Apr 09, 2018
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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A tale of two modules
Geordie Williamson (University of Sydney)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The spherical and anti-spherical modules are modules for the affine Hecke algebra obtained by inducing the trivial and sign modules from the finite Hecke algebra. They have have an illustrious history (Kazhdan-Lusztig isomorphism, Bezrukavnikov equivalence). They admit canonical and p-canonical bases. The canonical bases in the spherical and anti-spherical modules contain important information in the representation theory of reductive algebraic groups: the canonical basis in the spherical module appears in Lusztig's character formula for simple modules; the canonical basis in the anti-spherical module appears in Soergel's character formula for tilting modules for the quantum group. I will state p-versions of these results: the p-canonical basis in the spherical module controls simple characters; the p-canonical basis in the anti-spherical module controls tilting characters. The first statement is recent work with Riche (p >= 2h-2), the second statement was a conjecture with Riche, and was solved last year in joint work with Achar, Makisumi and Riche (p >= h). It doesn't seem unreasonable to hope that both statements are true for all p
- Supplements
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Notes
2.23 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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On Picard groups of blocks of finite group algebras
Radha Kessar (City University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will present joint work with Robert Boltje and Markus Linckelmann on the structure of the Picard group of self Morita equivalences of a block of a finite group algebra over a complete discrete valuation ring. In particular, I will give a description of the subgroup of endopermutation source Morita equivalences in terms of the underlying fusion system, Dade group, and source algebra.
- Supplements
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Notes
126 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Overgroups of regular unipotent elements, finite and algebraic
Donna Testerman (École Polytechnique Fédérale de Lausanne (EPFL))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will report a recent joint work with Cédric Bonnafé in which we compute equivariant cohomology of smooth Calogero-Moser spaces using representation theory of rational Cherednik algebras. We also give an application to the computation of equivariant cohomology of related symplectic resolutions.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Characters and Sylow 2-subgroups of maximal class
Benjamin Sambale (Universität Kaiserslautern)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In this joint work with G. Navarro and P. H. Tiep we characterize finite groups with a Sylow 2-subgroup of maximal class in terms of their character table. Secondly, we characterize groups with Sylow 3-subgroups of order 3 in terms of their principal block.
- Supplements
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Notes
141 KB application/pdf
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Apr 10, 2018
Tuesday
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09:30 AM - 10:30 AM
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Applications of representation theory to statistical problems
Persi Diaconis (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Lots of real world problems lead to group valued data...500 people ranking 5 types of chocolate chip cookies gives data on S(5). Daily birth rate data in New York City to data on C(365). There is even monster valued data- does the product replacement algorithm algorithm generate random looking elements. Fourier analysis on the groups calls for detailed knowledge of both characters and representations. It needs non-commutative analogs of the FFT. I will try to explain the statistics to people who know about groups (but not statistics). Again, there are many open problems
- Supplements
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Notes
4.12 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Character ratios for finite groups of Lie type
Martin Liebeck (Imperial College, London)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
A character ratio for a finite group G is a complex number of the form chi(g)/chi(1), where chi is an irreducible character of G. I shall discuss some recent results on character ratios for finite groups of Lie type, I shall also mention several diverse applications, to random walks, probabilistic generation, and representation varieties of some classes of infinite groups
- Supplements
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Notes
435 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Character methods and probabilistic methods in groups
Aner Shalev (The Hebrew University of Jerusalem)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Character methods, probabilistic methods, and their combination, have been instrumental in the solution of various problems in group theory over the years.
In the talk I will provide a brief background and then focus on some brand new results and conjectures, related to random walks and to word maps.
In particular I will discuss a solution (joint work with Larsen and Tiep) of the so-called Probabilistic Waring Problem for finite simple groups, which applies strong new character bounds.
- Supplements
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Notes
147 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Fock space categorification, Soergel bimodules, and modular representation representation theory in type A
Ben Elias (University of Oregon)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We give an executive summary of recent work with Ivan Losev, which connects modular representation theory in type A with a diagrammatic category of singular Soergel bimodules. As a consequence, various multiplicities in modular representation theory are encoded by p-Kazhdan-Lusztig polynomials. All this is in spite of the observation that the two sides of this story categorify different objects: one categorifies Fock space, while the other a space spanned by virtual partitions. We explain a magic trick due to Losev which allows one to compare this different categorical representations.
- Supplements
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Notes
1.49 MB application/pdf
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Apr 11, 2018
Wednesday
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09:30 AM - 10:30 AM
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Lie superalgebras and 2-representation theory
Jonathan Brundan (University of Oregon)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will talk about some connections between the representation theory of linear Lie superalgebras and the 2-representation theory of Kac-Moody 2-categories
- Supplements
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Notes
1.35 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Parallelotope tilings for symmetric groups
Joseph Chuang (City University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will describe Will Turner’s conjecture that part of the modular representation theory of the symmetric groups can be constructed from parallelotope tilings. Hyohe Miyachi, Kai Meng Tan and I have proved a numerical version of the conjecture for q-Schur algebras at complex roots of unity.
