
2021 CRMPIMS Summer School in Probability (CRM, Montreal)
Organizers: LEAD Louigi AddarioBerry (McGill University), Omer Angel (University of British Columbia), Alexander Fribergh (University of Montreal), Mathav Murugan (University of British Columbia), Edwin Perkins (University of British Columbia)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 largepopulation limit.
Updated on Aug 31, 2020 10:40 AM PDT 
Sparsity of Algebraic Points
Organizers: Philipp Habegger (University of Basel), LEAD Hector Pasten (Pontificia Universidad Católica de Chile)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 ominimality and bialgebraic geometry, leading to results in the context of the ManinMumford conjecture, the AndréOort 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 
Mathematics of Big Data: Sketching and (Multi) Linear Algebra
Organizers: LEAD Kenneth Clarkson (IBM Research Division), Lior Horesh (IBM Thomas J. Watson Research Center)This summer school will introduce graduate students to sketchingbased approaches to computational linear and multilinear 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 
Foundations and Frontiers of Probabilistic Proofs (Zurich, Switzerland)
Organizers: Alessandro Chiesa (University of California, Berkeley), Tom Gur (University of Warwick)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 complexitytheoretic study of proof verification has led to exciting reenvisionings of mathematical proofs. For example, probabilistically checkable proofs (PCPs) admit localtoglobal 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 ultrafast protocols that allow one party to check the correctness of the computation performed by another, untrusted, party. These protocols have even been realized within recentlydeployed technology, for example, as part of cryptographic constructions known as succinct noninteractive 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 cuttingedge research in this area.
Updated on Aug 04, 2020 09:47 AM PDT 
Metric Geometry and Geometric Analysis (Oxford, United Kingdom)
Organizers: LEAD Cornelia Drutu (University of Oxford), Panos Papazoglou (University of Oxford)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 nonsmooth 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 Aug 04, 2020 10:05 AM PDT 
Gauge Theory in Geometry and Topology
Organizers: Lynn Heller (Universität Hannover), Francesco Lin (Columbia University), LEAD Laura Starkston (University of California, Davis), Boyu Zhang (Princeton University)Figure 1. A rotationally symmetric solution to the selfduality equations on an open and dense subset of the torus. Singularities appear where the surface intersects the ideal boundary at infinity of the hyperbolic 3space 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 lowdimensional 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 lowdimensional topology. By the end of the twoweek program, the students will understand the relevant analytic and geometric aspects of several partial differential equations of current interest (including the YangMills ASD equations, the SeibergWitten equations, and the Hitchin equations) and some of their most impactful applications to problems in geometry and topology.
Updated on Aug 04, 2020 10:04 AM PDT 
Random Conformal Geometry
Organizers: Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), LEAD Fredrik Viklund (Royal Institute of Technology)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 semesterlong 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 
Recent Topics in Well Posedness (Taipei, Taiwan)
Organizers: Jungkai Chen (National Taiwan University), Yoshikazu Giga (University of Tokyo), Maria Schonbek (University of California, Santa Cruz), Tsuyoshi Yoneda (University of Tokyo)The purpose of the workshop is to introduce graduate students to fundamental results on the NavierStokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.
Updated on Jul 20, 2020 10:01 AM PDT

Upcoming Summer Graduate Schools 