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.

<p>Mapping tensors to their moment polytyopes to their asymptotic spectrum.</p>

Tensors play a central role in various areas of mathematics, physics and computer science: from constructing fast matrix multiplication algorithms in algebraic complexity theory to the study of entanglement in quantum information. While matrices (order-two tensors) are very well understood, tensors have a much more intricate structure, and much about them is still unknown, despite much interest. In this lecture series we will discuss foundations and recent advances in the theory of tensors and their applications across computer science and physics.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

MSRI-UP 2027 will focus on statistical models applied to real-world data using the free statistical software R. Students should have completed the calculus sequence, and prior coursework in statistics and programming experience in R or a similar language are desired. Beyond these prerequisites, students should be willing to learn quickly, work in a team, and communicate results clearly. The first two weeks of the program will consist of crash-course training in regression, inference, and R. Research projects will be chosen from three areas: epidemiological statistics, education statistics, and labor economics. The summer projects will be directed by Dr. Minho Kim (Colorado College).

<p>A real-world Flying Wing. Lai proved Hamilton&rsquo;s conjecture on the existence of Flying Wing Ricci solitons.</p>

Participants will learn the basic theory of Ricci flow and Ricci solitons—they will learn the fundamental equations and inequalities, how to do basic tensor calculations, how to control the potential function, how to obtain global curvature bounds, how to do bootstrap arguments, how to control the geometry at infinity, how to construct new Ricci solitons, the current cutting edge existence theory for Ricci solitons, including the existence and classification of flying wing solitons. Many arguments involve the maximum principle, integration by parts, and/or bootstrap arguments. In short, the student will learn the art of the a priori estimate. This is one of the main methods for understanding the geometry of Ricci solitons.

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 course will introduce graduate students to the many modern methods to construct minimal surfaces from a variational or min-max principle as well as applications to problems in geometry and topology. Topics include Almgren-Pitts min max theory, Allen-Cahn construction and PDE methods, the proof of the Willmore conjecture, the resolution of Yau’s conjecture on the infinitude of minimal hypersurfaces in general Riemannnian manifolds and corresponding Weyl law and equidistribution results.

Scheduled at ICARM in Pittsburgh, Computer-Aided Reasoning for Algebraic Geometry is a special summer school activity for 40–50 graduate students and junior researchers. This workshop positions algebraic geometry, formalization, and artificial intelligence as the three vertices of a connected framework, bridging the gaps between these distinct skill sets. Participants will explore topics beyond standard introductory courses through a rigorous schedule of blackboard-style foundations lectures, Lean formalization sessions, and hands-on machine learning tutorials. The program features a heavy emphasis on collaborative afternoon group projects led by expert mentors, allowing students to apply state-of-the-art autoformalization tools and ML pipelines directly to open problems in algebraic geometry.

The aim of this summer school is to introduce students to landmark results in mathematical ecology while emphasizing the modern techniques and ideas developed to address these emerging challenges. The school will begin with a review of classical questions and results, together with a discussion of new problems arising from both novel data sources and changing environmental conditions. Particular emphasis will be placed on identifying ecological questions that are amenable to mathematical formulation.

The core of the school will focus on modern dynamical systems approaches. A sequence of lectures will introduce nonautonomous systems, including their attractors, stability theory, and bifurcation structure, with a strong emphasis on transient dynamics and tipping phenomena. This will be complemented by an introduction to ergodic theory and random dynamical systems, providing tools to model and analyse stochastic influences and variability. The final part of the school will return to open problems in ecology, highlighting how these modern mathematical frameworks can be used to address unresolved questions and to formulate new ones.

<p>A K3 surface</p>

The study of complex projective surfaces is yet to attain the level of realization and widespread prevalence as that of curves. The classification of surfaces is far from complete, and many basic problems remain open. This is particularly pronounced in geometric topology, where understanding of algebraic curves (i.e., Riemann surfaces) is essential knowledge but, for just one example, applications of surface theory to studying the geometric topology of four-dimensional manifolds are very limited at the present moment.

The purpose of this summer school is to close the background gap and connect young researchers with different backgrounds that are, perhaps unbeknownst to themselves, well-poised to fruitfully collaborate on important problems at the interface of geometric topology and surface geometry.

<p>A simplification of the Hopf link in the quantum GL(N)-category</p>

The purpose of this summer school will be to introduce students to the world of tensor categories. We will approach this topic in a variety of ways. Beyond simply exploring the formal definitions, many examples will be introduced and studied, and the natural occurrences of tensor categories “in the wild” will be explored. One such example will be the Temperley-Lieb-Jones category, and the corresponding “quantum” knot invariants.

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. 

Blocks, but not as you know them

The summer school will provide a comprehensive introduction to the structure and classification of blocks. Participants will explore the algebraic, homological, combinatorial, categorical, geometric, and computational perspectives that shape the area. Beginning with basic concepts such as defect groups, decomposition numbers, and Cartan matrices, the lectures will develop Morita and derived equivalences and higher representation-theoretic methods, before turning to finite reductive groups and geometric constructions arising from Deligne–Lusztig varieties. Hands-on computational sessions will complement this theoretical material, illustrating how character tables, decomposition matrices, and block invariants can be explored using Julia, and in particular its Chevie package for computations involving finite reductive groups.

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.