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Members
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
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, Under revision for Invent. Math.
 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 symmetric pairs, arXiv:1806.02506
 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 2021, Semester 2
Organisers: Kari Vilonen, Ting Xue, Yaping Yang
Topics:
Time: Thursdays 2:15pm4:15pm.
Location: Peter Hall 213 and Zoom
Sep 23 Dougal Davis (Edinburgh) Hodge modules, LusztigVogan polynomials and unitary representations of real groups.
Abstract: I will discuss joint work in progress with Kari Vilonen on Kequivariant mixed Hodge modules on the flag variety for a symmetric pair (g, K). In close analogy with KazhdanLusztig theory, the Kgroup of mixed Hodge modules has two bases consisting of standard and irreducible objects; our first main theorem computes the change of basis matrix by adding an extra Hodge parameter to the LusztigVogan polynomials. Our second main theorem is a “polarised” version of the Jantzen conjecture; following ideas of Schmid and Vilonen, it allows the signature multiplicity polynomial of Adamsvan LeeuwenTrapaVogan for the corresponding real group representations to be read off from the Hodge polynomial (modulo a small claim). This recovers a key formula in the ALTV algorithm for the unitary dual.
Past Seminars:
Special lectures by Arun Ram on Reflection groups and the KZ functor
Lecture 1: 7 Dec Tuesday 1:002:15pm
Lecture 2: 10 Dec Friday 3:305:00pm
These first two lectures will be on Zoom (please contact organisers of the student seminar for link)
Lecture 3: 14 Dec 9:3011:30am
Lecture 4: 17 Dec 9:3011:30am
Lecture 5: 11 Jan 24pm Zoom
Abstract: This series of lectures will be focused on working through Section 3.1, the first three sentences of 5.1.1, and section 5.2.2 of the paper by GinzburgGuayOpdamRouquier (arXiv:math/0212036) entitled “On the category O for rational Cherednik algebras”. The goal is to do the examples of the KZfunctor for cyclic groups, dihedral groups and symmetric groups. The plan is for the lectures to be suitable for a GroupTheory and Linear Algebra student to build on their knowledge of what a group, vector space, and an eigenvalue are.
The TAs will be Yifan Guo and Haris Rao and anyone else who wishes to step up.
Depending on how we go, there may be further lectures in this series in January, on Tuesdays and Fridays, recommencing on 11th January.
Representation Theory Student Seminar 2021 Semester 2
Organisers: Ting Xue, Yaping Yang, Gufang Zhao
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 3:154:15pm
Location: Zoom (please contact the organisers for link)
Exercises
Schedule
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  Examples  [S] 1  Benjamin Gerraty 
Aug 11/18  [S] 2  Eskander Salloum  
Aug 25/Sep 1  Homogeneous spaces  [S] 3  Yifan Guo 
Sep 8  Polar decomposition, GrahamSchmidt, Bruhat decomposition  [S] 4  Grace Yuan 
Sep 22/29  Diagonalisation and maximal tori  [S] 4  Abraham Zhang 
Oct 6/20/27  Smooth manifolds, tangent space, one parameter subgroups and the exponential map  [S] 5  Adam Monteleone 
Nov 24/Dec1  Lie’s theorems  [S] 5  Linfeng Wei 
Fourier series and Representation theory  [S] 6  Joshua Culbert  
Compact groups and integration  [S] 7  Justin Tan  
Maximal compact subgroups  [S] 8  Beaudon Anasson  
The PeterWeyl theorem I  [S] 9  Haris Rao  
The PeterWeyl theorem II  [S] 9  Haris Rao  
Functions on and  [S] 10  
Induced representations  [S] 11  
The complexification of a compact group  [S] 12  
The unitary groups and the symmetric groups  [S] 13  Weiying Guo  
The BorelWeil theorem  [S] 14  
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
Dougal Davis (University of Edinburgh), Mar. 30, 2020Apr. 3, 2020
Xinwen Zhu (Caltech), November 24Dec 1, 2019
ChengChiang Tsai (Stanford University), Nov 19Dec 8, 2019
Geordie Williamson (Sydney), October 2628, 2019
Bill Casselman (University of British Columbia), Oct 2427, 2019
Emily Norton (MPIM), February 19March 2, 2019
Luca Migliorini (University of Bologna), December 39, 2018
ChengChiang Tsai (Stanford University), Nov 3Dec 1, 2018
Xinwen Zhu (Caltech), August 2330, 2018
Anthony Henderson (Sydney), August 2124, 2018
Jessica Fintzen (University of Michigan), November 27December 3, 2017
Sam Raskin (University of Texas at Austin) , September 1222, 2017
Takuro Mochizuki (RIMS, Kyoto University), September 415, 2017
Carl Mautner (UC Riverside), August 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.