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Members

Dougal Davis Yau Wing Li Peter McNamara Arun Ram

Kari Vilonen Ting Xue Yaping Yang Gufang Zhao

Affiliated Members

Jan de Gier Nora Ganter Alex Ghitza Christian Haesemeyer Jack Hall

Thomas Quella David Ridout Chenyan Wu Paul Zinn-Justin

We are part of the Pure mathematics group.

Peter McNamara‘s research is in categorical and geometric representation theory. He has worked on a range of topics including quantum groups and their categorifications, perverse sheaves and Schubert varieties, algebraic combinatorics and p-adic groups.

Arun Ram’s research is in the area of Combinatorial Representation Theory. Topics he has worked on include tableaux combinatorics, crystals, diagram algebras, Lie theory, quantum groups, Hecke algebras, Schubert calculus, K-theory and cohomology of flag varieties and affine flag varieties. Current projects include constructions of representations of quantum affine algebras and a study of the combinatorics of double affine Hecke algebras and Macdonald polynomials.

Kari Vilonen‘s research is in the areas of real groups, the Langlands program, and related algebraic geometry. He has worked on several aspects of the geometric Langlands program and on more foundational questions on perverse sheaves and D-modules from the microlocal point of view. His research on real groups, joint with Schmid, includes the proof of the Barbasch-Vogan conjecture and a conjectural theory of Hodge structures on representations of real groups.

Ting Xue‘s research is in the areas of representation theory and algebraic groups. She has worked on questions related to geometry of nilpotent orbits and Springer theory, including small or bad characteristics. She is also interested in combinatorics arising from representation theory.

Yaping Yang‘s research lies in the area of Lie algebras, geometric representation theory, quantum groups, and the related geometry and topology. Her current work includes Knizhnik-Zamolodchikov equations, Cohomological Hall algebras, and vertex algebras associated to toric Calabi-Yau 3-folds.

Gufang Zhao‘s research lies at the interface between algebraic geometry and representation theory. More specifically, he has been working on projects concerning derived category of coherent sheaves, oriented cohomology theories of algebraic varieties, and their applications in representation theory. He is also fond of varieties of local systems and instantons, quantum integrable systems, and related aspects in mathematical physics.

Selected papers

- - P. J. McNamara and A. Savage, The spin Brauer category. arXiv2312.11766.
  - P. J. McNamara, Cluster monomials are dual canonical. arXiv2112.04109.
  - M. Lanini and P. J. McNamara, Singularities of Schubert varieties within a right cell. arXiv2003.08616. SIGMA 17 (2021), 070, 9pp.
  - P. J. McNamara. Representations of Khovanov-Lauda-Rouquier Algebras III: Symmetric Affine Type. arXiv. Math. Z. 287 (2017), no. 1-2, 243–286.
  - Peter J. McNamara, Finite Dimensional Representations of Khovanov-Lauda-Rouquier Algebras I: Finite Type, arXiv. J. Reine Angew. Math. 707 (2015), 103–124.
  - D. George, Arun Ram, J. Thompson and R. Volkas, Symmetry breaking, subgroup embeddings and the Weyl group, arXiv1203.1048, Physical Review D 87 105009 (2013) [14 pages]
  - Z. Daugherty, Arun Ram and R. Virk, Affine and degenerate affine BMW algebras: The center, arXiv1105.4207, Osaka J. Math 51 (2014), 257-283.
  - A. Kleshchev, A. Mathas, and Arun Ram, Universal Specht modules for cyclotomic Hecke algebras , arXiv1102.3519, Proc. London Math. Soc. (3) 105 (2012) 1245-1289.
  - P. Diaconis and Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arXiv1007.4779, The Annals of Probability 40 (2012) Vol. 40 No. 5, 1861-1896.
  - Roman Bezrukavnikov and Kari Vilonen, Koszul Duality for Quasi-split Real Groups, arXiv:1510.08343, Inventiones Mathematicae, 226 (2021),139-193.
  - Masaki Kashiwara and Kari Vilonen, Microdifferential systems and the codimension-three conjecture. Ann. of Math. (2) 180 (2014) no. 2, 573-620.
  - Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups. Frontiers of mathematical sciences, 397-420, Int. Press, Somerville, MA, 2011, arXiv
  - Kari Vilonen and Ting Xue, Character sheaves for classical symmetric spaces. With an appendix by Dennis Stanton. Represent. Theory 26 (2022), 1097-1144.
  - Tsao-Hsien Chen, Kari Vilonen and Ting Xue, Springer correspondence for the split symmetric pair in type A , Compos. Math. 154 (2018), no. 11, 2403-2425. arXiv
  - Tsao-Hsien Chen, Kari Vilonen and Ting Xue, On the cohomology of Fano varieties and the Springer correspondence, With an appendix by Dennis Stanton. Adv. Math. 318 (2017), 515-533. arXiv.
  - Ting Xue, Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2. Adv. Math. 230 (2012) no. 1, 229–262.
  - Marc Levine, Yaping Yang, and Gufang Zhao, Algebraic Elliptic cohomology theory and flops 1, appendix by Joël Riou. Mathematische Annalen volume 375, pages 1823–1855 (2019).
  - Miroslav Rapcak, Yan Soibelman, Yaping Yang, and Gufang Zhao, Cohomological Hall algebras, vertex algebras and instantons. Communications in Mathematical Physics volume 376, pages 1803–1873 (2020)
  - Yaping Yang, and Gufang Zhao, The cohomological Hall algebras for a preprojective algebra. Proc. Lond. Math. Soc. 116, 1029-1074.

