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Upcoming Summer Graduate Schools

  1. 2021 CRM-PIMS Summer School in Probability (Virtual School)

    Organizers: LEAD Louigi Addario-Berry (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 Sherrington-Kirkpatrick model, aka the randomly-weighted complete graph. Edge weights are indicated using grayscale. Six distinguished vertices have been randomly chosen; edges between those vertices are shaded black to form a "hidden signal".

    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 Mar 24, 2021 01:52 AM PDT
  2. Sparsity of Algebraic Points (Virtual School)

    Organizers: Philipp Habegger (University of Basel), LEAD Hector Pasten (Pontificia Universidad Católica de Chile)
    The Corvaja-Zannier proof of Siegel's theorem using subspaces. Illustrated by Sofía Pastén Vásquez.

    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 Mar 05, 2021 11:34 AM PST
  3. Mathematics of Big Data: Sketching and (Multi-) Linear Algebra (Virtual School)

    Organizers: LEAD Kenneth Clarkson (IBM Research Division), Lior Horesh (IBM Thomas J. Watson Research Center), Misha Kilmer (Tufts University), Tamara Kolda (Sandia National Laboratories), Shashanka Ubaru (IBM Thomas J. Watson Research Center)
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    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 Mar 15, 2021 03:16 PM PDT
  4. [POSTPONED] 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 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
  5. Gauge Theory in Geometry and Topology (Virtual School)

    Organizers: Lynn Heller (Universität Hannover), Francesco Lin (Columbia University), LEAD Laura Starkston (University of California, Davis), Boyu Zhang (Princeton University)
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    Image by Nick Schmitt

    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 Mar 05, 2021 11:35 AM PST
  6. Random Conformal Geometry (Virtual School)

    Organizers: Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), LEAD Fredrik Viklund (Royal Institute of Technology)
    a random quasiconformal map obtained from Beltrami equation by randomly assigning the values of +-1/2 for the Beltrami coefficient on small squares subdividing the unit square

    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 Mar 19, 2021 03:03 PM PDT
  7. Foundations and Frontiers of Probabilistic Proofs (Virtual School)

    Organizers: Alessandro Chiesa (University of California, Berkeley), Tom Gur (University of Warwick)
    Proofs main logo
    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.

    Updated on Apr 19, 2021 06:23 PM PDT