Dirac operators in representation theory

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute

Abstract

The mathematical structure that describes the symmetries that appear in nature is a simple algebraic object called a group. For example, one can consider the symmetries of a circle in the plane, i.e., length-preserving transformations of the plane that map the circle to itself. Rotations by any angle preserve the circle. But this set of symmetries has some intrinsic additional structure, e.g., performing one rotation followed by another gives another rotation in the same set and every rotation has an inverse rotation. This set, together with the additional structure, is called the special orthogonal group in the plane, and it is an example of a Lie group. Lie groups, named after the Norwegian mathematician Sophus Lie, are mathematical objects underlying the continuous symmetries inherent in a system.

This proposals falls in the area of representations of Lie groups. Representations are ways in which Lie groups can manifest themselves, e.g., rather than regarding the special orthogonal group as an abstract object, one can think of its `representation' as transformations of the plane given by rotations. The study of representations of Lie groups has a long and illustrious history and has had transformative impact in number theory and theoretical physics. The main idea of the present project is to import and generalize a beautiful mathematical construction, called the Dirac operator. The Dirac operator originated in physics by the famous work of Paul Dirac in quantum mechanics, and subsequently, found a home in mathematics (geometry) by the seminal work of Atiyah and Singer, and in the representation theory of Lie groups in the work of Parthasarathy, Atiyah-Schmid, Kostant, and many others. The new algebraic approach to the theory was initiated by Vogan about 15 years ago with the introduction of Dirac cohomology and this has opened new and exciting perspectives of research in mathematics. The current project will extend the Dirac theory to an algebraic setting and apply the techniques of the Dirac operator and Dirac cohomology to the world of representations of p-adic Lie groups and of related algebraic structures (Hecke algebras) with applications to modern number theory and areas of mathematical physics.

Planned Impact

The proposal brings in a novel approach to algebraic representation theory, by exploring the applications of the Dirac operators to new algebraic settings. The proposal falls into an area of pure mathematics, called abstract harmonic analysis of Lie groups, that has been the subject of sustained and extraordinarily productive international research. The proposal will help to turn the UK into a leading place for research in this important mathematical field. The applications of the research in the proposal will have impact on modern number theory and on mathematical physics.

The results of the proposed research will be disseminated widely through talks at national and international conferences, graduate schools aimed at training postgraduate students and young researchers, at university seminars, and via papers published in top mathematical journals. The PI will also engage in outreach activities, for example, by giving public lectures targeted at senior secondary school students interested in mathematics and science.
 
Description I have made partial progress towards the verification of one of the conjectures listed in the main objectives for this grant. Particularly, I have verified an important conjecture in the field of unitary representations of reductive p-adic groups, and the proof has been published by one of the top journals in Mathematics. In addition, in joint work, we provided a complete and novel answer to a basic classification problem in the representation theory of affine Hecke algebras, certain algebraic objects that appear naturally in various areas of Mathematics. In joint work with a postdoctoral fellow, we have developed a theory of Dirac operators and dual pairs for rational Cherednik algebra. In a second project, with two postdoctoral fellows, we have introduced symplectic Dirac operators for Lie algebras, graded Hecke algebras, and rational Cherednik algebras. These discoveries are providing new lines of research inquiry that are likely to produce new activity in the my research field in the near and medium terms.
Exploitation Route The classification result is likely to be used by researchers in representation theory and abstract harmonic analysis as it provides a new and useful angle to the representation theory of reductive p-adic groups and Hecke algebras. The work on unitary representations is important to researchers in the field of automorphic forms, as well as to those in representation theory of reductive groups and abstract harmonic analysis. The algebraic results related to the representation theory of Cherednik algebras are of particular interest to the active community of researchers in the UK in this field.
Sectors Education

URL https://www.maths.ox.ac.uk/user/1691
 
Description Graduate course - representations of semisimple Lie algebras
Geographic Reach Local/Municipal/Regional 
Policy Influence Type Influenced training of practitioners or researchers
Impact I've created (2016) and taught (2016-2018) a graduate course in the representation theory of semisimple Lie algebras at Oxford. I wrote course notes and accompanying problem sheets, and these are freely available online.
 
Description National Science Foundation panel member, USA
Geographic Reach North America 
Policy Influence Type Participation in a advisory committee
Impact At the invitation of the NSF, I served on a grants panel at the NSF, Virginia, USA.
 
Description Ph.D. supervision (E. Okada)
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact I've been Mr. Emile Okada's DPhil supervisor towards a thesis in the representation theory of finite groups of Lie type and semisimple p-adic groups.
 
