<p>Homotopy measures how spheres can be tangled in spaces; the logo shows a sphere tangled in a grove of California redwoods<br />The background painting is &ldquo;Giant Redwood Trees of California&rdquo; by Albert Bierstadt in 1874</p>

The goal of this school will be to introduce students to several powerful cohomological tools that were brought to commutative algebra by Avramov in the 80s and 90s: Lie algebra methods from homotopy theory, and support theoretic methods from the representation theory of finite groups. These tools have have seen a huge array of applications that continue to grow, with several major developments in recent years, opening new connections to algebraic topology, noncommutative algebraic geometry, and representation theory.

This Summer Graduate School aims to introducing graduate students to some aspects of contemporary modeling and analysis of dynamical systems in their interactions with machine learning and artificial intelligence (AI) applications.

<p>A wall-crossing in a moduli problem</p>

One of the central problems in algebraic geometry is to classify so-called algebraic varieties: geometric shapes cut out by polynomial equations. Algebraic varieties are parametrized by certain moduli spaces (roughly: parameter spaces whose points correspond to these different varieties). The geometry of these moduli spaces encodes the ways of continuously deforming these shapes. Furthermore, classification questions for algebraic varieties often boil down to understanding the geometry of these moduli spaces. In the past few years, powerful new tools have been developed in moduli theory, especially for higher dimensional varieties – those which are of complex dimension at least two. The goal of this summer school is to provide an introduction to many of these recently emerging breakthroughs to enable graduate students to begin working in this area. The program will be motivated and often guided by examples and is intended to be accessible to a wide variety of students

The Simons Laufer Mathematical Sciences Institute (SLMath) will host the 2026 Modern Math Workshop (MMW) as part of the SACNAS Annual Conference. The Modern Math Workshop encourages undergraduates to pursue careers in the mathematical sciences, and builds research and networking opportunities among undergraduates, graduate students and recent PhDs in the mathematical sciences.

The goal of our school is to make connections between different flavors of geometry: euclidean geometry, hyperbolic geometry, translation surfaces and graph theory. We will focus on growth, asymptotics and extremal problems.

The PIMS–CRM Summer Schools in Probability are a highlight of Canadian probability. Launched by PIMS in 2004, they take the form of two main 4-week courses by top researchers in probability theory and related fields, along with a number of keynote addresses and shorter mini-courses. The school is geared primarily at doctoral students and postdoctoral scholars, who are also given the opportunity to present their own research, should they desire. The schools have played a major role in the development of an exceptionally strong community of young probabilists in Canada, North America and overseas. This will be the 14th time this school has run. 

The Whitney Umbrella. Courtesy of Herwig Hauser, Fac. Math., University of Vienna, Austria

The school will introduce students to the fundamental problem of resolution of singularities of varieties and schemes. This is a subject that is relevant for all areas of algebraic geometry and commutative algebra. Students will learn algorithms to resolve singularities, and learn how to apply them in specific situations and understand how these techniques can be used in their research.