- Address:
- Department of Mathematics and Statistics,
- Fylde College,
- Lancaster University,
- Lancaster,
- LA1 4YF
- United Kingdom
- Office:
- B66, Fylde College
- Email:
- j.grabowski@lancaster.ac.uk
- Website:
- http://www.maths.lancs.ac.uk/~grabowsj/

- August 2020-present
- Assistant Dean for Engagement (0.3FTE), Faculty of Science and Technology, Lancaster University.
- August 2019-present
- Senior Lecturer in Algebra and Geometry, Department of Mathematics and Statistics, Lancaster University.
- September 2011-July 2019
- Lecturer in Pure Mathematics, Department of Mathematics and Statistics, Lancaster University.
- October 2009-September 2011
- Fixed-Term Fellow by Special Election in Mathematics, Keble College, Oxford.
- October 2008-October 2009
- Stipendiary Lecturer in Mathematics, Keble College, Oxford.
- January-March 2008-2010
- Lecturer and class tutor, Mathematical Institute, Oxford.
- October 2005-September 2008
- Research Fellow and Tutor in Mathematics, Keble College, Oxford.
- October 2005-September 2008
- Lecturer in Mathematics, Somerville College, Oxford.

- December 2013
- Postgraduate Certificate in Academic Practice (PGCert), Lancaster University.
- April 2006
- PhD in Mathematics, Queen Mary, University of London (QMUL), under the supervision of Prof. Shahn Majid.
- July 2002
- Master of Mathematics with Honours (MMath(Hons)), First class, University of Warwick.

*A triple construction for Lie bialgebras*, Pacific Journal of Mathematics,**221**, no. 2 (2005): 281-301, arXiv:math.QA/0312507*On Lie induction and the exceptional series*, Journal of Algebra and Its Applications,**4**, no. 6 (2005): 707-737, Special Issue on Representation Theory and Its Applications, arXiv:math.QA/0409359.*Braided-Lie bialgebras associated to Kac-Moody algebras*, Journal of Lie Theory,**18**, no. 1 (2008): 125-140, arXiv:0708.4200.*Examples of quantum cluster algebras associated to partial flag varieties*, Journal of Pure and Applied Algebra,**215**(2011): 1582-1595, arXiv:0907.4922.- with Stéphane Launois.
*Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases*, International Mathematics Research Notices,**10**(2011): 2230-2262, arXiv:0912.4397. *Braided enveloping algebras associated to quantum parabolic subalgebras*, Communications in Algebra,**39**, no. 10 (2011): 3491-3514, arXiv:0706.0455.- with Justin Allman.
*A quantum analogue of the dihedral action on Grassmannians*, Journal of Algebra,**359**(2012): 49-68, arXiv:1102.0422. - with Sira Gratz.
*Cluster algebras of infinite rank*, with an appendix by Michael Groechenig, Journal of the London Mathematical Society,**89**, no. 2 (2014): 337-363, arXiv:1212.3528. - with Stéphane Launois.
*Graded quantum cluster algebras and an application to quantum Grassmannians*, Proceedings of the London Mathematical Socity,**109**, no. 3 (2014): 697-732, arXiv:1301.2133. *Graded cluster algebras*, Journal of Algebraic Combinatorics,**42**, no. 4 (2015): 1111-1134, arXiv:1309.6170.- with Sira Gratz.
*Graded quantum cluster algebras of infinite rank as colimits*, Journal of Pure and Applied Algebra,**222**, no. 11 (2018): 3395-3413, arXiv:1510.04109. - with Matthew Pressland.
*Graded Frobenius cluster categories*, Documenta Mathematica,**23**(2018): 49-76, arXiv:1609.09670. - with Nicholas Cooney.
*Automorphism groupoids in noncommutative projective geometry*, preprint, arXiv:1807.06383. - with Nicholas Cooney.
*Slices of groupoids are group-like*, preprint, arXiv:2002.02860.

