<p>This figure depicts dynamics of flows on convex cocompact hyperbolic 3-manifolds; where the girl is a traveller along a horocycle.</p>

This research program will bring together two intellectual communities that have made significant advances in the study of discrete subgroups of higher rank semisimple Lie groups: the homogeneous dynamics community and the community studying geometric structures and Anosov groups.

The stable and unstable foliations near a singular orbit of a pseudo- Anosov flow in 3 dimensions. Courtesy Michael Landry.

Low dimensional topology is a meeting place for many objects and ideas from diverse areas of mathematics, including foliation theory, geometry, and smooth and conformal dynamics.   For instance, many foliations on 3-manifolds admit transverse flows, connecting (local) leafwise homeomorphisms to flow dynamics and the mapping class groups of the leaves.  Leafwise conformal or hyperbolic structures can be approached through Teichmüller theory, and connect again to one-dimensional dynamics through "universal circles" organizing compactifications of all the leaves or of the flow space.  Many of these ideas originate in work of Thurston but in recent years have diverged and are ripe for reconnection.  

The program will bring together experts in all these fields together with younger researchers, who together can form new connections and open new areas for exploration.

This workshop features a series of invited talks by experts across the fields of low-dimensional topology, homogeneous dynamics, and the geometry of symmetric spaces. Spanning all career stages, the event aims to foster a collaborative and supportive environment, particularly for early-career mathematicians, encouraging engagement, learning, and exploration in a welcoming academic setting.

The joint introductory workshop for the programs in Geometry and Dynamics for Discrete Subgroups of Higher Rank Lie Groups and Topological and Geometric Structures in Low Dimensions will feature lectures introducing subjects of interest to both programs, including Teichmuller Theory, geometry in higher rank, foliations and flows, Anosov groups and thermodynamic formalism,  mapping class groups, counting and equidistribution, and related topics. Minicourses will be targeted at early career researchers as well as specialists looking to find connections between the different subjects.

Foliations around a pseudo-Anosov singularity (Image credit: Chi Cheuk Tsang)

This workshop will bring together ideas from diverse areas of mathematics that meet in the setting of geometry and topology in low dimensions.  This includes the study of flows, foliations, and fibrations of three-manifolds and the related study of geometry (e.g. hyperbolic or conformal structures) of the manifolds and of the leaves or fibers, and their mapping class groups.  This is a rich and interconnected area and many adjacent topics will also be featured.

Limit set of an Anosov representation

This workshop will focus on recent advances on geometric and dynamical approaches to the study of discrete subgroups of higher rank Lie groups and their deformation spaces. The goal will be to present results and exchange ideas from different areas of mathematics, and we hope to create bonds between several different mathematical communities.

Building on the vision Bob Moses set out in the 1993 Algebra Initiative Colloquium, the 2025 Critical Issues in Mathematics Education (CIME) Workshop celebrated the legacy of Moses’ call for a mathematics literacy that supports citizenship in the 21st century. A mathematics towards citizenship has proved elusive as a goal for mathematicians, mathematics educators, researchers, students, teachers, and leaders—even as efforts among different individuals, organizations, and communities labor for greater access to high-quality mathematics learning experiences. The 2026 CIME Workshop, Math Literacy for 21st Century Citizenship - Part II, endeavors to design opportunities for learning, collaborating, and struggling towards a mathematics for lived citizenship, as a theme and organizing principle for communities across the Nation.

For Moses, the notion of lived citizenship is not simply a matter of gaining legal status but of demanding that the legal status be meaningful. Thus mathematics literacy for lived or participatory citizenship might be thought of to include contributing to your community, not just being part of a community, and beyond belonging, to actively participate in making your community better, in making life better for others.

The conception of mathematics for lived citizenship is an important foundation for the goals for the 2026 CIME Workshop, in order to envision the development of coordinated actions in the short and long term within the mathematics and mathematics education communities. To ground and catalyze these efforts, we invite participants to consider a mathematics for lived citizenship through three foundational ideas:

• Claiming mathematics -- establishing agency in and ownership of math.
• Building Agency -- claiming and leveraging mathematics knowledge to overcome present day, generational, community, and national challenges.
• Collaboration and collective effort -- using mathematics literacy to make systemic change.

Three questions will guide our work:

• How does the unfolding history of mathematics impact lived citizenship?
• How do we understand the relationship between mathematical flourishing and the evolution of vibrant democracy in the United States? 
• In light of the above two questions, how do we reimagine our roles, responsibilities, and actions in the short and long term?

