Bio

I am a mathematician working on bridges between low-dimensional topology and representation theory. Two focus points of my research have been the development of functorial homology theories for knots and links, and the exploration of their deep connections to (higher) representation theory and mathematical physics. I also enjoy applying topological and combinatorial tools to solve problems in representation theory.

In October 2019 I have been appointed as a Hirzebruch Research Instructor affiliated with the Max Planck Institute for Mathematics and the Mathematical Institute of the University of Bonn. During the spring semester 2020, I visited the Mathematical Sciences Research Institute.

Links: A community hub for Geometric Representation Theory at Home. The Oberseminar Darstellungstheorie (representation theory) at the University of Bonn.

MSRI Postdoctoral Fellow

Mathematical Sciences Research Institute
02/2020-05/2020

Postdoctoral Fellow

The Australian National University
10/2017-09/2019

Research Associate

Imperial College London
10/2015-09/2017

PhD

University of Cambridge
10/2015, Advisor: Jake Rasmussen

Master of Advanced Study

University of Cambridge
06/2012

Bachelor of Science

Universität Wien
07/2011

Publications

All my research papers are available on the mathematics arXiv. They are also listed by Google Scholar, ORCID logo ORCiD and ResearchGate. Preprints come before published or accepted articles in the following list.

19) gl(2) foams and the Khovanov homotopy type

The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen.

With Vyacheslav Krushkal. Submitted.

18) Invariants of 4-manifolds from Khovanov-Rozansky link homology

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.

With Scott Morrison and Kevin Walker. Submitted.

17) Evaluations of annular Khovanov-Rozansky homology

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.

With Eugene Gorsky. Submitted.

16) Extremal weight projectors II

In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley–Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case of gl(N) for N > 1, with a view towards categorifying the corresponding torus skein algebras via Khovanov–Rozansky link homology. As by-products, we obtain compatible diagrammatic presentations of the representation categories of gl(N) and its Cartan subalgebra, and a categorification of power-sum symmetric polynomials.

With Hoel Queffelec. Submitted.

15) Tangle addition and the knots-quivers correspondence

We prove that the generating functions for the one row/column colored HOMFLY-PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. Our approach extends the previously established tangles-quivers correspondence for rational tangles to algebraic tangles by developing gluing formulas for HOMFLY-PT skein generating functions under Conway's tangle addition. As a consequence, we prove the conjectural links-quivers correspondence of Kucharski–Reineke–Stošić–Sułkowski for all arborescent links.

With Marko Stošić.

Journal of the London Mathematical Society published online.

14) The center of SL(2) tilting modules

In this note we compute the centers of the categories of tilting modules for G=SL(2) in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective G_gT-modules when g=1,2.

With Daniel Tubbenhauer.

Glasgow Mathematical Journal published online.

13) A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain

The hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N-1 types of Temperley-Lieb generators. This generalises a previous result for N=3 obtained by J. F. Fjelstad and T. Månsson [J. Phys. A 45 (2012) 155208]. A pictorial representation of a related coupled algebra is given for the N=3 case which involves a generalisation of the pictorial representation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. (shortened)

With Remy Adderton and Murray T. Batchelor.

Journal of Physics A: Mathematical and Theoretical 53-36 (2020)

12) Derived traces of Soergel categories

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type $A$. As an application we obtain a derived annular Khovanov--Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus.

With Eugene Gorsky and Matthew Hogancamp.

International Mathematics Research Notices published online

11) Quivers for SL(2) tilting modules

Using diagrammatic methods, we define a quiver algebra depending on a prime p and show that it is the algebra underlying the category of tilting modules for SL(2) in characteristic p. Along the way we obtain a presentation for morphisms between p-Jones-Wenzl projectors.

With Daniel Tubbenhauer.

Accepted for publication in Representation Theory

10) Algèbre diagrammatique et catégorification

Nous proposons une illustration diagrammatique abordable du concept de catégorification qui s’est développé au cours des vingt dernières années.