Representation Theory Student Seminar 2021 Semester 2 and 2022 Semester 1&2

Organisers: Ting Xue, Yaping Yang, Gufang Zhao

Student organiser (2022S2): Davood Nejaty

Overview

This is a learning seminar on representation theory and related topics, aimed at 3rd year undergraduate students, MSc students, as well as PhD students in mathematics or related fields.

In the learning seminar, participants are expected to learn a topic based on the reference given, and present their work to the other participants. The audience are expected to ask questions and make comments during the presentation. Discussions tangential to the topics are welcome. The idea is for all participants to actively learn and discuss together.

This semester the topic will be Lie groups and their representations. We will follow the chapter on Lie groups by Graeme Segal in the reference below.

Reference: Carter, Roger; Segal, Graeme; Macdonald, Ian, Lectures on Lie groups and Lie algebras. With a foreword by Martin Taylor. London Mathematical Society Student Texts, 32. Cambridge University Press, Cambridge, 1995.

Time: Wednesdays 11-12

Location: Peter Hall 107 and Zoom (please contact the organisers for zoom link)

Exercises

Schedule (2022S2)

Time: Wednesdays 11-12

Oct 19/26 Kevin Fergusson The unitary and symmetric groups

Oct 12 Ali Khalili The complexification of a compact group

Oct 4 Cancelled

Sep 21 Amit Ben Harim Induced representations, continued

Sep 14 Amit Ben Harim Induced representations

Sep 7 Haris Rao The Peter-Weyl theorem, continued

Aug 31 Yuhan Gai Functions on $\mathbb{R}^n$ and $S^{n-1}$

Aug 24 Cancelled

Aug 17 Haris Rao The Peter-Weyl theorem

Schedule (2022S1)

Time: Mondays 11-12

Location: Peter Hall 107 and Zoom (please contact the organisers for zoom link)

Student organisers (2022S1): Weiying Guo and Davood Nejaty

May 30 Beaudon Anasson Maximal compact subgroups

May 23 Zhongtian Chen Compact groups and integration, continued

May 16 Zhongtian Chen Compact groups and integration

May 9 Kevin Fergusson Fourier series and Representation theory

May 2 Davood Nejaty Review, continued

Grace Yuan Bruhat decomposition Notes

Apr 11 Organisational meeting and review of previous topics

Schedule (2021S2)

Time: Wednesdays 3:15-4:15pm

Location: Zoom (please contact the organisers for link)

Dec 1 Linfeng Wei Lie’s theorems, continued. Notes

Nov 24 Linfeng Wei Lie’s theorems

Oct 27 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map, continued. Notes

Oct 20 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map, continued.

