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Members

Affiliated Members

Jan de Gier Nora Ganter Alex Ghitza Christian Haesemeyer Jack Hall

Thomas Quella David Ridout Chenyan Wu Paul Zinn-Justin

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

- - 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 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, Under revision for Invent. Math.
  - 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 symmetric pairs, arXiv:1806.02506
  - 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 seminar 2021, Semester 2

Organisers: Kari Vilonen, Ting Xue, Yaping Yang

Topics:

Time: Thursdays 2:15pm-4:15pm.

Location: Peter Hall 213 and Zoom

Sep 23 Dougal Davis (Edinburgh) Hodge modules, Lusztig-Vogan polynomials and unitary representations of real groups.

Abstract: I will discuss joint work in progress with Kari Vilonen on K-equivariant mixed Hodge modules on the flag variety for a symmetric pair (g, K). In close analogy with Kazhdan-Lusztig theory, the K-group 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 Lusztig-Vogan 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 Adams-van Leeuwen-Trapa-Vogan 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:

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 3:15-4:15pm

Location: Zoom (please contact the organisers for link)

Exercises

Schedule

Sep 15 Grace Yuan Bruhat decomposition

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	Examples	[S] 1	Benjamin Gerraty
Aug 11/18	$SU_2,SO_3,SL_2\mathbb{R}$	[S] 2	Eskander Salloum
Aug 25/Sep 1	Homogeneous spaces	[S] 3	Yifan Guo
Sep 8/15	Polar decomposition, Graham-Schmidt, Bruhat decomposition	[S] 4	Grace Yuan
	Diagonalisation and maximal tori	[S] 4	Abraham Zhang
	Smooth manifolds, tangent space, one parameter subgroups and the exponential map	[S] 5	Adam Monteleone
	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 Peter-Weyl theorem I	[S] 9	Haris Rao
	The Peter-Weyl theorem II	[S] 9	Haris Rao
	Functions on $\mathbb{R}^n$ and $S^{n-1}$	[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 Borel-Weil 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

Student seminar 2020S2 and 2021S1

What are the groups $E_2,E_3$ ?
Compute the dimension of $SO(n,\mathbb{R})$ .
What are the Lie algebras of the matrix groups $GL(n,\mathbb{R}),SL(n,\mathbb{R}), SO(n,\mathbb{R})$ ?
What is $SO(3,\mathbb{R})$ as a manifold?
Verify that the map $T_g$ , $g=\cos\theta+u\sin\theta$ indeed is rotation about the axis $u$ through the angle $2\theta$ .
Show that the Lie algebras of $SU_2$ and $SO_3$ are isomorphic (as Lie algebras).
Construct the double coverings $SU_2\times SU_2\to SO_4$ , $SL_2\mathbb{C}\to SO_{1,3}^+,\ SL_2\mathbb{R}\to SO_{1,2}^+$ and $SO_4\to SO_3\times SO_3$ .
Consider the action of $SO(4)$ on $\wedge^2\mathbb{R}^4$ . Find a decomposition of $\wedge^2\mathbb{R}^4$ into the direct sum of two stable subspaces.
Explain the picture of $SL_2\mathbb{R}$ . What is the simply connected covering group of $SL_2\mathbb{R}$ ?
Show that the space of complex structures on $\mathbb{R}^{2n}$ compatible with the inner product is isomorphic to the isotropic Grassmannian of $\mathbb{C}^{2n}$ .
For the sphere $S^{n-1}$ , find an isometry $f_x$ which reverses geodesics through $x$ . Let $G=O(n)$ be the isometry group of $S^{n-1}$ . Consider the automorphism $\alpha:G\to G,\,g\mapsto f_{x_0}\circ g\circ f_{x_0}$ , where $x_0$ is a fixed point on the sphere. What is the fixed point subgroup $G^\alpha$ ? What is the stabiliser group $H$ of $x_0$ in $G$ ? What is the relation between $G^\alpha$ and $H$ ?
Show that the positive-definite symmetric matrices form an open subset of the vector space of symmetric matrices.
Prove that an open convex subset in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$ .
Describe the polar decomposition of $GL(n,\mathbb{C})$ . That is, every invertible complex $n\times n$ matrix has a unique factorisation into a positive-definite Hermitian matrix and a unitary matrix (or the other way around).
Describe the analogous decomposition in Theorem 4.2 for $GL(n,\mathbb{R})$ .

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

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

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

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

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

Emily Norton (MPIM), February 19-March 2, 2019

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

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

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

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

Jessica Fintzen (University of Michigan), November 27-December 3, 2017

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

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

Carl Mautner (UC Riverside), August 16-26, 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 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.