Description Ph.D. supervision - Ruben La
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
 
Description PhD supervision (K. Calvert)
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact I've supervised Mr Kieran Calvert's Oxford DPhil thesis in the representation theory of Hecke algebras and Dirac operators.
 
Description PhD supervision (X. Zhao)
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact I've been Miss Xin Zhao's DPhil supervisor towards a thesis in the representation theory of Cherednik and diagrammatic KLR algebras.
 
Description Bernstein projectors 
Organisation Hong Kong University of Science and Tech
Department Institute for Public Policy (IPP)
Country Hong Kong 
Sector Academic/University 
PI Contribution I have collaborated with Prof. Allen Moy at HKUST. I have contributed by expertise in the representation theory of reductive p-adic groups.
Collaborator Contribution Prof. Moy contributed his expertise in harmonic analysis for p-adic groups and Bruhat-Tits buildings. Prof Moy visited University of Oxford to work on this joint project and he gave a research talk in the Algebra seminar at Oxford (January 2017).
Impact This collaboration resulted in one preprint ``An Euler-Poincare formula for Bernstein projectors" (D. Barbasch, D. Ciubotaru, A. Moy), posted on the arXiv in Nov 2016, and submitted for journal publication. The paper is currently under review by the journal.
Start Year 2016
 
Description Cocenters of p-adic groups 
Organisation University of Maryland
Country United States 
Sector Academic/University 
PI Contribution I provide the expertise in representation theory of p-adic groups needed for this project.
Collaborator Contribution My collaborator is Xuhua He. He is an expert on geometric aspects of the structure of the algebraic objects (Hecke algebras) that occur.
Impact One paper ``Cocenters of p-adic groups: elliptic and rigid cocenters'' submitted for publication. Both He and I have given research talks about this work at international conferences, in my case at Representation Theory XV (Dubrovnik 2017) and in research seminars.
Start Year 2017
 
Description Cuspidal functions for reductive p-adic groups 
Organisation Tel Aviv University
Country Israel 
Sector Academic/University 
PI Contribution This project's goal is to describe the spaces of smooth cuspidal functions on reductive p-adic groups. This is an important problem in abstract harmonic analysis. We have verified a conjectural description of the answer in certain examples and we worked on how to generalise them.
Collaborator Contribution The collaborators are David Kazhdan from Tel Aviv and Eric Opdam from Amsterdam. Kazhdan is a world-leading expert in abstract harmonic analysis of reductive groups, and he has proposed this programme of understanding these spaces of functions. Opdam is a leading expert in the representation theory of reductive groups, which is one essential ingredient in the methods employed.
Impact Several interesting examples were calculated supporting conjectures by Kazhdan and Bezrukavnikov and Kazhdan. Eric Opdam reported on this collaboration in a research talk at Hebrew University in 2018.
Start Year 2017
 
Description Cuspidal functions for reductive p-adic groups 
Organisation University of Amsterdam
Department Korteweg-de Vries Institute for Mathematics
Country Netherlands 
Sector Academic/University 
PI Contribution This project's goal is to describe the spaces of smooth cuspidal functions on reductive p-adic groups. This is an important problem in abstract harmonic analysis. We have verified a conjectural description of the answer in certain examples and we worked on how to generalise them.
Collaborator Contribution The collaborators are David Kazhdan from Tel Aviv and Eric Opdam from Amsterdam. Kazhdan is a world-leading expert in abstract harmonic analysis of reductive groups, and he has proposed this programme of understanding these spaces of functions. Opdam is a leading expert in the representation theory of reductive groups, which is one essential ingredient in the methods employed.
Impact Several interesting examples were calculated supporting conjectures by Kazhdan and Bezrukavnikov and Kazhdan. Eric Opdam reported on this collaboration in a research talk at Hebrew University in 2018.
Start Year 2017
 
Description Dirac cohomology and unitary representations 
Organisation Cornell University
Country United States 
Sector Academic/University 
PI Contribution With Dan Barbasch and Eric Opdam, we have been investigating the use of Dirac operators and Dirac cohomology to the problem of classifying unitary modules in families for the affine Hecke algebras of types B/C with unequal parameters. Due to my expertise in the field, I have been a driving factor for this approach to the problem.
Collaborator Contribution Barbasch and Opdam are experts in unitary representations and Hecke algebras.
Impact We are working on extending the preliminary results and on organising the results so far into a paper.
Start Year 2019
 