A friendly outreach article in French, written mostly by Hoel Queffelec, inspired by our joint work.

La Gazette des mathématiciens 163 (janvier 2020)

9) Khovanov homology and categorification of skein modules

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the gl(2) skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunctor on the target category and conjecture that it extends to a monoidal structure. This would give rise to a canonical basis of the associated gl(2) skein algebra and verify an analogue of a positivity conjecture of Fock–Goncharov and Turston. We provide evidence towards the monoidality conjecture by checking several instances of a categorified Frohman-Gelca formula for the skein algebra of the torus. Finally, we recover a variant of the Asaeda–Przytycki–Sikora surface link homologies and prove that surface embeddings give rise to spectral sequences between them.

With Hoel Queffelec

Quantum Topology 21-1 (2021) 129-209.

8) Rational links and DT invariants of quivers

We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. This shows that the conjectural links-quivers correspondence of Kucharski–Reineke–Stošić–Sułkowski as well as the LMOV conjecture hold for rational links. Along the way, we extend the links-quivers correspondence to tangles and, thus, explore elements of a skein theory for motivic Donaldson-Thomas invariants.

With Marko Stošić.

International Mathematical Research Notices 6 (2021) 4169-4210.

7) Functoriality of colored link homologies

We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.

With Michael Ehrig and Daniel Tubbenhauer.

Proceedings of the London Mathematical Society 117-5 (2018), 996-1040

6) Extremal weight projectors

We introduce a quotient of the affine Temperley-Lieb category that encodes all weight preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl projectors, and that they categorify the Chebyshev polynomials of the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus, which is well adapted to the task of categorifying the multiplication on the Kauffman bracket skein module of the torus.

With Hoel Queffelec.

Mathematical Research Letters 25-6 (2018), 1911-1936

5) Exponential growth of colored HOMFLY-PT homology

We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT homologies.

Advances in Mathematics 353 (2019), 471-525

4) Super q-Howe duality and web categories

We use super q–Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of gl(N)–modules (and, more generally, gl(N|M)–modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation-theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY–PT polynomials.

With Daniel Tubbenhauer and Pedro Vaz.

Algebraic & Geometric Topology 17-6 (2017), 3703-3749

3) Deformations of colored sl(N) link homologies via foams

We prove a conjectured decomposition of deformed sl(N) link homology, as well as an extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To this end, we use foam technology to give a completely combinatorial construction of Wu’s deformed colored sl(N) link homologies. By studying the underlying deformed higher representation-theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.

With David Rose.

Geometry & Topology 20-6 (2016), 3431-3517

2) q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links

We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.

Journal of Pure and Applied Algebra 223-4 (2019), 1434-1439

1) Categorified sl(N) invariants of colored rational tangles

We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links and discuss consequences for the corresponding decategorified invariants.

Algebraic & Geometric Topology 16-1 (2016), 427-482

Events

Upcoming

  1. Workshop: "Perspectives on Knot Homology", Banff International Research Station, 16th - 21st May 2021.
  2. I will give a lecture series at the combined student workshop and research conference "Perspectives on quantum link homology theories" at the University of Regensburg, 9th - 15th August 2021.
  3. Workshop: "Foam Evaluation", ICERM, 5th - 7th November 2021.

Organisation

  1. I co-organised a workshop on "Monoidal and 2-categories in representation theory and categorification" at the Hausdorff Research Institute for Mathematics, Bonn, 30th November - 4th December 2020. Video recordings
  2. I co-organised the Felix Klein Lectures 2020 by Markus Reineke at the Hausdorff Center for Mathematics, Bonn, 20th - 29th October 2020. Video recordings.
  3. I co-organised a Lecture series on modified traces in algebra and topology at the Hausdorff Institute for Mathematics, Bonn, 12th - 22th October 2020. Video recordings.
  4. I was a group leader in the HIM Junior Trimester Program: "New Trends in Representation Theory", September - December 2020.
  5. I coordinated the Oberseminar Darstellungstheorie (representation theory) at the University of Bonn, winter semester 2019/20.
  6. I co-organised a Workshop on "Categorification in quantum topology and beyond" at the Erwin Schrödinger Institut, Vienna, 7th - 18th January 2019. Video recordings.
  7. I co-organised a Workshop on "Classical and quantum three-manifold topology" at Monash University, Melbourne, 14th - 21st December 2018.
  8. I co-organised the "Mathematical Sciences Institute Colloquium" at the Australian National University, 2018 - 2019.
  9. I was part of the Junior Trimester Program on "Symplectic Geometry and Representation Theory" at the Hausdorff Institute for Mathematics, Bonn, September - December 2017.

