Home page
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 ZinnJustin
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 padic 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, Ktheory 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 Dmodules from the microlocal point of view. His research on real groups, joint with Schmid, includes the proof of the BarbaschVogan 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 KnizhnikZamolodchikov equations, Cohomological Hall algebras, and vertex algebras associated to toric CalabiYau 3folds.
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

 A full list of each member’s publications can be found on their individual webpages.

 M. Lanini and P. J. McNamara, Singularities of Schubert varieties within a right cell. arXiv2003.08616.
 P. J. McNamara, Representation Theory of Geometric Extension Algebras, arXiv:1701.07949.
 P. J. McNamara. Representations of KhovanovLaudaRouquier Algebras III: Symmetric Affine Type. arXiv. Math. Z. 287 (2017), no. 12, 243–286.
 Peter J. McNamara, Finite Dimensional Representations of KhovanovLaudaRouquier 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), 257283.
 A. Kleshchev, A. Mathas, and Arun Ram, Universal Specht modules for cyclotomic Hecke algebras , arXiv1102.3519, Proc. London Math. Soc. (3) 105 (2012) 12451289.
 P. Diaconis and Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arXiv1007.4779, The Annals of Probability 40 (2012) Vol. 40 No. 5, 18611896.
 Roman Bezrukavnikov and Kari Vilonen, Koszul Duality for Quasisplit Real Groups, arXiv:1510.08343, Inventiones Mathematicae, 226 (2021),139193.
 Masaki Kashiwara and Kari Vilonen, Microdifferential systems and the codimensionthree conjecture. Ann. of Math. (2) 180 (2014) no. 2, 573620.
 Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups. Frontiers of mathematical sciences, 397420, 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), 10971144.
 TsaoHsien Chen, Kari Vilonen and Ting Xue, Springer correspondence for the split symmetric pair in type A , Compos. Math. 154 (2018), no. 11, 24032425. arXiv
 TsaoHsien 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), 515533. 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, 10291074.

Representation theory seminar 2023 Semester 1
Organisers: Dougal Davis, Kari Vilonen, Ting Xue
Place and time: Peter Hall 162, Thursdays 2:15 – 4:15pm
Please contact one of the organisers to be added to the mailing list.
Upcoming Seminars
May 18: Arun Ram (University of Melbourne) Lusztig varieties and Macdonald polynomials
Place and time: Peter Hall 162, 2:15pm – 3:15pm
In recent works AbreuNigro and Xuhua He have introduced the term Lusztig variety. I like this term, as Lusztig has many papers about these varieties. In 1997 Halverson and I computed the number of points of Type A nilpotent Lusztig varieties over finite fields in connection to characters of Hecke algebras. Recently, my study of Macdonald polynomials and central elements in Hecke algebras have led me to look at these computations again.
Past Seminars
May 11: Naoki Genra (University of Tokyo) Reduction by stages on Walgebras
Place and time: Peter Hall 162, 2:15pm – 3:15pm
Let be a Poisson variety with a Hamiltonian action and be a normal subgroup of . Then is obtained by a (Hamiltonian) reduction of by the induced action under suitable assumptions, called reduction by stages. We apply for the Slodowy slices and show that the Slodowy slice associated to is obtained by a reduction of the Slodowy slice associated to for a simple Lie algebra and nilpotent orbits , such that with some conditions. The quantum cases imply that the finite/affine Walgebras associated to are obtained by Walgebras associated to , which proves a conjecture of Morgan in finite cases and gives a conjectural generalization of results of Madsen and Ragoucy in affine cases. This is a joint work with Thibault Juillard.
May 4: Peter Fiebig (Erlangen) Representations and Binomial Coefficients
Place and time: Peter Hall 162, 2:15pm –
The talk is motivated by the „generational phenomena” that occur in the representation theory of algebraic groups in positive characteristics. The representation theory of quantum groups is known to provide a first step approximation to modular representations. Lusztig was the first to suggest that there should be „algebraic structures” that provide further steps towards modular representations beyond quantum groups. None of these structures are known today, even though some candidates have been suggested by several authors. In the talk I want to motivate the generational idea and then introduce a model category that makes the proximity of modular and quantum representations quite transparent. Using this category I want to show that the generational problem seems to be closely connected to finding generalizations of binomial coefficients.
April 27: Alexandre Minets (University of Edinburgh) A proof of P=W conjecture
Place and time: Peter Hall 162, 2:15pm – 4:15pm
Let C be a smooth projective curve. The nonabelian Hodge theory of Simpson is a homeomorphism between the character variety M_B of C and the moduli of (semi)stable Higgs bundles M_D on C. Since this homeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de CataldoHauselMigliorini conjectured that the weight filtration on H^*(M_B) gets sent to the perverse filtration on H^*(M_D), associated to the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on H^*(M_D). Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.