Description Dirac cohomology and unitary representations 
Organisation University of Amsterdam
Department Korteweg-de Vries Institute for Mathematics
Country Netherlands 
Sector Academic/University 
PI Contribution With Dan Barbasch and Eric Opdam, we have been investigating the use of Dirac operators and Dirac cohomology to the problem of classifying unitary modules in families for the affine Hecke algebras of types B/C with unequal parameters. Due to my expertise in the field, I have been a driving factor for this approach to the problem.
Collaborator Contribution Barbasch and Opdam are experts in unitary representations and Hecke algebras.
Impact We are working on extending the preliminary results and on organising the results so far into a paper.
Start Year 2019
 
Description Elliptic representations and affine Hecke algebras 
Organisation University of Amsterdam
Department Korteweg-de Vries Institute for Mathematics
Country Netherlands 
Sector Academic/University 
PI Contribution I have collaborated with Prof. Eric Opdam at KdV Institute. This collaboration has resulted in two accepted papers that can be attributed to this grant. I have contributed my expertise in the use of Dirac methods in representation theory.
Collaborator Contribution Prof Opdam contributed his expertise in harmonic analysis for Hecke algebras and p-adic groups. He visited Oxford once in order to work on these joint projects.
Impact This collaboration resulted in several papers, two of which can be attributed to this grant. They are listed under publications.
Start Year 2016
 
Description On the unramified spherical spectrum 
Organisation Aix-Marseille University
Department Institute of Mathematics of Marseille
Country France 
Sector Academic/University 
PI Contribution I have collaborated with Prof. Eric Opdam at KdV Institute (University of Amsterdam) and Prof. Volker Heiermann at I2M (Universite d'Aix Marseille).
Collaborator Contribution Prof. Opdam contributed his expertise in harmonic analysis for Hecke algebras and Prof. Heiermann contributed with his expertise on automorphic forms. I have visited the University of Amsterdam twice to work on this joint project.
Impact This collaboration is in progress. There is already a pre-print from 2015 (in the arXiv) which will be updated in the next few months.
Start Year 2014
 
Description On the unramified spherical spectrum 
Organisation University of Amsterdam
Department Korteweg-de Vries Institute for Mathematics
Country Netherlands 
Sector Academic/University 
PI Contribution I have collaborated with Prof. Eric Opdam at KdV Institute (University of Amsterdam) and Prof. Volker Heiermann at I2M (Universite d'Aix Marseille).
Collaborator Contribution Prof. Opdam contributed his expertise in harmonic analysis for Hecke algebras and Prof. Heiermann contributed with his expertise on automorphic forms. I have visited the University of Amsterdam twice to work on this joint project.
Impact This collaboration is in progress. There is already a pre-print from 2015 (in the arXiv) which will be updated in the next few months.
Start Year 2014
 
Description Reducibility of induced representations of p-adic groups 
Organisation Aix-Marseille University
Department Institute of Mathematics of Marseille
Country France 
Sector Academic/University 
PI Contribution I provided the expertise in the representation theory of reductive p-adic groups and Iwahori-Hecke algebras that was needed to successfully complete the project.
Collaborator Contribution My collaborator, Volker Heiermann, provided his expertise in automorphic forms and representations.
Impact We have written one paper ``On the reducibility of induced representations'' which is submitted for publication. I visited Aix-Marseille in 2017 for this collaboration and during my visit, I gave a minicourse in representations of affine Hecke algebras to an audience consisting of PhD students, postdocs and researchers in the Mathematics Institute in Marseille.
Start Year 2017
 
Description The nonabelian Fourier transform 
Organisation Pierre and Marie Curie University - Paris 6
Country France 
Sector Academic/University 
PI Contribution I have begun a collaboration with Anne-Marie Aubert at Paris 6, Jussieu, building on previous work with Eric Opdam (Amsterdam). This is related to a fundamental question in the character theory of finite groups of Lie type and reductive p-adic groups. I provide expertise in the character theory of p-adic groups.
Collaborator Contribution Aubert is an expert in abstract harmonic analysis on p-adic groups.
Impact This collaboration is in the preliminary stages and the problem is difficult and ambitious. We are hopeful to verify an important example this year.
Start Year 2020
 
Description Algebra seminar Birmingham 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Professional Practitioners
Results and Impact I gave a seminar talk in the algebra seminar at University of Birmingham, November 2018.
Year(s) Of Engagement Activity 2018
 
Description An online piece about my research 
Form Of Engagement Activity Engagement focused website, blog or social media channel
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Public/other audiences
Results and Impact At the request of the Mathematical Institute in Oxford, I prepared a description of my new research aimed at the general public and general mathematicians. The piece was posted on the institute's website, and disseminated by the institute on twitter and Facebook. The piece collected thousands of views.
Year(s) Of Engagement Activity 2018
URL https://www.maths.ox.ac.uk/node/27666
 