Conference talks and colloquia

  1. Online conference "QUAntum groups, Categorification, Knot invariants, and Soergel bimodules", University of Oregon, 10th - 14th August 2020. Video
  2. Workshop: "Soergel Bimodules and Categorification of the Braid Group", ICERM, 28th February - 1st March 2020. Video
  3. Conference: "Mathematics and Physics of Knots", Institute Mittag Leffler, 24th - 28st June 2019.
  4. Conference: "Hilbert schemes, categorification and combinatorics", UC Davis, 19th - 23rd June 2019.
  5. Conference: "Quantum Topology and hyperbolic geometry III", Da Nang, Vietnam, 27th - 31st May 2019. Video
  6. Workshop: "Hidden Algebraic Structures in Topology", Caltech, 13th-16th March 2019.
  7. Conference: Kioloa, 18th - 22nd February 2019.
  8. Conference: "Aspects in higher representation theory", Brussels, 21st - 25th January 2019.
  9. Workshop: "Categorified Hecke algebras, link homology, and Hilbert schemes", American Institute of Mathematics, 1st - 5th October 2018.
  10. Conference: "Interactions of low-dimensional topology and higher representation theory", Universität Zürich, 17th - 21st September 2018.
  11. Conference: "Categorification and Higher Representation Theory", Institute Mittag Leffler, 9th - 13th July 2018.
  12. Conference: TQFT and categorification, IESC, Corsica, 15th - 20th April 2018.
  13. Workshop: Categorification in mathematical physics, SCGP, Stony Brook, 9th - 13th April 2018. Video
  14. Workshop: Modular Forms and Quantum Knot Invariants, Banff International Research Station, 11th - 16th March 2018.
  15. Conference: Quantum knot homology and supersymmetric gauge theories, Aspen Center for Physics, 4th - 10th March 2018.
  16. Junior Trimester Programme: Symplectic Geometry and Representation Theory, Hausdorff Research Institute for Mathematics, Bonn, 20th November - 1st December 2017. Video
  17. Conference: Representation Theory and Combinatorics of Torus Links, University of Massachusetts, Amherst, 8th July 2017.
  18. INI workshop: Quantum topology and categorified representation theory, Isaac Newton Institute, Cambridge, 26th - 30th June 2017. Video
  19. Symposium in Mathematical Physics, Universität Zürich, 24th - 25th April 2017.
  20. INI workshop: Physics and knot homologies, Isaac Newton Institute, Cambridge, 10th - 13th April 2017. Video
  21. Quantum invariants and low-dimensional topology, MATRIX, Melbourne, Australia, 14th - 17th December 2016, Slides.
  22. Workshop on Homological Methods in Algebra and Geometry, African Institute for Mathematical Sciences, Ghana, August 2016.
  23. WARTHOG, University of Oregon, 25th - 29th July 2016.
  24. Conference: Knots in Hellas, Keynote talk, Olympic Academy, Greece, 17th - 23rd July 2016, Slides.
  25. Workshop: Categorification, Universität Bonn, Germany, 16th - 20th May 2016.
  26. SwissMAP Conference, Engelberg, Switzerland, 24th - 30th January 2016.
  27. Colloquium, George Washington University, Washington DC, 13th November 2015.
  28. Workshop: Physics and mathematics of knot homologies, SCGP, Stony Brook, 1st June - 3rd June 2015. Video, Slides.
  29. Conference: Winter Braids V, Université de Pau, 16th February - 19th February 2015.
  30. Young Topology Meeting, Imperial College London, 6th June 2014.