Oct 6 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map

Sep 29 Abraham Zhang Diagonalisation and maximal tori, continued. Notes

Sep 22 Abraham Zhang Diagonalisation and maximal tori

Sep 8 Grace Yuan Polar decomposition, Graham-Schmidt

Sep 1 Yifan Guo Homogeneous spaces, continued. Notes

Aug 25 Yifan Guo Homogeneous spaces Notes

Aug 18 Eskander Salloum $SU_2,SO_3,SL_2\mathbb{R}$ , continued. Notes

Aug 11 Eskander Salloum $SU_2,SO_3,SL_2\mathbb{R}$

Aug 4 Benjamin Gerraty Examples


Date	Topics	Reference	Speaker
Aug 4, 2021	Examples	[S] 1	Benjamin Gerraty
Aug 11/18, 2021	$SU_2,SO_3,SL_2\mathbb{R}$	[S] 2	Eskander Salloum
Aug 25/Sep 1, 2021	Homogeneous spaces	[S] 3	Yifan Guo
Sep 8, 2021	Polar decomposition, Graham-Schmidt, Bruhat decomposition	[S] 4	Grace Yuan
Sep 22/29, 2021	Diagonalisation and maximal tori	[S] 4	Abraham Zhang
Oct 6/20/27, 2021	Smooth manifolds, tangent space, one parameter subgroups and the exponential map	[S] 5	Adam Monteleone
Nov 24/Dec1, 2021	Lie’s theorems	[S] 5	Linfeng Wei
May 9/2022	Fourier series and Representation theory	[S] 6	Kevin Fergusson
May 16, 23/2022	Compact groups and integration	[S] 7	Zhongtian Chen
May 30/2022	Maximal compact subgroups	[S] 8	Beaudon Anasson
Aug 17, Sep 7, 2022	The Peter-Weyl theorem	[S] 9	Haris Rao
Aug 31, 2022	Functions on $\mathbb{R}^n$ and $S^{n-1}$	[S] 10	Yuhan Gai
Sep 14/21, 2022	Induced representations	[S] 11	Amit Ben Harim
Oct 12, 2022	The complexification of a compact group	[S] 12	Ali Khalili
Oct 19/26, 2022	The unitary groups and the symmetric groups	[S] 13	Kevin Fergusson
	The Borel-Weil theorem	[S] 14	Ali Khalili
References	[S] Graeme Segal, Lie groups, in Carter, Roger; Segal, Graeme; Macdonald, Ian, *Lectures on Lie groups and Lie algebras.* With a foreword by Martin Taylor. London Mathematical Society Student Texts, 32. Cambridge University Press, Cambridge, 1995.

Past seminar

Special lectures by Arun Ram summer 2021/22

Student seminar 2020S2 and 2021S1

Ryo Fujita (Kyoto University), Oct 2-31, 2024

Jingren Chi (Morningside Center of Mathematics), Oct 6-26, 2024

Wille Liu (Academia Sinica), August 23-September 6, 2024

Emile Okada (National University of Singapore), August 4-10, 2024

Elijah Bodish (MIT), July 30-August 8, 2024

Matt Emerton (U Chicago), July 11-Aug 2, 2024

Hiraku Nakajima (Kavli), June 30-July 4, 2024

Shilin Yun (Xiamen), June 9-16, 2024

Anne Dranowski (USC), August 9-11, 2023

Ryo Fujita (U Kyoto), August 8-24, 2023

Cheng-Chiang Tsai (Academia Sinica), August 6-18, 2023

Wille Liu (Academia Sinica), July 26-August 18, 2023

Peter Fiebig (Friedrich-Alexander-Universität) Apr 30-May 6, 2023

Alexandre Minets (University of Edinburgh), April 23-30, 2023

Xuhua He (Chinese University of Hong Kong), April 12-18, 2023

Dennis Gaitsgory (MPIM), April 5-17, 2023

Alistair Savage (University of Ottawa), March 7-10 (TBC), 2023

Changlong Zhong (State University of New York at Albany), Feb 28-Mar 28, 2023

Ben Webster (University of Waterloo), Jan 16-20, 2023

Andrea D’Agnolo (Padova), May 20-Jun 17, 2022

Wille Liu (MPIM), May 30-Jun 10, 2022

Cheng-Chiang Tsai (Academia Sinica), May 29-Jun 11, 2022

Dougal Davis (University of Edinburgh), Mar 30-Apr 3, 2020

Xinwen Zhu (Caltech), Nov 24-Dec 1, 2019

Cheng-Chiang Tsai (Stanford University), Nov 19-Dec 8, 2019

Geordie Williamson (Sydney), Oct 26-28, 2019

Bill Casselman (University of British Columbia), Oct 24-27, 2019

Emily Norton (MPIM), Feb 19-Mar 2, 2019

Luca Migliorini (University of Bologna), Dec 3-9, 2018

Cheng-Chiang Tsai (Stanford University), Nov 3-Dec 1, 2018

Xinwen Zhu (Caltech), Aug 23-30, 2018

Anthony Henderson (Sydney), Aug 21-24, 2018

Jessica Fintzen (University of Michigan), Nov 27-Dec 3, 2017

Sam Raskin (University of Texas at Austin) , Sep 12-22, 2017

Takuro Mochizuki (RIMS, Kyoto University), Sep 4-15, 2017

Carl Mautner (UC Riverside), Aug 16-26, 2017

Representation theory seminar 2024 Semester 2

Organisers: Dougal Davis, Kari Vilonen, Ting Xue

Place and time: Peter Hall 162, 2:15-4:15PM

Please contact one of the organisers to be added to the mailing list.