My research is concerned with the algebraic structures related to quantum groups and noncommutative geometry. These include Hopf algebras and Lie bialgebras and are especially important for their links to the study of the Yang-Baxter equation, also known as the braid relations. I have studied inductive constructions for Lie bialgebras and Hopf algebras based on the addition of nodes to Dynkin diagrams and their implications for the classification of simple Lie algebras and other structural questions. More recently, I have examined quantum cluster algebra structures associated to partial flag varieties. These structures encode much of the intricate combinatorics related to the algebraic geometry of these varieties.

- January 2020-present
- Supervising Dr Raquel Coelho Simões on Marie Skłodowska-Curie Actions Individual Fellowship COMBGEOREP.
- September 2015-present
- Working with Matthew Pressland, Max-Planck-Institut für Mathematik, Bonn.
- December 2014-present
- Working with Nicholas Cooney, Université Blaise Pascal - Clermont-Ferrand II.
- October 2013-December 2017
- Supervision of PhD student Thomas Booker-Price. Thesis:
*Applications of Graded Methods to Cluster Variables in Arbitrary Types* - May 2011-July 2011
- Supervised visiting student Paul Schneider, ENS Cachan, France.
- October 2010-present
- Working with Sira Gratz, Leibniz Universität Hannover, Germany. (Supervised Masters thesis October 2010-July 2011.)
- May 2010-November 2011
- Worked with Justin Allman, University of North Carolina at Chapel Hill, USA.
- June 2009-present
- Working with Stéphane Launois, University of Kent.

- March 2017
- Hosted Charlie Beil for research visit to Lancaster University.
- September 2015
- Hosted Matthew Pressland for research visit to Lancaster University.
- February 2015, March 2016
- Hosted Sian Fryer for research visits to Lancaster University.
- June 2014
- 3-day research visit to work with Sira Gratz at the Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Germany.
- October 2013
- Hosted Stéphane Launois for three-day research visit to Lancaster University.
- November 2011
- Hosted Sira Gratz for four-day research visit to Lancaster University, funded by Departmental Research Incentivisation Award.
- June 2009
- Visit to work with Stéphane Launois at the University of Kent.
- Summer 2001
- Research Project entitled "Orthogonal representations of generalisations of the Temperley-Lieb algebras", supported by a Nuffield Foundation Undergraduate Research Bursary, under the supervision of Dr. Bruce Westbury, University of Warwick.

- January 2022
- Principal Investigator on EPSRC Small Grant EP/W017881/1 (£33,573), 12-month project entitled "Schubert calculus via cluster categories"
- January 2020
- Principal Investigator on Marie Skłodowska-Curie Actions Individual Fellowship COMBGEOREP (€212,934), 2-year project, to be held by Dr Raquel Coelho Simões.
- November 2014
- Principal Investigator on EPSRC First Grant EP/M001148/1 (£98,663), 16-month project entitled "Reconstructing broken symmetry".
- October 2014
- Co-Investigator on EPSRC Standard Grant EP/M017001/1 (£6,367), part-funding for Workshop on Homotopical Algebra and Geometry, to be held at Lancaster in April 2015.
- January 2014
- Faculty Travel Grant (£282), to fund attendance at meeting "Cluster algebras and combinatorics", February 2014, Münster, Germany.
- February 2012
- London Mathematical Society Scheme 1 Conference Grant (£2,567.80), to part-fund conference "Future directions for quantum groups" held in Lancaster in September 2012.
- February 2012
- Faculty Research Grant (£1,000), to part-fund conference "Future directions for quantum groups" held in Lancaster in September 2012.
- November 2011
- Faculty Travel Grant (£348), to fund attendance at British Mathematical Colloquium, April 2012, Canterbury, Kent.
- October 2011
- Departmental Research Incentivisation Grant (£550), to fund research visit of Sira Gratz to Lancaster University.

- September 2021
- External Examiner of PhD thesis of Majed Almubark, University of Sheffield.
- March 2018
- External Examiner of MRes thesis of Robert Brown, University of York.
- July 2015
- Internal Examiner of PhD thesis of Heather Johnson, Lancaster University.
- July 2011
- Examiner of Masters thesis of Sira Gratz, ETH Zürich.