Seminar talks

  1. Online seminar talk, University of Hamburg, 2nd February 2021.
  2. Online seminar talk, Topology Seminar, MPIM, 21st December 2020.
  3. Two online seminar talks, Learning seminar on categorification, 9th and 16th July 2020. Video 1, Slides 1, Video 2, Slides 2
  4. Online seminar talk, Séminaire de Topologie LAGA/IMJ-PRG, Paris, 16th June 2020.
  5. Online seminar talk, Kansas State University, 8th May 2020.
  6. Online seminar talk, MSRI, 6th May 2020.
  7. Online seminar talk, Joint Los Angeles Topology Seminar, 4th May 2020.
  8. Seminar talk, George Washington University, 28th April 2020.
  9. Online seminar talk, Representation Theory and Mathematical Physics Seminar, UC Berkeley, 21st April 2020.
  10. Online Seminar talk, Stanford University, 14th April 2020.
  11. Seminar talk, University of Massachusetts, Amherst, 2nd March 2020.
  12. Seminar talk, UC Davis, 18th February 2020.
  13. Research talk, University of Birmingham, 13th January 2020.
  14. Seminar talk and research visit, Instituto Superior Técnico 9th - 12th December 2019.
  15. Seminar talk, University of Stuttgart, 19th November 2019.
  16. Seminar talk, University of Kaiserslautern, 7th November 2019.
  17. Seminar talk, University of Bielefeld, 25th October 2019.
  18. Seminar talk and research visit, Universität Zürich, 24th - 27th September 2019.
  19. Seminar talk, The Australian National University, 26th July 2019.
  20. Seminar talk and research visit, Université de Montpellier, 14th - 18th April 2019.
  21. Seminar talk and research visit, Universität Zürich, 10th - 14th April 2019.
  22. Research talk, University of Vienna, 6th March 2019.
  23. Research talk, University of Leicester, 14th February 2019.
  24. Research talk, University of Oklahoma, 5th February 2019.
  25. Research talk, Montana State University, 31st January 2019.
  26. Research talk, University of Essex, 28th January 2019.
  27. Research talk, University of Hamburg, 25th January 2019.
  28. Seminar talk, Claremont topology seminar, 6th December 2018.
  29. Seminar talk, UCSB, 4th December 2018.
  30. Seminar talk, Uppsala Universitet, 16th July 2018.
  31. Research talk, TU München, 16th July 2018.
  32. Seminar talk, The Australian National University, 18th May 2018.
  33. Seminar talk and research visit, Monash University, 7th - 9th May 2018.
  34. Seminar talk, University of Melbourne, 7th May 2018.
  35. Seminar talk, University of Southern California, 19th March 2018.
  36. Seminar talk, The Australian National University, 14th November 2017.
  37. Seminar talk, Algebra Seminar, Sydney University, 27th October 2017.
  38. Seminar talk, The Australian National University, 20th October 2017.
  39. Seminar talk and research visit, QGM, Aarhus University, 28th August - 1st September 2017.
  40. Seminar talk and research visit, Instituto Superior Técnico 22nd - 24th May 2017.
  41. Seminar talk, Universität Wien, 2nd May 2017.
  42. Seminar talk, Symplectic cut, London, 22nd March 2017.
  43. Seminar talk, Geometry Tea, Cambridge, 17th March 2017.
  44. Seminar talk, Symplectic cut, London, 15th February 2017.
  45. Seminar talk, Université de Montpellier, 1st December 2016.
  46. Seminar talk, Université catholique de Louvain, 28th November 2016.
  47. Seminar talk, Universität Wien, 30th June 2016.
  48. Seminar talk, Oberseminar Darstellungstheorie, Universität Bonn, Germany, 15th April 2016.
  49. Seminar talk, MPI Oberseminar, Max-Planck-Institut für Mathematik Bonn, Germany, 14th April 2016.
  50. Seminar talk and research visit, Physics and Geometry Seminar, Caltech, 15th November - 25th November 2015.
  51. Seminar talk, Columbia University, New York, 12th November 2015.
  52. Seminar talk and research visit, Université catholique de Louvain, 22nd April - 25th April 2015.
  53. Seminar talk and research visit, Universität Zürich 17th March - 20th March 2015.
  54. Seminar talk, Durham University, 29th January 2015.
  55. Seminar talk, Differential Geometry and Topology Seminar, University of Cambridge, 28th January 2015.
  56. Seminar talk and research visit, QGM, Aarhus University, 12th January - 17th January 2015.
  57. Seminar talk, Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), 4th December 2014.
  58. Seminar talk, Université de Genève, 27th November 2014.
  59. Seminar talk and research visit, QGM, Aarhus University, 8th July - 12th July 2014.
  60. Seminar talk, Erwin Schrödinger Institut, 30th June 2014.