One of the hottest topics in analytic number theory involves the use of statistics and probability in understanding different aspects of algebraic and analytic number theory, through various new lenses. This is reflected in some of the most exciting number theory research of the last few years (for example, of Bhargava, of Ellenberg and Venkatesh, of Alexander Smith, of Sawin and Wood, of Adam Harper, of Koukoulopoulos and Maynard, of Helfgott and Radziwill, of Pilatte,....). As a consequence the CRM will host a thematic semester Mar 2-July 3, 2026 on these topics involving some of the world leaders in the subject. Since this new area can roughly be split in two into Algebraic and Analytic, we will focus for two months on each, with the SMS school placed in the middle. The 2026 SMS will introduce junior mathematicians to important trends in number theory. 

PROOF (Promoting Research Opportunities and Open Forums) is a two-week summer program designed to provide research opportunities for U.S.-based mathematicians, statisticians, and their collaborators in the U.S. and abroad, whose ongoing research may have been impacted by factors such as heavy teaching loads, professional isolation, limited access to funding, heavy administrative duties, personal obligations, or other constraints.

The program welcomes in-person participation from faculty and advanced graduate students at all U.S. colleges and universities. It aligns with SLMath’s broader goal to open doors to mathematics by creating environments where researchers from a wide range of institutional settings can thrive, pursue projects, and reach their full academic/scientific potential.

The school will consist of three interrelated courses, aimed to introduce the main concepts in Geometric Measure Theory.

(1) Sets and measure in the Euclidean space, Guido De Philippis (Courant Institute of Mathematical Sciences)
(2) Theory of currents, Annalisa Massaccesi (University of Padua)
(3) Allard regularity theory, Camillo De Lellis (Institute of Advanced Study)

<p>The five largest clusters in critical site percolation on a large three-dimensional box</p>

Percolation and spin models such as the Ising model have a history that goes back over 100 years. The subject has taken a central role in probability theory over the last few decades, in particular through interactions with various other areas of mathematics. These include graph theory, theoretical
computer science, statistical physics, quantum field theory, complex analysis, partial differential equations, and geometric group theory. Through examples, the summer school aims to illustrate some of the successful techniques and ideas in the subject area.

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.

In 2026, MSRI-UP will focus on Numerical Methods for Differential Equations The research program will be led by Dr. Johnny Guzman of Brown University.

The aim of this summer school is to provide an introduction to theoretical ideas that have been developed with the objective of understanding machine learning methods and their domain of applicability. The focus will be on proof technique and general mathematical tools. The lecturers are two worldwide experts in the area and the material is regularly taught in Mathematics and Statistics Departments of the top world Universities.

Models of random growth and of the separation of phases occurring when one substance is suspended in another often evince universal features, in which scaling exponents are shared among a broad class of such models. A foothold for understanding such features is often offered by studying a few special models that are exactly solvable, which means that exact formulas of algebraic or integrable origin are available. Showing that a broader range of models also have the features is a task that may rely on a range of robust probabilistic or geometric tools. The summer school will offer an introduction to random growth and phase separation, with an emphasis on tools that offer the prospect of proving universality for a wider class of models.

Lasting Alliance Through Team Immersion and Collaborative Exploration (LATTICE)  is a yearlong program which provides opportunities for U.S. mathematicians to conduct collaborative research on topics at the forefront of the mathematical and statistical sciences. The goal of the program is to advance mathematical research by offering a structured environment where participants can concentrate on their work, make substantial progress, build professional connections, and contribute to the overall growth of the mathematical sciences. This program is designed for faculty whose research momentum may be affected by demanding teaching loads, limited access to research infrastructure, professional isolation, administrative responsibilities, or personal obligations.

<p>Artificial image generation by flow-based generative models starting from noise</p>

The overarching goal of this summer school is to expose students to the latest developments in the mathematics of generative models. Our ultimate goal is to teach them how to conduct research in this exciting area in machine learning and use their knowledge to make contributions to applied mathematics using these new tools.

This two week summer school, jointly organized by SLMath with OIST, will offer the following two mini-courses:

1. Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form
   This course will present some recent developments in the theory of divergence-measure fields via measure-theoretic analysis and its applications to the analysis of nonlinear PDEs of conservative form – nonlinear conservation laws.
2. Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs
   This course will present some recent developments precisely characterizing the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extending the role of the Wiener test in classical analysis.

<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

<p>Illustrated by Rok Gregoric</p>

The upcoming SLMath program is organized around key themes of “higher” quantization and mirror symmetry as they impact and elucidate a wide variety of questions in representation theory. The program will bring together experts and young researchers from algebra, geometry, physics and number theory to help develop and disseminate this unified vision of a rapidly evolving field, exploring the mathematical consequences of the examples, structures, and dualities discovered in physics.