Description Colloquium University East Anglia 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Professional Practitioners
Results and Impact I gave an invited colloquium talk in the Maths department at University of Anglia attended by members of the department.
Year(s) Of Engagement Activity 2019
 
Description Colloquium and seminar at University of Maryland 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I gave one departmental colloquium and one seminar talk (Lie groups seminar) at University of Maryland, College Park, September 2018.
Year(s) Of Engagement Activity 2018
 
Description Colloquium talk (Durham) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact I gave an invited colloquium talk in the Mathematics department at Durham University. The talk was based on my research but aimed at a more general audience.
Year(s) Of Engagement Activity 2017
 
Description Croatia - Workshop Research Talk 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a research talk in front of an audience of experts at a large international workshop in Dubrovnik, Croatia. The results were well received by the experts and there was significant interest from PhD students attending the workshop.
Year(s) Of Engagement Activity 2017
 
Description Croatia - Workshop Research Talk 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact A research talk about the project related to the "Dirac operators in representation theory" project at an international workshop in Dubrovnik, Croatia. After the talk, I received an invitation from one of the workshop participants, Dr M Balagovic, to visit her in Newcastle.
Year(s) Of Engagement Activity 2017
 
Description Grant reviewer 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I have reviewed Mathematics grant applications for the Royal Society UK, the German Research Foundation (DFG), the Canadian Natural Sciences and Engineering Research Council (NSERC), and Chile's (CONICYT - Chile).
Year(s) Of Engagement Activity 2018,2019
 
Description Israel - Conference Research Talk 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I presented new research results in front of an audience of experts at a large international conference (Weizmann Institute, Israel). The results were very well received and sparked interesting discussions and new ideas.
Year(s) Of Engagement Activity 2017
 
Description Lie Groups conference (TSIMF, Sanya, China) 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact This was an international workshop on harmonic analysis for reductive Lie groups. It brought together international experts in the field, as well as postdocs and PhD students. I made a research presentation about findings associated with this grant.
Year(s) Of Engagement Activity 2016
URL http://msc.tsinghua.edu.cn/sanya/2016/HALG2016/synopsis_and_organizers.aspx
 
Description Manchester - Workshop research talk 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a research talk at a workshop at University of Manchester with an audience of about 30 people. This allowed me to publicise my results and programme to an audience consisting of national experts and several international experts.
Year(s) Of Engagement Activity 2017
 
Description National Science Foundation grant panel 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact At the invitation of the National Science Foundation in the US, I served on a grants panel at the NSF building in Alexandria, Virginia.
Year(s) Of Engagement Activity 2019
 
Description Newcastle Algebra and Geometry Seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact Following an invitation by Dr M. Balagovic during a conference in Dubrovnik, Croatia, I gave a talk in the local seminar in Newcastle regarding a recent preprint (Dirac induction for rational Cherednik algebras, https://arxiv.org/abs/1710.06847) related to the "Dirac operators in representation theory" project. After the talk, some question arose which motivates further research.
Year(s) Of Engagement Activity 2017
URL http://www.ncl.ac.uk/maths-physics/seminars/pure/algebra/
 
Description Oberwolfach, Germany 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Research talk at the MPIM Oberwolfach, Germany in October 2019. The talk was very well received.
Year(s) Of Engagement Activity 2019
 
Description Open Days Somerville College 
Form Of Engagement Activity Participation in an open day or visit at my research institution
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Schools
Results and Impact About 30 pupils visited Somerville College in June 2017 (one day) to learn about the Mathematics course at Oxford and the application system at Oxford and Somerville College. Judging by the feedback received from the participants and from the number of applications to Mathematics at Somerville in Autumn, this was a very successful event.
Year(s) Of Engagement Activity 2017
 
Description Oxford - Workshop expository talk 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk at an international workshop in Oxford, December 2017. The talk sparked increased interest in the subject of representation theory of real reductive groups.
Year(s) Of Engagement Activity 2017
 
Description Oxford Algebra Seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Professional Practitioners
Results and Impact I have helped to organise the Oxford Algebra Seminar which meets weekly during term time. I have invited speakers for this seminar. The result is a more vibrant research community in the algebra group at Oxford and a platform for researchers to present their newest results.
Year(s) Of Engagement Activity 2016,2017,2018
 
