I am a Professor and NSERC, NFRF, and CFI-supported researcher in the Department of Mathematics and Statistics at the University of Saskatchewan, which is located on Treaty 6 Territory and the Homeland of the Métis. I am a faculty member of both the College of Arts and Science and the College of Graduate and Postdoctoral Studies. I sereve as the Lead for the University's Signature Area of Research in Quantum Innovation and as the Director of the Centre for Quantum Topology and Its Applications (quanTA).  I am also the University's Director of Interdisciplinary Programming, a position housed in the College of Graduate and Postdoctoral Studies.

I am a pure mathematician influenced in large part by physics. My primary areas of research are:

• algebraic geometry, topology, and representation theory of moduli spaces
• geometric and topological techniques in mathematical physics and quantum theory

Within the Quantum Innovation Signature Area, I also serve as the Co-Lead for the Quantum Computing theme.

You can read an overview of my research or you can view my papers directly. If you want to learn about some of my research and also see cartoons of hedgehogs at the same time, please have a look at Higgs Bundles without Geometry, a pamphlet that I wrote with Laura Schaposnik for the Oberwolfach Snapshots of Modern Mathematics series, which is "designed to promote the understanding and appreciation of modern mathematics and mathematical research in the interested public world-wide." You can also check out my TED^x talk about the mathematics and art behind quantum magic.

I was invited by the Canadian Science Policy Centre to write about grand challenges around quantum innovation in the current decade. You can read my article on page 39 of the 2020 edition of the Canadian Science Policy Magazine.

I am also an editor of the journal Research in the Mathematical Sciences, an affiliate faculty member of the Perimeter Institute for Theoretical Physics, and a member of the NSF GEAR Network and Geometry Labs United.

Papers

• October 2022: A new preprint, TQFTs and quantum computing, written with student M. Azam, is online.
• September 2022: My article Hyperbolic band theory through Higgs bundles, written with my student E. Kienzle, now appears in Advances in Mathematics.
• August 2022: A new preprint, On the hyperbolic Bloch transform, written with Á. Nagy, is online.
• July 2022: My article The Kapustin-Witten equations and nonabelian Hodge theory, written with C.-C. Liu and Y. Tanaka, now appears in European Journal of Mathematics.
• March 2022: My article Crystallography of hyperbolic lattices, written with I. Boettcher, A. Gorshkov, A. Kollár, J. Maciejko, and R. Thomale, now appears in Physical Review B.  The article was selected as an Editor's Pick!
• February 2022: My article Automorphic Bloch theorems for hyperbolic lattices, written with J. Maciejko, now appears in Proceedings of the National Academy of Sciences.
• December 2021: A new preprint, All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces, written with G. Bellamy, A. Craw, T. Schedler, and H. Weiss, is online.
• September 2021: My article Hyperbolic band theory, written with J. Maciejko, now appears in Science Advances. My article Explicit Poincaré duality in the cohomology ring of the SU(2) character variety of a surface, written with L. Jeffrey and my student A. Lindberg, now appears in Expositiones Mathematicae.
• June 2021: My article Moduli spaces of generalized hyperpolygons, written with L. Schaposnik, now appears in a special double issue of Quarterly Journal of Mathematics dedicated to Sir Michael Atiyah.
• April 2021: My article Toric co-Higgs bundles on toric varieties, written with I. Biswas, A. Dey, and M. Poddar, now appears in Illinois Journal of Mathematics.
• March 2021: My article Twisted cyclic quiver varieties on curves, written with my student E. Sundbo, now appears in European Journal of Mathematics.
• January 2021: My article What is...a hyperpolygon?, written with L. Schaposnik, now appears in Notices of the American Mathematical Society.
• December 2020: My article Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces, written with I. Biswas, now appears in International Journal of Mathematics.
• August 2020: My recent paper Nonabelian Hodge theory and vector-valued modular forms, written with C. Franc, now appears in AMS Contemporary Mathematics (special issue on Vertex Operator Algebras, Number Theory, and Related Topics in honour of Geoffrey Mason).
• May 2020: My recent paper Principal co-Higgs bundles on P¹, written with I. Biswas, O. García-Prada, and J. Hurtubise, now appears in Proceedings of the Edinburgh Mathematical Society.
• March 2020: My recent paper The Calogero-Françoise integrable system: algebraic geometry, Higgs fields, and the inverse problem, written with T. Stanley and J. Szmigielski, now appears in Integrable Systems and Algebraic Geometry, a refereed LMS volume in honour of Emma Previato.
• January 2020: My recent paper Ginzburg-Landau equations on Riemann surfaces of higher genus, written with D. Chouchkov, N. Ercolani, and I. M. Sigal, now appears in Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire.