Upcoming Talks

Dec 5: Shigenori Nakatsuka (FAU Erlangen-Nurnberg)

Past Talks

Oct 24: Ting Xue (Melbourne) Character sheaves, affine Springer fibres, and d-Harish-Chandra series

I will explain how level-rank duality (conjecturally) arises from three pictures: character sheaves for graded Lie algebras, Oblomkov-Yun’s construction of rational Cherednik algebra modules using affine Springer fibres, and d-Harish-Chandra series introduced by Broué-Malle-Michel. This is based on joint work with various co-authors, Grinberg, Liu, Trinh, Tsai, and Vilonen.

Oct 17: Ryo Fujita (RIMS, Kyoto University) Singularities of normalized R-matrices and E-invariants for Dynkin quivers

The singularities of normalized R-matrices between modules over quantum loop algebras contain some information about the non-commutativity of the modules with respect to the tensor product. In this talk, we relate the pole orders of normalized R-matrices between simple modules in a certain nice subcategory (so-called Hernandez-Leclerc’s level-one subcategory) to the E-invariants (analogs of $Ext^1$ ) for decorated representations of Dynkin quivers. This may be seen as a correspondence of numerical characteristics between monoidal and additive categorifications of cluster algebras of finite ADE type.

Oct 10: Jingren Chi (Morningside Center of Mathematics) Geometry of affine Springer fibers and generalizations

Affine Springer fibers are analogues of Springer fibers for the loop Lie algebras of reductive groups. They were first studied by Kazhdan and Lusztig and they have played important roles in various problems from geometric representation theory and automorphic representation theory.
In this talk I will review the basic geometric properties of affine Springer fibers and report on recent work on some of their generalizations, including the group version and the mixed characteristic analogue.

Aug 29: Wille Liu (Academia Sinica) Deligne–Langlands correspondence for affine Hecke algebras at roots of unity

The affine Hecke algebra attached to a root system depends on a complex parameter q. When q is not a root of unity, Kazhdan and Lusztig established in 1987 a geometric parametrisation of simple modules of the affine Hecke algebra (called Deligne–Langlands correspondence). A modified version of this correspondence for root-of-unity q was announced without proof by Grojnowski in 1994. I will explain a proof of it that I discovered, which is based on the study of character sheaves on a graded Lie algebra.

Aug 8: Emile Okada (National University of Singapore) Character sheaves, Hecke algebras, and the wavefront set

The wavefront set is a distribution theoretic invariant first introduced to the study of representations of linear algebraic groups by Roger Howe in the early 80s. It associates to a representation a closed subset of the characteristic variety, and for groups over a local field it contains important arithmetic information. In this talk I will discuss recent progress toward understanding the wavefront set for groups over p-adic fields – the setting which is least well understood. In particular I will present recent developments in harmonic analysis which reduce the study to an arithmetic component, governed in depth 0 by character sheaves, and an algebraic components, governed in depth 0 by branching problems for modules of Hecke algebras.

Aug 6: Elijah Bodish (Massachusetts Institute of Technology) Spin link homology

Reshetikhin-Turaev define a Laurent polynomial invariant of knots for each simple Lie algebra “colored” by a finite dimensional irreducible representation. In the case of sl(2) and the defining representation, this polynomial invariant is the Jones polynomial.

Khovanov discovered that the Jones polynomial is the Euler characteristic of a complex of graded vector spaces. Thus, Khovanov’s homology categories the Jones polynomial. Many other definitions of categorified Reshetikhin-Turaev invariants have appeared since. The most notable works are: Khovanov-Rozansky’s generalization of Khovanov homology to sl(n), and Webster’s uniform construction for an arbitrary simple Lie algebra. However, very little is known (e.g. no examples are computed) unless the Lie algebra is sl(n) and the representation is a fundamental representation.