- November 2019
- Algebra, Geometry and Topology seminar, University of Kent.
- October 2018
- Oberseminare Algebra, Darstellungstheorie und Homologische Methoden, Institut für Algebra und Zahlentheorie, Universität Stuttgart, Germany.
- March 2018
- MAXIMALS seminar, School of Mathematics, University of Edinburgh.
- February 2018
- Algebra seminar, School of Mathematics and Statistics, University of Glasgow.
- February 2017
- Algebra and Geometry seminar, School of Mathematics, University of Bristol.
- January 2017
- Algebra seminar, Department of Mathematics, University of York.
- November 2016
- Algebra seminar, Mathematical Institute, Oxford.
- October 2016
- Algebra seminar, University of Kent.
- October 2016
- London Algebra Colloquium, Department of Mathematics, City University London.
- March 2016
- Pure Mathematics seminar, Department of Mathematics and Statistics, Lancaster University.
- March 2016
- Algebra-Geometry seminar, School of Mathematics and Statistics, Newcastle University.
- March 2015
- Pure Mathematics Colloquium, School of Mathematics and Statistics, University of Sheffield.
- February 2015
- 67th Bristol-Leicester-Oxford Colloquium (BLOC), City University, London.
- Janaury 2015
- Algebra seminar, Department of Pure Mathematics, University of Leeds.
- June 2014
- Oberseminar zur Algebra und Algebraischen Kombinatorik, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Germany.
- April 2014
- Pure Mathematics Colloquium, Department of Mathematical Sciences, Durham University.
- March 2014
- London Algebra Colloquium, Queen Mary, University of London.
- November 2013
- "Gradings on cluster algebras and associated combinatorics", Algebra-Geometry Seminar, School of Mathematics and Statistics, Newcastle University.
- October 2013
- "Gradings on cluster algebras and associated combinatorics", Pure Mathematics Seminar, Department of Mathematics, University of Leicester.
- November 2011
- "Quantizing cluster algebras associated to coordinate rings", Algebra seminar, University of Glasgow.
- November 2011
- "A dihedral group action on Grassmannians and its quantization", Pure Mathematics seminar, Lancaster University.
- October 2011
- "Some properties of quantum Grassmannians", ARTIN, Newcastle University.
- May 2011
- "Some quantum analogues of properties of Grassmannians", MAXIMALS, University of Edinburgh.
- November 2010
- "An introduction to cluster algebras and their quantum analogues", Mathematics seminar, University of Kent.
- October 2010
- "Quantizing Grassmannians, Schubert cells and cluster algebras", Algebra seminar, University of Manchester.
- May 2009
- "Partial flag varieties and quantum cluster algebras", Algebra and Geometry seminar, University of Bristol.
- March 2009
- "Partial flag varieties and quantum cluster algebras", 44th Bristol-Leicester-Oxford Colloquium (BLOC), University of Leicester.
- November 2008
- "Cluster algebras, partial flag varieties and quantum groups", London Algebra Colloquium.
- April 2008
- "Making modules into algebras", Mathematics research seminar, City University, London.
- March 2008
- "Making modules into algebras", Pure and Applied Mathematics seminar, University of Kent.
- September 2007
- "Some algebras associated to quantum parabolic subalgebras", Algebra seminar, University of Leeds.
- January 2004
- "Lie bialgebras and Lie induction", Topology and Geometry seminar, National University of Singapore.
- December 2004
- "On Lie induction and the exceptional series", Séminaire sur les Algèbres Enveloppantes, Institut de Máthematiques de Jussieu, Paris.

Also several talks to the Quantum Groups seminar at Queen Mary and the Representation Theory seminar, Kinderseminar and to the Invariants Society at Oxford.

