To maximize interactions between research areas covered by the conference, we will have three introductory short courses and an introductory lecture on the following topics.
Noncommutative Algebraic Geometry, Adam-Christaan van Roosmalen, Hasselt University
In algebraic geometry, one uses algebraic techniques to study geometry, and one uses geometric ideas and intuition to describe algebra. Even though the same interplay cannot be duplicated naïvely to the noncommutative world, many techniques and definitions can still be carried over to the setting of noncommutative rings.
In these lectures, we will concentrate on projective algebraic geometry as introduced by Artin–Zhang. Here, one starts with a reasonably nice graded ring R and studies the category qgr(R) as the category of coherent sheaves on a “nonexistent” noncommutative projective space Proj(R).
Around 2002, Polishchuk and Schwarz constructed a category of coherent sheaves on a noncommutative 2-torus, starting from a given Dolbeault dg algebra. They subsequently showed that the category of coherent sheaves is (derived equivalent to) the category of coherent sheaves on an elliptic curve. This provides an example of a link between noncommutative differential geometry and noncommutative algebraic geometry.
Recent progress in understanding the vector bundles on quantum flag manifolds (specifically, the introduction of a noncommutative Kähler structure and a noncommutative version of the Kodaira vanishing theorem) seem to make a similar result tractable.
In this lecture series, we will study some techniques of noncommutative algebraic geometry, in particular those related to quantum flag manifolds (based on Heckenberger and Kolb’s construction). We will then discuss the category of coherent sheaves on quantum flag manifolds (or specifically, quantum projective space) and show how one can obtain this category as qgr(R) where R is the q-polynomial ring. This establishes another instance where one has a correspondence between noncommutative algebraic geometry and noncommutative differential geometry
Parabolic Geometry, Katharina Neusser, Masaryk University
This mini-course will give an introduction to Cartan geometries, which provide a uniform approach to a large variety of differential geometric structures.It will focus on parabolic geometries which are Cartan geometries infinitesimally modelled on flag manifolds. The most prominent examples of geometric structures admitting descriptions as parabolic geometries are conformal manifolds (dim>2), projective structures, almost quaternionic manifolds, and certain types of CR-structures. After having introduced the basic concepts and discussed some examples, we will (as time permits) give some applications of Cartan connections to symmetries and/or to the construction of invariant differential operators for parabolic geometries.
Quantum semisimple groups and quantum flag varieties, Robert Yuncken, Université Clermont Auvergne
Compact and complex semisimple Lie groups admit quantizations which have been discovered twice—first by Drinfeld and Jimbo in the form of quantized enveloping algebras, and later by Woronowicz in the form of quantum matrix pseudogroups. These quantum groups have a rich representation theory which echoes that of their classical cousins. At least in the compact case, setting the classical group in its family of quantum groups has given us new tools in representation theory.
Meanwhile, for the noncommutative geometer, quantized semisimple groups and their homogeneous spaces give some of the most natural examples of noncommutative spaces, although they have been frustratingly difficult to incorporate precisely into Connes’ philosophy of noncommutative differential geometry. In these lectures, we’ll give a rapid tour of compact and complex semisimple quantum groups and their flag varieties, from both a representation theoretic and a geometric point of view.
The plan of the lectures will be as follows, time permitting:
Lecture 1: A survey of quantum groups; definitions of quantized enveloping algebras, compact semisimple quantum groups, and quantum flag varieties; finite dimensional representation theory and Verma modules.
Lecture 2: Complex semisimple quantum groups as quantum doubles in the framework of algebraic quantum groups; bundles over flag varieties as principal series representations; duality with Verma modules; intertwining operators.
Lecture 3: The Bernstein–Gelfand–Gelfand complex, algebraically and geometrically; application to noncommutative geometry (the fundamental class of the flag variety of SLq(3,C)) and representation theory (the Plancherel formula for a complex semisimple Lie group).
References:
Klimyk, A & Schmüdgen, K, Quantum groups and their representations.
Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997
Voigt, C & Yuncken R, Complex semisimple quantum groups
https://arxiv.org/abs/1705.05661
Quaternion-Kähler Manifolds and Where to Find Them, Henrik Winther, Masaryk University
In this lecture we discuss the history, origins and theory of quaternion-Kähler manifolds. We introduce the two main classes of examples, the so-called Wolf spaces and Alekseevskian spaces. The former class consists of the quaternion-Kähler symmetric spaces. Slides
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Quantum Flag Manifolds in Prague 