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The workshop will focus on geometric and homological aspects of categorification and symplectic duality in the context of representation theory. The conference will concentrate on various representation theoretic aspects of the Langlands program.

University of Notre Dame

Geometrie und Topologie,    Zahlentheorie, Arithmetik

Lie groups are central objects in modern mathematics; they arise as the automorphism groups of many homogeneous spaces, such as flag manifolds and Riemannian symmetric spaces. Often, one can construct manifolds locally modelled on these homogeneous spaces by taking quotients of their subsets by discrete subgroups of their automorphism groups. Studying such discrete subgroups of Lie groups is an active and growing area of mathematical research. The objective of this summer school is to introduce young researchers to a class of discrete subgroups of Lie groups, called Anosov subgroups. These subgroups are central objects in the study of higher Teichmuller theory, convex projective geometry, and character varieties. In this summer school, there will be an emphasis on discussing some of the dynamical tools that have recently been successfully used to study Anosov subgroups. The required background in dynamics, hyperbolic geometry, and Lie theory will also be discussed. Aside from providing a stimulating academic environment for learning about Anosov subgroups, this summer school also aims to be a relaxed and friendly space for participants to interact with fellow aspiring mathematicians.

Geometrie und Topologie,    Kurse und Veranstaltungen für Studenten der Mathematik

American Institute of Mathematics, Pasadena, California (AIM)

Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI (ICERM)

The diagrammatic approach has its origins in the discovery of new quantum invariants of knots and links in the 1980s. Crane and Frenkel raised the idea of categorifying quantum groups, hence, link invariants, already in the 1990s, and this vision prompted the development of powerful link homology theories such as Khovanov-Rozansky homology. The categorification of quantum groups involves Khovanov-Lauda-Rouquier algebras, which are often presented diagrammatically. They are a building block for Kac-Moody 2-categories, which categorify Lusztig’s modified integral form for quantized enveloping algebra. There have been many remarkable developments in this direction in the last few years, including the introduction by Webster of more general algebras categorifying tensor products, DG versions of these algebras which categorify Verma modules, and p-DG versions which are being used to categorify link invariants at roots of unity. The field of diagrammatic categorification is still in its early stages, but it has already had a significant impact on more traditional mathematics. This workshop aims to unite both established experts and emerging scholars across various domains of diagrammatic categorification, including representation theory, combinatorics, and link homology.

Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI (ICERM)

Webs are diagrammatic tools for representing complex calculations graphically. These diagrams first arose from the representation theory of classical groups, and they have since become important in disparate areas of mathematics. In representation theory, they encode morphisms of quantum groups. In topology, webs give rise to powerful link invariants. In algebra and geometry, Kuperberg's \(mathrm{sl}(3)\) web bases have important relationships with the theory of cluster algebras and affine buildings. In combinatorics, they explain certain dynamics on Young tableaux. Recent work by Gaetz--Pechenik--Pfannerer--Striker--Swanson introduced an \(\mathrm{sl}(4)\) web basis that has exploited and extended exciting connections between webs, plabic graphs, and crystals. There are further connections to total positivity, duality conjectures for cluster algebras and mirror symmetry.

Graphentheorie und Kombinatorik,    Algebra