- September 2019
- Quantum Flag Manifolds, Charles University, Prague, Czech Republic.
*Invited talk "Leveraging cluster structures".* - June 2015
- Special Session on "Combinatorics and Geometry of Quantum Algebras", International Meeting of the American, European and Portuguese Mathematical Societies, Porto, Portugal.
*Invited talk "Recovering automorphisms of quantum spaces".* - March 2015
- "Algebraic Interfaces of Integrability", Workshop on Classical and Quantum Integrability, University of Leeds.
*Invited talk "Clusters, combinatorics, grading, twisting and quantization".* - October 2014
- LMS Workshop on Cluster Algebras and Preprojective Algebras, School of Mathematics, Cardiff University.
*Invited talk "Quantizing cluster algebra constructions".* - February 2014
- "Cluster algebras and combinatorics", Mathematical Institute, University of Münster, Germany.
*Gave invited talk "Graded cluster algebras and quantum Grassmannians".* - September 2013
- "Mirror Symmetry and Cluster Algebras", School of Mathematics, University of Leeds.
*Presented poster "Graded cluster algebras".* - August 2012
- XV International Conference on Representations of Algebras and Workshop (ICRA XV), Universität Bielefeld, Germany.
*Gave talk entitled "A quantum analogue of a dihedral group action on Grassmannians".* - August 2011
- Kent Algebra Days 2011, University of Kent.
*Gave invited talk entitled "Quantum cluster algebras: a survey".* - December 2009
- 2nd Belgian and London Mathematical Societies Conference, Katholieke Universiteit Leuven, Belgium.
*Gave invited talk entitled "Quantizing Grassmannians, Schubert cells and cluster algebras".* - June 2009
- Geometric Representation Theory and Extended Affine Lie Algebras, University of Ottawa, Ontario, Canada.
*Presented poster "Examples of quantum cluster algebras associated to partial flag varieties".* - March 2009
- Algebraic Lie Structures with Origins in Physics, Isaac Newton Institute for Mathematical Sciences, Cambridge.
*Presented poster "Braided enveloping algebras and quantum cluster algebras related to partial flag varieties".* - January 2008
- LieGrits workshop, Mathematical Institute, University of Oxford.
- August 2007
- XII International Conference on Representations of Algebras and Workshop (ICRA XII), Nicolaus Copernicus University, Toruń, Poland.
*Gave 20 minute talk entitled "On the inductive construction of quantized enveloping algebras".* - June 2007
- Workshop on Cherednik Algebras, International Centre for Mathematical Sciences (ICMS), Edinburgh.
- December 2006
- Trends in Noncommutative Geometry, Isaac Newton Institute for Mathematical Sciences, Cambridge.
- June 2005
- Algebraic Groups and Finite Reductive Groups, École Polytechnique Fédérale de Lausanne, Switzerland.
- May 2005
- Algebras and Representation Theory, Symposium on Noncommutative Algebra and its Applications, University of Warwick.
- June 2004
- Fourth European Congress of Mathematics, Stockholm, Sweden.
- June 2004
- Representation Theory and its Applications, Uppsala University, Sweden.
*Gave 25 minute talk entitled "Lie induction: constructing Lie bialgebras".* - November 2003
- LMS Workshop on Contemporary Aspects of Mathematical Physics and Regional Meeting, Gregynog Hall, Wales.
*Presented poster "Fuss-Catalan recoupling theory".* - April 2002
- B(A)MC, University of Warwick.

In addition, I have chaired sessions at

- the Workshop on Noncommutative Geometry held as part of the British Mathematical Colloquium 2012, at the University of Kent; and
- the aforementioned meeting "Mirror Symmetry and Cluster Algebras" (September 2013, University of Leeds).

I participated in the Isaac Newton Institute programme "Cluster algebras and representation theory", visiting Cambridge for one week in November 2021.

I have attended meetings of the ABC-KLM Network, taken part in the LMS Workshop series on Quantum Groups and Noncommutative Geometry (two one-day meetings) and meetings of BLOC (Bristol-Leicester-Oxford Colloquium). I spoke at a meeting of ARTIN (Algebra and Representation Theory in the North) in October 2011.

