Conferences  >  Mathematics  >  Geometry and Topology  >  United States

American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to studying arithmetic dynamics of multiple maps. In classical arithmetic dynamics, we consider the iteration of a single endomorphism of a variety defined over a field of arithmetic interest — typically a number field or the function field of a curve. An exciting new direction in arithmetic dynamics that holds particular promise for striking new results is what we call "dynamics of multiple maps": dynamical behavior arising from the interaction of two or more endomorphisms on the same space. This includes iteratively applying rational functions on P1 chosen at random from a family according to some probability distribution; a correspondence from a variety X to itself; and forming words from a non-commuting family of involutions on a K3-surface.

American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to non-Archimedean methods in complex geometry.

Institute for Computational and Experimental Research in Mathematics

Geometric and topological methods are reshaping the study of complex, high-dimensional data, yet their theoretical foundations still lag behind their accelerating adoption in practice. This workshop will bring together researchers in computational geometry, applied topology, and machine learning to fortify and extend these foundations while developing scalable, interpretable approaches for data-driven science. By bridging theory and computation, the program aims to clarify core principles, overcome algorithmic bottlenecks, and establish rigorous frameworks that can guide future applications. The workshop will feature talks from senior leaders and rising scholars, collaborative discussions, and opportunities for junior participants to engage with the community and help shape the evolving landscape of the field.

Phone: [4018635030];     Email: info@icerm.brown.edu

Institute for Computational and Experimental Research in Mathematics

Spheres are the basic building blocks of geometry. More complicated geometric objects can be built by attaching spheres to each other along continuous maps. For many purposes, such constructions depend only on the homotopy classes of these continuous maps. A fundamental problem in algebraic topology is to compute these homotopy classes, i.e., to compute the homotopy groups of spheres. Machines can be used to great effect in the exhaustive computation of these fundamental invariants of homotopy theory. The workshop will study several software packages that are specifically designed for this purpose. Participants will have the opportunity to interact directly with codebases in work groups led by experienced programmers. The workshop will also introduce a variety of projects in homotopy theory that rely on computers.

Phone: [4018635030];     Email: info@icerm.brown.edu

Institute for Computational and Experimental Research in Mathematics

This workshop will explore the rich and rapidly evolving interface between high-dimensional geometry, probability, and computational methods. Recent years have witnessed major break-throughs, bringing tools from stochastic processes to resolve long-standing problems in geometry, and using geometry to investigate the limits of computation and efficiency of sampling algorithms.

The workshop will bring together researchers from the communities of geometry, probability, and computational complexity to promote further synergy and continue the progress on the major open problems of the field.

Phone: [4018635030];     Email: info@icerm.brown.edu

On the occasion of 40+ years after the seminar paper of Gross--Zagier, we bring together experts to deliver lectures on a broad range of topics connected with the Gross-Zagier formula, its generalizations, related future directions, and other works that it has inspired.

The program will unite researchers from a number of areas: algebraic and complex geometry, arithmetic geometry, Hodge theory, and mathematical physics. It will bring theoretically and computationally oriented researchers together, expecting that computations will illuminate conjectures made by the theorists and that theory will enlarge the range of what can be computed. We intend to develop databases of certain types of K3 surfaces for the L-Functions and Modular Forms Database and promote the development of software for computations on K3 surfaces in Magma, SageMath, or other systems for public release.

Number Theory, Arithmetic,    Courses and Events for Math Students and Early Career Researchers

Institute for Computational and Experimental Research in Mathematics

Cubic fourfolds and Gushel-Mukai fourfolds are bound to K3 surfaces through geometry and Hodge theory; when they contain unexpected surfaces, their middle cohomology contains a Hodge structure matching that of a K3 surface. This association contains rich geometry and involves certain Hyperkähler manifolds associated to the cubic fourfold. Hyperkähler manifolds are higher-dimensional generalizations of K3 surfaces, whose second cohomology group with integer coefficients has a lattice structure that allows us to study them with tools analogous to those used to understand K3 surfaces. The past 30 years have seen significant advances and new avenues develop in the study of the geometry of cubic fourfolds, Gushel-Mukai fourfolds, and high-dimensional Hyperkähler manifolds. The aim of this workshop is to bring together experts from algebraic and arithmetic geometry to exchange mutually beneficial points of view. For example, are there computations over finite fields that could inform or lead to a Honda-Tate theory for K3 surfaces? Are there exotic automorphism groups of high-dimensional Hyperkähler manifolds in positive characteristic? What geometric insights can be descended to number fields for applications to the study of rational points and the Brauer-Manin obstruction? We hope to study these and many other similar questions.

Phone: [4018635030];     Email: info@icerm.brown.edu

Institute for Computational and Experimental Research in Mathematics

Moduli spaces of K3 surfaces can be constructed using many points of view in algebraic geometry, including Hodge theory, birational geometry, and invariant theory. These different perspectives often yield related but distinct compactifications of the moduli of K3 surfaces. The current understanding of these spaces is quite explicit for low degree polarized K3 surfaces but requires a significant increase in computational complexity in higher degrees. Furthermore, studying moduli of K3 surfaces informs the study of moduli of other K-trivial varieties, such as abelian varieties and Calabi-Yau manifolds, related to many open questions in algebraic geometry. The goal of this workshop is to explore connections between different perspectives on moduli of K3 surfaces and applications to other K-trivial varieties. This workshop intends to unite experts in Hodge theory, birational geometry and moduli of Calabi Yau manifolds, singularity theory, mirror symmetry, and anyone working on questions related to degenerations and deformations of K3 surfaces.

Phone: [4018635030];     Email: info@icerm.brown.edu