- Supplements
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Notes
528 KB application/pdf
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Apr 12, 2018
Thursday
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09:30 AM - 10:30 AM
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Finite groups of Lie type and (q,t)-polynomials
Raphael Rouquier (University of California, Los Angeles)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We will explain how double affine structures and two-variable polynomials arise from the modular representation theory of finite groups of Lie type in non-describing characteristic
- Supplements
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Notes
3.66 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Deligne—Lusztig induction and almost characters
Jay Taylor (University of Manchester)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
One of the few tools one has when studying the characters of a finite group $G$ is given by induction $\mathrm{Ind}_H^G$ from a subgroup $H \leqslant G$. If $G$ is a finite reductive group and $L \leqslant G$ is a Levi subgroup then, in 1976, Deligne and Lusztig have defined a geometric analogue of this construction, which is typically referred to as Deligne—Lusztig induction $R_L^G$. A fundamental problem is to understand the decomposition of $R_L^G(\chi)$ into its irreducible constituents for any irreducible character $\chi$ of $L$. If $L$ is a torus then this problem was completely settled by Lusztig in the mid to late 80s. Using Shintani descent Asai solved this problem when $\chi$ is a unipotent character of $L$. Assuming $G$ comes from an algebraic group with connected centre then Shoji obtained generalisations of Asai’s results for arbitrary characters. In this talk we describe a new approach to this problem which relies on the validity of the Mackey formula
- Supplements
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Notes
193 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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The McKay conjecture and Galois action on characters
Carolina Vallejo Rodríguez (Università di Firenze)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The McKay conjecture was refined in 2004 by G. Navarro
taking into account the action of Galois automorphisms on characters.
This refinement is known as the Galois McKay conjecture, and it is one
of the last counting conjectures without a reduction theorem. In this
talk I will present a possible reduction theorem for the Galois McKay
conjecture to a question on finite simple groups. This is joint work
with G. Navarro and B. Späth.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Fourier matrices for unipotent characters
Olivier Dudas (Université de Paris VII (Denis Diderot))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
[This is a work in progress with Bonnafé-Broué-Malle-Michel-Rouquier.] In this talk I will explain how to construct a non-trivial action of SL2(Z) on the vector space spanned by unipotent characters. The existence of such an action follows from Lusztig's classification of unipotent characters, but I will present a global construction relying on traces of automorphisms of Deligne--Lusztig varieties. The advantage of this approach is that it can be naturally generalized to Spetses, thus giving candidates for Fourier matrices and unipotent character sheaves for Spetses.
- Supplements
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Notes
1.29 MB application/pdf
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Apr 13, 2018
Friday
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09:30 AM - 10:30 AM
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The Loewy structure of certain fixed point algebras
Burkhard Kuelshammer (Friedrich-Schiller-Universität)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
My talk is motivated by questions concerning the Loewy structure of finite group algebras, their blocks and their centers. More generally, one may want to look at the Loewy structure of fixed point algebras where a finite group acts on a finite-dimensional algebra. I will report on some recent results and open problems. This is ongoing joint work with T. Breuer, L. Héthelyi and E. Horváth.
- Supplements
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Notes
163 KB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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More counting for the counting conjectures
Britta Späth (Bergische Universität-Gesamthochschule Wuppertal (BUGH))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The action of automorphisms on the irreducible characters of finite quasi-simple groups of Lie type is a well-known open problem. For symplectic groups this problem could be solved via a technical counting argument. We give a more structural explanation of this counting that applies to other types. Using this the inductive McKay condition can be verified for certain simple groups of Lie type. This is joint work with M. Cabanes.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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On equivariant cohomology of Calogero-Moser spaces
Peng Shan (Tsinghua University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will report a recent joint work with Cédric Bonnafé in which we compute equivariant cohomology of smooth Calogero-Moser spaces using representation theory of rational Cherednik algebras. We also give an application to the computation of equivariant cohomology of related symplectic resolutions.
- Supplements
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Notes
190 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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A Drinfeld center approach to character sheaves
Roman Bezrukavnikov (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The talk will be based on joint work with Finkelberg, Ostrik and also with Kazhdan and Varshavsky. Character sheaves (or D-modules) can be described as categorical (Drinfeld) center of the Hecke category (this idea was also independently worked out by Ben-Zvi and Nadler in a different context). This provides a new way to derive their classification and partly generalize the theory to loop groups."
- Supplements
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