- January-June 2015
- Led
*Derived Categories Reading Group*, Lancaster. Topics including representations of quivers, derived categories of these and derived categories of coherent sheaves on varieties. - July 2014
*Cluster algebras and their quantum analogues*(4 lectures), Kent Algebra Days PGR, University of Kent.

- September 2004
*Noncommutative spaces: their topology and measure theory*, LMS/EPSRC Short Course, University of Southampton.- September 2003
*Derived Categories in Algebra and Geometry*, LMS/EPSRC Short Course, University of Warwick.

- July 2020
- Effective Doctoral Supervision training session.
- May 2018
- Autism Awareness training session.
- November 2015
- Attended 2-day residential course "Communication and Media Skills", Royal Society at Chicheley Hall.
- October 2012-2013
- Completed Certificate in Academic Practice, Lancaster University.
- May 2012
- Attended course "Supervising Research Students", Lancaster University.
- November 2011
- Attended course "Using Audio Visual Equipment in Lecture Theatres" and completed online modules "Diverstiy in the workplace" and "Information security", Lancaster University.
- November 2011
- Attended course "Welcome to Lancaster", Lancaster University. (0.5 days)
- Spring 2008
- Completed programme on "Developing Teaching and Learning" (reading seminar and submission of portfolio), University of Oxford.
- May 2007
- Workshops on "Planning Lectures" and "Delivering a Course of Lectures", University of Oxford. (Two 3-hour sessions)
- October 2005
- Attended course "Introduction to Academic Practice", University of Oxford. (3 days)
- November 2004
- UK Research Councils' GRADSchool, Brighton. (3 days)

Lead organiser of 71st British Mathematical Colloquium, held in Lancaster in April 2019. Funded by grants from the London Mathematical Society, Clay Mathematics Institute and the Heilbronn Institute for Mathematical Sciences.

Organiser of 49th ARTIN (Algebra and Representation Theory in the North) network meeting in Lancaster in November 2016.

Co-organiser of 5-day LMS 150th Anniversary Northern Regional Meeting and Workshop on Homotopical Algebra, held in Lancaster in April 2015. Funded by grants from the London Mathematical Society and the EPSRC.

Organised 3-day meeting entitled "Future directions for quantum groups", held in Lancaster in September 2012. Funded by grants from the London Mathematical Society and the Faculty of Science and Technology, Lancaster University.

Subject Editor (Algebra) for the Glasgow Mathematical Journal.

Member of the EPSRC Peer Review College (from September 2013, renewed September 2016).

EPSRC Prioritisation Panel Member, June 2018.

Carnegie Research Assessor (by invitation), Carnegie Trust for the
Universities of Scotland (from February 2018).

Referee for Advances in Mathematics (2), the Journal of Algebraic Combinatorics, Frontiers of Mathematics in China, Algebras and Representation Theory (2), the Journal of Algebra (2), the Journal of the London Mathematical Society (2) and Advances in Applied Mathematics.

Reviewer of research proposals for EPSRC, FQRNT (Le fonds québécois de la recherche sur la nature et les technologies, Canada) and NWO (De Nederlandse Organisatie voor Wetenschappelijk Onderzoek; Netherlands Organistion for Scientific Research).

Reviewer for Mathematical Reviews and Zentralblatt.

Member, London Mathematical Society.

Member, European Mathematical Society.

Senior Fellow of the Higher Education Academy (from July 2018); previously Fellow (from December 2013) and Associate Fellow (from July 2008).

Fellow of the Institute of Mathematics and its Applications.