Tremendous progress has been made using motivic techniques in geometric questions for affine algebraic varieties, especially those involving algebraic vector bundles. Computations in classical algebraic topology have been improved by motivic techniques, e.g., related to the problem of computing homotopy groups of spheres. Moreover, the theory has identified new structures of interest in arithmetic situations. Transformative recent progress in motivic homotopy theory has only broadened the scope for potential applications of motivic techniques, as well as new avenues of interaction with other areas of mathematics. This program will build on previous successes, explaining the tools that have been developed and how to use them, analyzing questions of the sort described above and identifying new domains where motivic techniques will be successful.

Image by Elliot Kienzle

This workshop will bring together theoretical physicists, representation theorists, algebraic geometers and symplectic geometers interested the connections between quantum field theory and geometric representation theory. The main topics to be discussed are mathematical aspects of 2d, 3d and 4d supersymmetric field theories, such as: topological twists and the resulting Higgs and Coulomb branches, relations to quantization and categorification, representations of vertex operator algebras, connections to enumerative geometry and quantum K-theory and elliptic cohomology, relations to knot homology and, finally, connections to the (relative) geometric Langlands program.

The workshop will bring together researchers at the forefront of ongoing work in motivic homotopy theory.  Topics will include the application of motivic techniques to: geometry of affine algebraic varieties and algebraic vector bundles; computations in classical algebraic topology such as homotopy groups of spheres; and enumerative geometry.  The workshop will also consider the internal foundational development of motivic homotopy theory itself.

Tropical surfaces. Images courtesy of Lars Allermann.

Tropical geometry can be viewed as a degenerate version of algebraic geometry,where the role of algebraic varieties is played by certain polyhedral complexes. As the degeneration process, called tropicalization, preserves many fundamental properties, tropical geometry provides important bridges and an exchange of methods between algebraic geometry, symplectic geometry and convex geometry; these links have been extremely fruitful and gave rise to remarkable results during the last 20 years. The main focus of the program will be on the most significant recent developments in tropical geometry and its applications. The following topics are particularly influential in the area and will be central in the program:

• real aspects of tropical geometry;
• tropical mirror symmetry and non-Archimedean geometry;
• tropical phenomena in symplectic geometry;
• matroids, combinatorial and algebraic aspects;
• tropical moduli spaces;
• tropical geometry and A1-homotopy theory.

Picture of an amplituhedron

Algebraic combinatorics is an area of mathematics that employs methods in abstract algebra in combinatorial contexts, and that uses combinatorial methods to approach problems in algebra. Some important topics are symmetric functions, Young tableaux, matroids, Coxeter combinatorics. There are links to computer algebra (sage-combinat), number theory (L-functions), representation theory, and mathematical physics through Macdonald processes and integrability. The work on the totally positive Grassmannian also gave rise to beautiful results in mathematical physics: for example KP solitons and Scattering Amplitudes. Schubert calculus is an important part of algebraic combinatorics and is now at the frontier with k-Schur functions, which first came up in the theory of Macdonald polynomials.

This workshop presents introductory talks and new trends in tropical geometry and algebraic combinatorics, including interactions between tropical geometry and enumerative, logarithmic, nonarchimedean, and real algebraic geometry, mirror symmetry and symplectic geometry, moduli spaces in tropical geometry, the geometry of matroids, integrable systems and dynamical combinatorics, combinatorial representation theory, geometry of polynomials, combinatorial varieties and connections to symmetric function theory and cluster algebras.

Inverse problems (IP) arise in all fields of science and technology where a cause for an observed or desired effect is sought. In the last 50 years or so there has been substantial progress in the mathematical understanding of these problems but many questions remain open. The mathematics of these problems involves many areas in Mathematics including PDE, differential geometry, integral geometry, probability, statistics, complex analysis, numerical analysis, mathematical physics, data science, etc. Since the 2010 program at the then-Mathematical Sciences Research Institute (now Simons Laufer Mathematical Sciences Institute), there has been significant progress in inverse problems; many of the advances can be traced back to that program. However, there are still deep open questions remaining as well, some of which are discussed in this proposal. New research topics include the connection between IP and machine learning, IP for nonlinear equations, IP for nonlocal operators, and connections between statistics and IP.

The field of Geoemtric Measure Theory (GMT)has become vast and many gaps between disparate areas have emerged. This thematic semester will bring together researchers from every corner of the field, to kick-start new interactions and discoveries. In light of the many exciting advancements and potential for future breakthroughs, this as a crucial moment to bring old and new members of the GMT community together. The program will encourage interactions between established experts, emerging researchers and students, allowing for the sharing of key idea that brought to the recent developments and helping to shape a research agenda for the future.