In my talk I will describe how to equip the sl(2n) link homology, colored by the n-th fundamental representation, with an involution such that the (super) Euler characteristic is the so(2n+1) Reshetikhin-Turaev link polynomial. This construction, which is apriori unrelated to Webster’s, is inspired by folding, categorified skew Howe duality, diagrammatics for centralizer algebras, and iquantum groups.

This is based on arXiv:2407.00189 — joint work with Ben Elias and David Rose.

July 23: Matt Emerton (U Chicago) Modular forms and Galois representations.

1-2PM Peter Hall 162

This is a pre-talk for the Thursday talk aimed at graduate students.

I will begin by recalling the basic facts about Tate modules of elliptic curves. Building on this discussion, I will recall the basic facts about Galois representations attached to modular forms, and sketch the proofs of some of them.

July 25: Matt Emerton (U Chicago) From modular curves to categorification

The categorical form of the local Langlands correspondence conjectures (and in some cases proves) that derived categories of smooth representations of p-adic reductive groups can be realized inside the derived categories of coherent sheaves on suitable moduli spaces of Langlands parameters. The goal of this lecture is to motivate this categorical local Langlands conjecture from an arithmetic point of view, with the cohomology of modular curves as the starting point.

June 12: Shilin Yu (Xiamen University) Duality of nilpotent orbit covers

3:15 – 4:15pm, Evan Williams Theatre

In their study of special unipotent representations for complex semisimple groups, Barbasch and Vogan constructed a duality map between the nilpotent orbits of $G$ and that of its Langlands dual group $G^\vee$ (also discovered by Lusztig and Spaltenstein), which allows them to describe the special unipotent ideals and representations of $G$ in terms of $G^\vee$ . This duality was later generalized by Sommers, Achar and Losev-Mason-Brown-Matvieievskyi.

In collaboration with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853), we reinterpret these duality maps in terms of covers of nilpotent orbits. This not only enables the definition of generalized unipotent representations, but also leads to interesting observations and conjectures regarding the birational geometry of the affinizations of the nilpotent orbit covers. If time permits, I will also discuss the connection of these findings to symplectic duality/3d mirror symmetry.

Past Seminars:

Kari Vilonen was awarded a highly prestigious ARC Laureate Fellowship on Real groups and the Langlands program.
Peter McNamara gave a talk on June 25, 2020 at the informal Friday seminar at U Sydney.
Nora Ganter, Peter McNamara, Yaping Yang and Gufang Zhao are co-organising (with Masoud Kamgarpour and Peng Shan) the MATRIX workshop Frontiers in Representation Theory, 14-25 February 2022.
Ting Xue will be a speaker at AMSI Winter School 2020, New directions in representation theory, University of Queensland. (Postponed.)
Yaping Yang gave a talk titled “Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds” at the GRT at Home seminar on 23 June, 2020.
Nora Ganter, Yaping Yang, and Gufang Zhao co-organized with Daniel Berwick Evans and Theo Johnson-Freyd the workshop on elliptic cohomology and physics 25-28 May 2020.
Arun Ram gave a performance of “Mendelssohn Salon 1828” with pianist Michael Leslie on
12 March 2020 at Tempo Rubato.
From Dec 2019 to Feb.29 2020, Yaping Yang and Gufang Zhao visited the Kavli Institute for the Physics and Mathematics of the Universe (IPMU), Japan. During the visit, Gufang Zhao gave a seminar talk titled “Cohomological Hall algebras and their representation theories” at the Mathematics and String Theory Seminar at IPMU.
The week of December 16-20 2019, Yaping Yang and Gufang Zhao co-organised the workshop on “Geometric Representation Theory and Quantum Field Theory” together with Hiraku Nakajima (IPMU), Peng Shan (Tsinghua), Wenbin Yan (Tsinghua) at TSIMF, Sanya, China.
Yaping Yang and Gufang Zhao visited the Perimeter Institute for Theoretical Physics, Waterloo, Canada during Feb-Mar 2019. During the week of February 25-March 1, 2019, Yaping Yang co-organised the workshop “Cohomological Hall algebras in Mathematics and Physics” at Perimeter Institute (with Kevin Costello (PI) and Yan Soibelman (KSU)).
Yaping Yang received an ARC Discovery Early Career Award (DE 190101231) in Dec 2018.
Gufang Zhao received an ARC Discovery Early Career Award (DE 190101222) in Dec 2018.