- 2019-2020
- Lecturing and workshop tutoring for MATH225 Abstract Algebra and supervision of MATH491 Pure Mathematics Dissertation students.
- 2018-2019
- Lecturing and workshop tutoring for MATH225 Abstract Algebra.
- 2017-2018
- Lecturing and workshop tutoring for MATH112 Discrete Mathematics and MATH225 Abstract Algebra.
- 2015-2017
- Lecturing and workshop tutoring for MATH112 Discrete Mathematics and MATH225b Rings.
- 2014-2015
- Lecturing and workshop tutoring for MATH112 Discrete Mathematics.
- 2011-2015
- Supervision of MATH390 Group Project and MATH491 Pure Mathematics Dissertation students.
- 2013-2014
- Lecturing and workshop tutoring for MATH112 Discrete Mathematics and MATH225b Rings.
- 2012-2013
- Lecturing and workshop tutoring for MATH112 Discrete Mathematics, MATH328 Number Theory and MATH225b Rings.
- 2011-2012
- Lectured MATH328 Number Theory and MATH114 Discrete Mathematics and workshops for module MATH110.
- January-March 2008-2010
- Lecturer for third year course "Group Theory", Mathematical Institute, Oxford. Also class tutor for same.
- October 2005-2011
- Tutorial teaching of first and second year undergraduates, Keble College and Somerville College, Oxford.

Courses taught:- Introduction to Pure Mathematics
- Linear Algebra
- Introduction to Groups, Rings and Fields
- Algebra (Ring Theory, Further Linear Algebra)
- Groups in Action
- Introduction to Fields
- Geometry
- Topology
- Number Theory
- Calculus of One Variable
- Calculus of Two or More Variables
- Multivariable Calculus

- October 2006-April 2007
- Tutor for one-term course entitled "Number Theory" for students on exchange programme between Exeter College, Oxford and Williams College, MA, USA (two groups: two students and one student).
- March 2005
- Waltham Forest AimHigher masterclass for year 10 students, entitled "What is a number?"
- October 2004-April 2005
- Undergraduate exercise classes (average: 30 students), QMUL.
- October 2002-April 2005
- Undergraduate coursework marking and administration, QMUL.
- October 2002-July 2003
- Private one-to-one tuition, A-level and GCSE.
- October 2001-June 2002
- First year undergraduate mathematics supervision tutor, University of Warwick. Small group teaching (four students) and marking for core first year courses.

- September 2021-present
- External Examiner for Pure Mathematics, University of Newcastle.

- June 2019
- Interviewed for video project on programme focus in curriculum design, Lancaster University.
- June 2019
- Assessor for HEA Senior Fellowship applications through the ATLAS programme, Lancaster University.
- October 2015
- Gave talk
*"What does that mean, precisely?" - learning to work with definitions*and led discussions at departmental Teaching and Learning sharing practice event. - July 2015
- Talk
*"What does that mean, precisely?" - learning to work with definitions*, Transition in year 1: Opportunities and challenges, Sharing Practice Day 2015, Lancaster University. - December 2013
- Completed Postgraduate Certificate in Academic Practice (PGCert), Lancaster University.

- September 2011-2014
- Deputy Admissions Tutor, Department of Mathematics and Statistics, Lancaster University.
- December 2005-2010
- Admissions interviewing, Keble College, Oxford.
- September 2003 & 2004
- Clearing assistant, QMUL.

- May 2013-present
- STEM Ambassador.
- September 2011-July 2016
- Outreach Officer, Department of Mathematics and Statistics, Lancaster University.

- December 2019
- Highly Commended, Dean's Award for Outstanding Contribution to the Faculty of Science and Technology, Lancaster.
- July 2008
- Departmental Teaching Development Award, Mathematical Institute, Oxford.

- September 2016-August 2019
- Part I Director of Studies (shared).
- March 2015-present
- Chair of university-level Web User Group, member of Web Transformation Steering Group.
- October 2012-March 2015
- Member of Web User Group.
- September 2012-August 2018
- Chair of Publicity, Web and Outreach Committee, Department of Mathematics and Statistics, Lancaster University.
- April 2010-September 2011
- Director of Studies for Mathematics, Keble College, Oxford.
- October 2009-11
- Member, Domestic Committee, Keble College, Oxford.
- October 2007-08
- Member, Finance Committee, Keble College, Oxford.
- May 2006-August 2006
- Acting Senior Mathematics Tutor, Keble College, Oxford.