Monday April 17: Xuhua He (Chinese University of Hong Kong) Affine DeligneLusztig varieties and affine Lusztig varieties
Place and time: Peter Hall 162, 4pm – 5pm
Abstract: Roughly speaking, an affine DeligneLusztig variety describes the intersection of a given Iwahori double coset with a Frobeniustwisted conjugacy class in the loop group; while an affine Lusztig variety describes the intersection of a given Iwahori double coset with an ordinary conjugacy class in the loop group. The affine DeligneLusztig varieties provide a grouptheoretic model for the reduction of Shimura varieties and play an important role in the arithmetic geometry and Langlands program. The affine Lusztig varieties encode the information of the orbital integrals of IwahoriHecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. In this talk, I will explain a close relationship between affine Lusztig varieties and affine DeligneLusztig varieties, and consequently, provide an explicit nonemptiness pattern and dimension formula for affine Lusztig varieties in most cases. This talk is based on my preprint arXiv:2302.03203.
April 13: Dennis Gaitsgory (Max Planck Institute for Mathematics) Categorical geometric Langlands for Dmodules
Place and time: Peter Hall 162, 2:15pm – 4:15pm
Abstract: In the talk we will describe the recently obtained proof of GLC in the context of Dmodules. This is a joint project with D.Arikinkn, D. Beraldo, L.Chen, J.Faegerman, K. Lin, and S.Raskin, who made the most crucial contributions.
April 12: Dennis Gaitsgory (Max Planck Institute for Mathematics) Geometric Langlands with nilpotent singular support
Place and time: Peter Hall 162, 2:15pm – 4:15pm
Abstract: This talk will summarize the recent series of papers by ArinkinGaitsgoryKazhdanRaskinRozenblyumVarshavsky.
We’ll introduce a version of the categorical geometric Langlands conjecture that makes sense for ladic sheaves over a base of positive characteristic. We will explain the mechanism of the categorical trace of Frobenius, which allows to pass from the geometric assertion to a classical one.
March 23: Changlong Zhong (State University of New York at Albany) Oriented cohomology of the affine Grassmannian and the Peterson subalgebra
Place and time: Peter Hall 162, 2:15pm – 3:15pm
Abstract: Peterson subalgebra is a subalgebra of the (small) torusequivariant homology of the affine Grassmannian. It was proved by Peterson and LamShimozono that certain localization of this algebra is isomorphic to the quantum cohomology of the flag variety. Ktheoretic analogue of this result was recently proved by Kato, as part of his study of semiinfinite flag varieties. In this talk I will talk about generalization of the Peterson subalgebra to the oriented cohomology in the sense of LevineMorel. I will then talk about a certain localization of this subalgebra in the case of connective Ktheory.
March 16: Aritra Bhattacharya (Institute of Mathematical Sciences, Chennai) Haglund’s positivity conjecture for Macdonald polynomials
Place and time: Peter Hall 162, 2:15pm – 3:15pm
Abstract: The Macdonald symmetric functions are an incredible family of symmetric functions that simultaneously generalize many known bases of symmetric functions, such as the Schur functions and the HallLittlewood functions. The transition matrix between the HallLittlewood and the Schur functions are very well studied – they are given by the KostkaFoulkes polynomials which are polynomials in with nonnegative integer coefficients. However very little is known about the transition matrix between the Macdonald functions and the Schur functions.
Haglund conjectured that the Schur coefficients of the integral form Macdonald Polynomials have the positivity property that . We present some new partial results about this conjecture.
March 9: Alistair Savage (University of Ottawa) Diagratification
Place and time: Peter Hall 162, 2:15pm – 3:15pm
Abstract: We will explain how one can construct diagrammatic presentations of categories of representations of Lie groups and their associated quantum groups using only a small amount of information about these categories. To illustrate the technique in concrete terms, we will focus on the exceptional Lie group of type F4.
March 2: Kari Vilonen (University of Melbourne) Introductory remarks on geometric Langlands
Place and time: Peter Hall 162, 2:15pm – 4:15pm
Abstract: In preparation for the talks of Gaitsgory in April I will explain some aspects of the geometric Langlands program.
Some related notes by Ed Frenkel (sent in by Lance Gurney)
January 19: Ben Webster (University of Waterloo) Noncommutative resolutions and Coulomb branches
Place and time: Peter Hall 107, 2pm – 4pm
Abstract: Work of Braverman, Finkelberg and Nakajima provides a new window onto the world of noncommutative algebras, by constructing both new and previously known algebras as Coulomb branches. While beautiful, their construction involves some very infinite dimensional geometry. I’ll explain how we can replace this with a combinatorial construction, and how this leads us to a new perspective on noncommutative resolutions and categories of coherent sheaves.
