The Summer Schools in Probability are a highlight of Canadian probability and are internationally significant.  Launched by PIMS in 2004, the school takes the form of two main 4-week courses along with three mini-courses. The schools have played a major role in the development of an exceptionally strong community of young probabilist in Canada, North America and overseas. This will be the 13th time this school has run.

<p>A minimal free resolution supported on a simplicial complex</p>

The 2025 SMS will allow graduate students to learn about a number of recent trends and advances in the field of commutative algebra. The aim of the SMS is to provide an “on-ramp” for graduate students interested in algebra, combinatorics, and/or algebraic combinatorics to learn more about commutative algebra’s interaction with these fields. The introductory courses will introduce fundamental skills in commutative algebra, the more intermediate courses will expose students to cutting-edge research in the field. The school will focus on four topics within commutative algebra: Combinatorial Methods, Homological Methods, Computational Methods, and Characteristic p Methods. The SMS will provide both a series of introductory lectures and intermediate/advanced lectures from leaders in one of the four areas. The lectures will include a series of problem sessions that will allow participants to develop and hone their skills in these areas, which will be especially helpful for new people to the field. Participants will be encouraged to work collaboratively, both to enhance their own mathematical networks as well as to promote future collaborations beyond the school.

<p>A display of the evolution of an Erdos-Renyi random graph .&nbsp;</p>

Random graphs are ubiquitous in modern probability theory. Besides their intrinsic mathematical beauty, they are also used to model complex networks. In the early 2000’s, I. Benjamini and O. Schramm introduced a mathematical framework in which they endowed the set of locally finite rooted connected graphs with the structure of a Polish space, called the local topology. The goal of this summer school is to introduce the framework of local limits of random graphs, the concepts of Benjamini-Schramm (or unbiased) limits and unimodularity, as well as the most important applications. The lectures will be delivered by Nicolas Curien (Prof. Paris-Saclay University) and Justin Salez (Prof. Université Paris-Dauphine) and will be complemented by many problem sessions, where students will work in small groups under the guidance of teaching assistants, who are researchers in the field. 

This summer school offers an exceptional opportunity for participants to delve into the intricate realm of statistical optimal transport theory. This captivating field stands at the crossroads of multiple disciplines, drawing from a rich tapestry of mathematical insights from diverse subjects, including partial differential equations, stochastic analysis, convex geometry, statistics, and machine learning, crafting a vibrant and interdisciplinary landscape. The foremost objective of this summer school is to create a dynamic learning environment that unites students from diverse backgrounds such as PDE theory, probability, or optimal transport. 

From micro to macro, sea ice supports life in the polar regions. Left: illustration of the porous sea ice microstructure, home to a microbial community of ice endemic organisms including bacteria and algae. Right: two penguins on Antarctic sea ice.

In this summer school, students will be introduced to mathematical and computational modeling of sea ice and polar ecosystems in a warming climate. As a material, sea ice is a multiscale composite structured on length scales ranging from tenths of millimeters to tens of kilometers. From tiny brine inclusions and surface melt ponds of increasing complexity, to ice floes of varying sizes in a seawater host, a principal challenge is how to find sea ice effective properties that are relevant to larger scale climate and process models, given data on smaller scale structures. Similarly, the sea ice ecosystem ranges from algae living in the brine inclusions to charismatic megafauna like penguins and polar bears, whose diets depend critically, down the line, on the tiny sea ice extremophiles. The dynamics of sea ice microbial communities are regulated by the physics of the ice microstructure, and, in turn, many of these microbes modify their environment by secreting extracellular polymeric substances. In addition to sea ice and its ecosystems, we will consider mathematical modeling of the broader climate system, including energy balance models, climate and ecological tipping points, and global circulation models.

<p>A plane partition and an hourglass plabic graph</p>

This school will introduce students to a range of powerful combinatorial tools used to understand algebraic objects ranging from the homogeneous coordinate ring of the Grassmannian to symmetric functions.   The summer school will center around two main lecture series "Webs and Plabic Graphs" and "Vertex Models and Applications".   While the exact applications differ, both courses will center on graphical models for algebraic problems closely related to Grassmannian and its generalizations.  This school will be accessible to a wide range of students.  Students will leave the school with a solid grasp of the combinatorics of webs, plabic graphs, and the six-vertex model, an understanding of their algebraic applications, and a taste of current research directions.