Other events

Teaching:

Past:

  • S4A2 Graduate Seminar on Representation Theory - Knot homology (with Catharina Stroppel), University of Bonn, winter semester 2020/21. Schedule of talks:
    • 29.10.2020: Introduction part 1
    • 05.11.2020: Introduction part 2
    • 12.11.2020: From Hecke to HOMFLYPT
    • 19.11.2020: Soergel bimodules
    • 26.11.2020: Diagrammatic calculus for Soergel bimodules
    • 03.12.2020: Rouquier complexes
    • 10.12.2020: Triply-graded link homology
    • 17.12.2020: Diagrammatric representation theory of quantum gl(N)
    • 07.01.2021: Foams and gl(N) link homology
    • 14.01.2021: Deformation theory for gl(N) link homologies
    • 21.01.2021: Functoriality under link cobordisms
    • 28.01.2021: Categorified Jones-Wenzl projectors
    • 04.02.2021: From triply-graded to gl(N) link homology
    • 11.02.2021: The Rasmussen invariant
  • Mini-course on skein theory, workshop at Quy Nhon University, Vietnam, 2th - 4th June 2019.
  • Mini-course on "Conjectures in Quantum Topology", workshop on "Classical and quantum three-manifold topology" at Monash University, Melbourne, 14th - 21st December 2018.
  • MATH1013, first year linear algebra, The Australian National University, Semester 2, 2018.
  • INI winter school tutorial, Isaac Newton Institute, Cambridge, January 2017.
  • London Taught Course Center Intensive Course Introduction to Khovanov homology, November 2016.
  • ESI Simons Lecture Series Knot homologies and higher representation theory, October 2016.
  • Series of talks on Khovanov homology, TAKTIC Seminar, Imperial College London, November - December 2015.
  • Supervisor for undergraduate Geometry (Part IB) and Algebraic Topology (Part II), University of Cambridge, 2012-2015.
  • Repetitorium Analysis, Universität Wien, 2010-2011.
  • Seminars on measure theory and analysis, Universität Wien, 2010-2011.

Research student supervision

  • Jack Brand, Honours thesis (co-supervisor), The Australian National University, 2018.
  • Hazel Browne, undergraduate summer research (co-supervisor), The Australian National University, 2018.
  • Sam Osborne, MSc project, Imperial College London, 2017.

Extracurricular activities:

  • Sailing: I am a skipper in the Cambridge University Yacht Club and I was a main sail trimmer in the Cambridge University Yacht Racing Team, for which I got a Cambridge Half Blue.
  • Running: In October 2017 I was part of a team from the Mathematical Sciences Institute which won Division 5 of Inward Bound, an overnight adventure footrace over roughly 50km in mountainous terrain.
  • Other stuff: I am interested in nuclear nonproliferation and disarmament. Here is a great podcast about these topics.
MSI Team for Inward Bound
Punt sailing in Cambridge
Updated: 15th April 2021 or later