are expected to be sparse.
This summer school will introduce the participants to two of the main techniques in the subject: (i) the filtration method to prove algebraic degeneracy of integral points by means of the subspace theorem, leading to special cases of conjectures by Bombieri, Lang, and Vojta, and (ii) unlikely intersections through o-minimality and bi-algebraic geometry, leading to results in the context of the Manin-Mumford conjecture, the André-Oort conjecture, and generalizations. This SGS should provide an entry point to a very active research area in modern number theory.Updated on Sep 15, 2020 04:26 PM PDT
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.Updated on Aug 04, 2020 09:38 AM 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 Feb 03, 2021 03:04 PM PST
Figure 1. A rotationally symmetric solution to the self-duality equations on an open and dense subset of the torus. Singularities appear where the surface intersects the ideal boundary at infinity of the hyperbolic 3-space visualized by the wireframe.
Gauge theory is a geometric language used to formulate many fundamental physical phenomena, which has also had profound impact on our understanding of topology. The main idea is to study the space of solutions to partial differential equations admitting a very large group of local symmetries. Starting in the late 1970s, mathematicians began to unravel surprising connections between gauge theory and many aspects of geometric analysis, algebraic geometry and low-dimensional topology. This influence of gauge theory in geometry and topology is pervasive nowadays, and new developments continue to emerge.
The goal of the summer school is to introduce students to the foundational aspects of gauge theory, and explore their relations to geometric analysis and low-dimensional topology. By the end of the two-week program, the students will understand the relevant analytic and geometric aspects of several partial differential equations of current interest (including the Yang-Mills ASD equations, the Seiberg-Witten equations, and the Hitchin equations) and some of their most impactful applications to problems in geometry and topology.Updated on Dec 23, 2020 12:30 PM PST
This Summer Graduate School will cover basic tools that are instrumental in Random Conformal Geometry (the investigation of analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics) and are at the foundation of the subsequent semester-long program "The Analysis and Geometry of Random Spaces". Specific topics are Conformal Field Theory, Brownian Loops and related processes, Quasiconformal Maps, as well as Loewner Energy and Teichmüller Theory.Updated on Aug 04, 2020 10:24 AM 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 Feb 11, 2021 01:48 PM PST
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