January 12: Emanuel Scheidegger (Peking University) Aspects of modularity for CalabiYau threefolds
Place and time: Peter Hall 107, 2pm – 3pm
Abstract: We give an overview of some mostly conjectural aspects of modularity for CalabiYau threefolds. We focus on one parameter families of hypergeometric type and give computational results in terms of classical modular forms. If time permits, we show an explicit correspondence in one case. This is based on work with K. Boenisch, A. Klemm, and D. Zagier, 2203.09426.
January 5: Lisa Carbone (Rutgers University) A Lie group analog for the monster Lie algebra
Place and time: Peter Hall 107, 2pm – 3pm
Abstract: Let be the monster Lie algebra, a generalized Kac–Moody algebra. We describe various approaches to constructing an analog of a Lie group associated to . We construct a group , associated to , given by generators and relations. We also construct a group of automorphisms of a completion of . The subgroup of generated by all positive imaginary roots embeds in .
Past Seminars:
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 1112
Location: Peter Hall 107 and Zoom (please contact the organisers for zoom link)
Exercises
Schedule (2022S2)
Time: Wednesdays 1112
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 PeterWeyl theorem, continued
Aug 31 Yuhan Gai Functions on and
Aug 24 Cancelled
Aug 17 Haris Rao The PeterWeyl theorem
Schedule (2022S1)
Time: Mondays 1112
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 3:154:15pm
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, GrahamSchmidt
Sep 1 Yifan Guo Homogeneous spaces, continued. Notes
Aug 25 Yifan Guo Homogeneous spaces Notes
Aug 18 Eskander Salloum , continued. Notes
Aug 11 Eskander Salloum
Aug 4 Benjamin Gerraty Examples
Date  Topics  Reference  Speaker 
Aug 4, 2021  Examples  [S] 1  Benjamin Gerraty 
Aug 11/18, 2021  [S] 2  Eskander Salloum  
Aug 25/Sep 1, 2021  Homogeneous spaces  [S] 3  Yifan Guo 
Sep 8, 2021  Polar decomposition, GrahamSchmidt, 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 PeterWeyl theorem  [S] 9  Haris Rao 
Aug 31, 2022  Functions on and  [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 BorelWeil 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
Wille Liu (Academia Sinica), July 26August 18, 2023
Peter Fiebig (FriedrichAlexanderUniversität) Apr 30May 6, 2023
Alexandre Minets (University of Edinburgh), April 2330, 2023
Xuhua He (Chinese University of Hong Kong), April 1218, 2023
Dennis Gaitsgory (MPIM), April 517, 2023
Alistair Savage (University of Ottawa), March 710 (TBC), 2023
Changlong Zhong (State University of New York at Albany), Feb 28Mar 28, 2023
Ben Webster (University of Waterloo), Jan 1620, 2023
Andrea D’Agnolo (Padova), May 20Jun 17, 2022
Wille Liu (MPIM), May 30Jun 10, 2022
ChengChiang Tsai (Academia Sinica), May 29Jun 11, 2022
Dougal Davis (University of Edinburgh), Mar 30Apr 3, 2020
Xinwen Zhu (Caltech), Nov 24Dec 1, 2019
ChengChiang Tsai (Stanford University), Nov 19Dec 8, 2019
Geordie Williamson (Sydney), Oct 2628, 2019
Bill Casselman (University of British Columbia), Oct 2427, 2019
Emily Norton (MPIM), Feb 19Mar 2, 2019
Luca Migliorini (University of Bologna), Dec 39, 2018
ChengChiang Tsai (Stanford University), Nov 3Dec 1, 2018
Xinwen Zhu (Caltech), Aug 2330, 2018
Anthony Henderson (Sydney), Aug 2124, 2018
Jessica Fintzen (University of Michigan), Nov 27Dec 3, 2017
Sam Raskin (University of Texas at Austin) , Sep 1222, 2017
Takuro Mochizuki (RIMS, Kyoto University), Sep 415, 2017
Carl Mautner (UC Riverside), Aug 1626, 2017
 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 coorganising (with Masoud Kamgarpour and Peng Shan) the MATRIX workshop Frontiers in Representation Theory, 1425 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 CalabiYau 3folds” at the GRT at Home seminar on 23 June, 2020.
 Nora Ganter, Yaping Yang, and Gufang Zhao coorganized with Daniel Berwick Evans and Theo JohnsonFreyd the workshop on elliptic cohomology and physics 2528 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 1620 2019, Yaping Yang and Gufang Zhao coorganised 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 FebMar 2019. During the week of February 25March 1, 2019, Yaping Yang coorganised 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.