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Simons Foundation

A conference marking the 50th anniversary of the publication of Barry Mazur's "Modular curves and the Eisenstein ideal" and Ken Ribet's "A modular construction of unramified p-extensions of Q(μ_p)". The talks will feature recent advances in areas of research that have been influenced by these two papers.

Institute for Computational and Experimental Research in Mathematics

A conjecture of Malle predicts an asymptotic formula for the quantity of number fields with a given Galois group and whose discriminant is bounded by a growing parameter. This conjecture is foundational to the area of Arithmetic Statistics and has received considerable interest over the past 20 years, notably including Bhargava receiving the Fields Medal in part for his work on quartic and quintic field extensions.

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

Algebraic distance optimization, a family of optimization problems defined by polynomial equations, is becoming a key approach for solving complex optimization tasks in computational algebraic geometry, computer graphics, image processing, and machine learning. Its primary focus is on solving algebraic optimization problems concerning the distance from a data point to a given algebraic variety. This workshop will bring together leading experts from algebraic geometry, algebraic statistics, optimization, and application areas to explore and develop advances in this research area.

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

Optimization

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

Geometrie und Topologie,    Wissenschaftliches Rechnen und Datenwissenschaft