Konferenzen  >  Mathematik  >  Geometrie und Topologie  >  Vereinigte Staaten

American Mathematical Society and Georgia Southern University-Armstrong Campus

The interaction between topology and curvature is a fundamental theme in modern mathematics. The study of scalar curvature plays an increasingly important role not only in geometry and topology, but also in general relativity and theoretical physics more generally through the theory of Dirac operators. This special session will bring together experts and young researchers who study these topics from many different points of view. The aim of this session is to share viewpoints and progress on understanding the scalar curvature and topology of manifolds, and to establish new connections among the culturally diverse groups spread worldwide, but particularly in the USA.

Email: ekanshjauhari@ufl.edu

Global Riemannian geometry, curvature restrictions, differential geometric analysis, index theory, spin geometry, algebraic topology of manifolds.

Mathematische Physik,    Analysis

Texas Tech University

The minisymposium will consist of talks from five confirmed speakers (Jacob Bernstein--Johns Hopkins University, Baltimore; Vladimir Dragovic--The University of Texas at Dallas; Jr-Shin Li--Washington University in St. Louis; Zhiqin Lu--University of California, Irvine; Francesco Maggi--The University of Texas at Austin) and also include a poster session.

Geometric Analysis, Differential Geometry, Applications

Analysis,    Infinitesimalrechnung, Differentialgleichungen, Integralrechnung

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.

Tel.: [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.

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

Western Washington University

Join us for a five-day workshop hosted by Western Washington University, contingent on NSF support. Mornings will be dedicated to talks, and in the afternoons, participants will work in small groups on open problems related to connections between 4-manifold trisections and computational topology. The workshop will be preceded by introductory mini-courses on Zoom. Participants may register for either or both workshop components. The workshop will conclude at noon on Friday, 7/24.

Email: zupan@unl.edu

trisection, 4-manifold, computational topology

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.

Tel.: [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.

Zahlentheorie, Arithmetik,    Kurse und Veranstaltungen für Studenten der Mathematik

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.

Tel.: [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.

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

Institute for Computational and Experimental Research in Mathematics (ICERM)

This Semester Program will explore the emerging research field of Metric Algebraic Geometry, which is about the geometry induced by metric and probabilistic structures on sets defined by polynomial equations. The philosophy of this program is that regarding spaces and models as algebraic varieties opens up powerful – notably, global - approaches for problems involving distances and/or approximation. Instead of viewing spaces through local charts or via simplicial pieces, Metric Algebraic Geometry proposes finding global descriptions. This brings new computational tools to the table to establish strong theoretical properties over the geometric domain. Target applications include the nonlinear function spaces of neural networks, reduced-order models for large datasets, and configuration spaces of cameras or images in computer vision. The program will develop this area both within pure mathematics and in its applications. Activities will encourage experts in theoretical fields (such as algebraic geometry, differential geometry, and integral geometry) to commingle with experts in application-driven domains (like in machine learning, computer vision, optimization, nonlinear inverse problems and probabilistic modeling) and to all mentor the next generation. The results will set the research agenda for Metric Algebraic Geometry for years to come.

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

Institute for Computational and Experimental Research in Mathematics (ICERM)

Integral geometry is the subject of averaging geometric properties of a submanifold of a homogeneous space. This is a classical topic, with its origin in the so-called Buffon's needle problem: drop a random needle on a floor made of planks and calculate the probability the needle lies across a crack. This is solved formulating accurately what “random” means, so that expectations become integrals with respect to a certain measure (on the Grassmannian of lines in this case). Generalizations of this problem lead to beautiful progress in the theory of valuations on convex bodies and stochastic processes. In this workshop we will focus on more recent developments, especially on integral geometry of (real) algebraic varieties. This allows to establish connections with intersection theory and convex geometry. At the same time, restricting to the algebraic framework, opens the possibility of effective studies of average properties, with applications to sampling methods.

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

Institute for Computational and Experimental Research in Mathematics (ICERM)

The goal of this workshop is to deepen connections between algebraic geometry and data science. Behind many optimization methods is the modeling assumption that data lie on, or close to, an algebraic variety. We will discuss methods to sample from varieties or distributions centered on them, extending the methods to adaptive techniques and for sampling from level sets. We will investigate consequences and applications of these methods for computations in Bayesian statistics, likelihood inference, and topological data analysis. A complementary area of focus will be on learning functions and varieties from samples, and its uses to test goodness of fit, to investigate the geometric and topological structure of data, and for structured matrix and tensor decompositions. The workshop is at the interface of applied algebraic geometry with its applications in numerical computations, machine learning, and statistics. It will bring researchers from these communities together to learn from each other and exchange ideas.

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

Algebra,    Wissenschaftliches Rechnen und Datenwissenschaft