Description Oxford Algebra seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact I have invited researchers in the areas related to my grant (representations of p-adic groups, Hecke and Cherednik algebras) to give research presentations in the Algebra seminar at Oxford which I am co-organizing. Between October 2016-March 2017, the invited speakers included: Niamh Farrell (City University London), Dr Simon Goodwin (Birmingham), Prof. Allen Moy (Hong Kong), Prof Shaun Stevens (University of East Anglia), Dr Max Gurevich (Weizmann, Israel). This led to more interactions between the members of the Algebra group at Oxford, including DPhil students and postdocs and to the dissemination of new research.
Year(s) Of Engagement Activity 2016,2017
URL https://www.maths.ox.ac.uk/events/past/623/
 
Description Oxford Algebra seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Postgraduate students
Results and Impact I gave a talk at the local seminar at the University of Oxford to communicate to the local colleagues the contents of a recent preprint (Dirac induction for rational Cherednik algebras, https://arxiv.org/abs/1710.06847) related to the "Dirac operators in representation theory" project. After the talk, some question arose which motivates further research.
Year(s) Of Engagement Activity 2017
 
Description Oxford Mathematics Open Days 
Form Of Engagement Activity Participation in an open day or visit at my research institution
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Schools
Results and Impact I was involved with the open days at the Maths Institute. I spoke with many prospective applicants about admissions and Maths teaching at Somerville.
Year(s) Of Engagement Activity 2018
 
Description Oxford algebra seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact I have reported over an ongoing collaboration related to the grant and to my PhD degree. This allowed me to introduce myself to the algebra group in the University of Oxford allowing me to establish communication with some of the members of the group.
Year(s) Of Engagement Activity 2016
URL https://www.maths.ox.ac.uk/node/16611
 
Description Paris workshop "Geometry and representation theory" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave an invited talk at the international workshop "Geometry and representation theory" in January 2020, in Paris, at the Institute Henri Poincare.
Year(s) Of Engagement Activity 2020
 
Description RGR (Representations des Groupes Reductifs) Seminar in Marseille 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk at the local seminar at the Math Institute of the University of Marseille to communicate about the contents of a recent preprint (Dirac induction for rational Cherednik algebras, https://arxiv.org/abs/1710.06847) related to the "Dirac operators in representation theory" project. After the talk, some question arose which motivates further research. I also used the activity to be in touch with a collaborator about another project.
Year(s) Of Engagement Activity 2017
URL https://www.i2m.univ-amu.fr/Seminaire-Representations-des-Groupes-Reductifs?lang=fr&date_debut=1900-...
 
Description Royal Institution Masterclass 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Schools
Results and Impact In March 2019, at the Maths Institute, I conducted one session of the Royal Institution Masterclass in Mathematics for year 9 school children from the Oxford area.
Year(s) Of Engagement Activity 2019
 
Description Seminar talk (Cambridge) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact I gave an invited seminar talk in the Number Theory seminar at the Mathematics Institute, Cambridge University. I presented new research results related to this grant objectives.
Year(s) Of Engagement Activity 2016
 
Description Somerville College Open Days 
Form Of Engagement Activity Participation in an open day or visit at my research institution
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Schools
Results and Impact I have been involved with the open days admissions at Somerville College, Oxford, in 2018. I spoke with many prospective applicants (and their teachers or family) about the admissions process in Oxford and the teaching of Mathematics in Oxford.
Year(s) Of Engagement Activity 2018
 
Description Talk - Representation Theory XVI, Dubrovnik, Croatia 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Research talk at an international workshop in Dubrovnik, Croatia. The new results have been well received.
Year(s) Of Engagement Activity 2019
 
Description Talk - Somerville undergraduates in Mathematics 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact In May 2017, I gave a talk to our Mathematics undergraduates at Somerville College, about 20 students attended. The talk aimed to explain my own research and the general research area called Representation Theory. There was a lot of enthusiasm among our students and several ended up taking Oxford courses in the subject subsequently.
Year(s) Of Engagement Activity 2017
 
Description Talk Rutgers University 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I gave a talk in the Lie groups and quantum algebra seminar at Rutgers University, New Brunswick, NJ, September 2018.
Year(s) Of Engagement Activity 2018
 
Description Talk conference Hangzhou Chine 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Gave a talk on my research at an internation workshop at Zhejiang University, China, June 2018.
Year(s) Of Engagement Activity 2018
URL http://www.math.zju.edu.cn/redir.php?catalog_id=23&object_id=34053
 
Description Workshop Zagreb, Croatia 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Gave a talk on my research at an international workshop "Dirac operators in representation theory" in Zagreb, Croatia, June 2018.
Year(s) Of Engagement Activity 2018