Fall 2026
Damiano Rossi
Subtitle:
Deligne-Luszitg theory
Course Description:
We will study the representation theory of finite reductive groups using ideas originated by the work of Deligne and Lusztig. The principal aim of the course is to provide a classification of the ordinary irreducible representations of this class of groups. Time permitting, and depending on the class' interest, we will cover some aspects of the modular representation theory as well (e.g. classification of blocks, generalized Harish-Chandra theory and more).
Text:
M. Geck, G. Malle - The Character Theory of Finite Groups of Lie Type
Prerequisites:
Basic knowledge of representation theory of finite groups and the structure of linear algebraic groups
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Spring 2022
Anders Buch
Subtitle:
Enumerative Geometry
Course Description:
Enumerative geometry concerns determining the number of geometric objects of a specified type that satisfy a list of conditions. Classical examples include the two lines meeting four general lines in projective 3-space, and the 27 lines contained in a smooth cubic surface. The standard strategy for solving enumerative problems is to start with a moduli space parameterizing the type of objects to be counted, and then hope that each condition corresponds to a subvariety of the moduli space defined by polynomial equations. In this case the problem is reduced to finding the number of points in the intersection of a list of subvarieties, which often amounts to a calculation in the cohomology ring of the moduli space. We will start with an introduction to intersection theory and Chow cohomology, and then examine the subproblems one encounters when employing the standard strategy. This includes understanding the cohomology ring of the moduli space, as well as the classes in this ring defined by the subvarieties corresponding to the conditions. The last question will lead us to degeneracy locus formulas such as the classical Thom-Porteous formula and generalizations.
Text:
Fulton, Intersection Theory (for the start of the course).
Prerequisites:
Some knowledge of algebraic geometry is an advantage, but students willing to take some technical facts for granted will be able to follow the course if they are familiar with any type of geometry, including manifolds.
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Spring 2021 - Simon Thomas
Subtitle:
Geometric Group Theory
Course Description:
This course will be an introduction to Geometric Group Theory. There are no prerequisites, except for the most basic notions of group theory. In Geometric Group Theory, finitely generated groups are viewed as metric spaces via the path metrics on their Cayley graphs and their large-scale geometry is studied. The topics covered in this course will include: (i) Quasi-isometries and the large-scale geometry of finitely generated groups. (ii) Growth rates of finitely generated groups, including the construction of groups of intermediate growth. (iii) The basic theory of amenable groups.
Text:
Pierre de la Harpe, Topics in Geometric Group Theory, Chicago, 2000.
Prerequisites:
Basic group theory
Schedule of Sections:
Previous Semesters:
- Spring 2020 Prof. Retakh
- Fall 2019 Prof. Lepowsky
- Fall 2018 Prof. Lepowsky