<p>The discriminant ∆ detects singular varieties. The picture shows three different scenarios: solutions of quadratic polynomials, cubic plane curves and cubic surfaces.</p>

This summer school will offer a hands-on introduction to discriminants, with a view towards modern applications. Starting from the basics of computational algebraic geometry and toric geometry, the school will gently introduce participants to the foundations of discriminants. A particular emphasis will be put on computing discriminants of polynomial systems using computer algebra software. Then, we will dive into three applications of discriminants: algebraic statistics, geometric modeling, and particle physics. Here, discriminants contribute to the study of maximum likelihood estimation, to finding practical parametrizations of geometric objects, and to computations of scattering amplitudes. We will explain recently discovered unexpected connections between these three applications. In addition to lectures, the summer school will have daily collaborative exercise sessions which will be guided by the teaching assistants and will include software demonstrations.

The school will consist of two courses: Homological Mirror Symmetry and Algebraic Models for Spaces. These courses will be planned and taught by organisers with the help of teaching assistants for the problem sessions. The school will be aimed at a wide range of graduate students, from students with a Bachelor degree to beginning PhD students. The lectures and problem sessions will be complemented by a poster session in week one and a total of four introductory research talks on Friday afternoons. 

One of the core elements of applied mathematics is mathematical modeling consisting of nonlinear equations such as ODEs, and PDEs. A fundamental difficulty which arises is that most nonlinear models cannot be solved in closed form. Computer assisted proofs are at the forefront of modern mathematics and have led to many important recent mathematical advances. They provide a way of melding analytical techniques with numerical methods, in order to provide rigorous statements for mathematical models that could not be treated by either method alone. In this summer school, students will review standard computational and analytical techniques, learn to combine these techniques with more specialized methods of interval arithmetic, and apply these methods to establish rigorous results in otherwise intractable problems

<p>The traditional scientific method cycle, with Francis Bacon and Rene Descartes, its concievers in the center, alongside formal and statistical AI machinary, as a propsective evolution of the method.&nbsp;<br />&nbsp;</p>

The summer school aims to expose participants to formal methods that can facilitate principled scientific discovery. The school will cover some of the basic automated statistical inference (in the form of machine learning techniques) and reasoning methods that are commonly used in scientific discovery, as well as novel techniques developed to tackle open questions and issues. This summer school will address novel computational methods for scientific discovery and focus on fusing axiomatic knowledge and experimental data to enable principled derivations of models of natural phenomena along with certificates of the consistency of these models with background knowledge specified as axioms.

<p>Flats and hyperbolic planes in a higher rank symmetric space</p> Drawn by Steve Trettel.

Lie groups are central objects in modern mathematics; they arise as the automorphism groups of many homogeneous spaces, such as flag manifolds and Riemannian symmetric spaces. Often, one can construct manifolds locally modelled on these homogeneous spaces by taking quotients of their subsets by discrete subgroups of their automorphism groups. Studying such discrete subgroups of Lie groups is an active and growing area of mathematical research. The objective of this summer school is to introduce young researchers to a class of discrete subgroups of Lie groups, called Anosov subgroups.

<p>Laminations arise naturally in hyperbolic geometry and (pseudo-) Anosov flows [Image by Jeffrey Brock]</p>

This school will serve as an introduction to the SLMath semester “Topological and Geometric Structures in Low-Dimensions”.  The school consists of two mini-courses: one on Teichmüller Theory and Hyperbolic 3-Manifolds and the other on Anosov Flows on Geometric 3-Manifolds.  Both topics lie at the interface of low-dimensional geometric topology (specifically, surfaces, foliations, and 3-manifolds) and low-dimensional dynamics.  The first course will be targeted towards students who have completed the standard first year graduate courses in geometry, topology, and analysis while the second course will geared towards more advanced students who are closer to beginning research. However, we expect that all students will benefit from both courses.