Talks

• April: I made a triple visit to the Pure Mathematics Department at the University of Waterloo, the Institute for Quantum Computing, and the Perimeter Institute.  While in Waterloo, I gave invited talks in the Geometry & Topology Seminar at the University and in the Mathematical Physics Seminar at Perimeter, on crepant resolutions of finite quotient singularities and on hyperbolic quantum matter, respectively.
• February: I gave an invited talk in the Algebraic Geometry Seminar at the University of Toronto about algebro-geometric approaches to condensed and quantum matter, in particular in regards to moduli of vector bundles and Higgs bundles.  I also gave an invited talk in the Algebraic Geometry Seminar at UC Davis on resolutions of singularities for Nakajima quiver varieties, preceded by an informal preparatory talk for graduate students.
• January 2023: I was invited to speak in the Special Session on Resolutions of Singularities and Cohomology in Geometry and Representation Theory at the Joint Mathematics Meetings in Boston.  In my talk, I discussed the McKay correspondence, Nakajima quiver varieties, and gravitational instantons from the point of view of crepant resolutions of complex 2-dimensional finite quotient singularities, and discussed a remarkable and unexpected extension of this circle of ideas in the case of a particular order-32 finite group acting on C⁴ as discovered recently in joint work with G. Bellamy, A. Craw, T. Schedler, and H. Weiss.  I also delivered an invited TED^x talk for a general audience at the University of Saskatchewan titled Quantum Magic: Using Mathematics and Art to Make the Impossible Possible.  You can watch it here!
• October 2022: I gave an invited presentation at the Quantum Alberta Summit in Calgary, where I gave a broad overview of quantum activities and innovation in the province of Saskatchewan.
• June 2022: I gave an invited talk at the Conference on the Quantum Hall Effect and Topological Phases at the Institut de recherche mathématique avancée (IRMA) in Strasbourg.  My talk focused on my recent work concerning the algebraic geometry of two-dimensional quantum materials modelled on hyperbolic lattices.  I focused especially on hyperbolic band theory, nonabelian Bloch states, and the intersection of these materials with moduli spaces in algebraic and symplectic geometry.
• March 2022: I delivered the Mathematical and Statistical Sciences Colloquium at the University of Alberta.  My talk introduced Higgs bundles from a pure-mathematical point of view, focusing on the algebra, topology, and combinatorics of their moduli spaces, utilizing matrix-valued polynomials as a toy model that anticipates most of the properties of the moduli space. I aimed to give a sense of where the subject is today and where it is going.
• January 2022: I was invited to speak in the Special Session on Intersections of Geometric Analysis and Mathematical Physics at the Joint Mathematics Meetings in Seattle.  The talk would have reported on new results on quantum matter that make use of moduli spaces of stable bundles on curves. However, due to the outbreak of the omicron variant of Covid-19, the JMM, and subsequently the special session, has been postponed.
• November - December 2021: I gave an invited talk in the Minisymposium on Applications of Algebra in Mathematical Physics and Integrable Systems at TXLA21: 4th Annual Meeting of the SIAM Texas-Louisiana Section.  I spoke about Gelfand-Tsetlin-type integrable systems on Nakajima quiver varieties. I also delivered two talks, one on problems in quantum matter that may be addressed by neutron scattering and the other more broadly on new models of quantum matter inspired by hyperbolic geometry, locally at the University of Saskatchewan.  These were delivered in the Fedoruk Centre Neutron Coffee Break and the Department of Biology's Mathematical Modelling Seminar, respectively.
• July - August 2021: In July, I gave an invited talk in the Special Session on Moduli Spaces in Algebraic Geometry and Applications at the Mathematical Congress of the Americas in Buenos Aires.  The talk, which was delivered virtually, concerned symplectic duality for hyperpolygon spaces and their connection with Higgs bundle modui spaces.  In August, I gave a talk at the International Symposium on Mathematical Dialogues held at the Chern Institute of Mathematics in Nankai, held in honour of the 110th anniversary of the birth of S.-S. Chern and the 90th birthday of Sir R. Penrose.  My talk concerned new constructions of quantum matter from algebraic geometry and number theory.
• May - June 2021: In May, I gave the WiSe Seminar in the School of Mathematics and Physics at the University of Queensland. I was invited to speak on my recent AMS Notices article (written with L. Schaposnik) concerning hyperpolygons. In my 2-hour virtual talk, I gave an overview of GIT and symplectic reduction (including the Kempf-Ness theorem), hyperkähler reduction, and Nakajima quiver varieties, before specializing to (generalized) hyperpolygon spaces and recent results concerning their integrability. Also in May, I gave a talk in the Differential Geometry Seminar at UC Santa Barbara on the differential topology of ordinary, twisted, and wild Hitchin systems, reporting on joint work in recent years with several students and other collaborators. In June, I gave a talk in the Scientific Session on Symplectic Geometry at the Canadian Mathematical Society's (75+1)th Anniversary Meeting. I reviewed results from recent work of mine on complete integrability for (generalized) hyperpolygon spaces and presented a conjecture regarding the Couloumb branch and symplectic duality for these spaces.
• March - April 2021: I gave the Geometry Labs United Seminar. My talk, which was broadcast online, featured an overview of the work of the quanTA Centre, the hyperbolic band theory proposal put forward by J. Maciejko and myself, and the potential for students to contribute to this emerging field. I also gave the Colloquium of the Quantum Science and Engineering Research Center (QSEC) at George Mason University, going into further detail on the same topic.
• January 2021: I gave the Mathematics & Statistics Colloquium at Utah State University. My talk, which was held online, was a survey of Hitchin systems together with an announcement of my recent results in hyperbolic band theory, emphasizing connections between the two.
• November - December 2020: I gave an invited online talk in the Caltech High Energy Theory Seminar. My talk concerned relations between quiver varieties and Hitchin systems, the construction of Higgs bundles from hyperpolygons, and the definition of BPS hyperpolygons as a step towards computing explicit Seiberg-Witten curves for higher-genus N=1 Coulomb branch dynamics. I gave a similar invited online talk in the Clusters and Geometry Seminar at Yale University. This talk focused more on the construction of the quiver variety, the Hitchin systems correspondence, and the integrability theorem that I proved with L. Schaposnik.
• September 2020: I gave an invited talk in the Séminaire physique mathématique at the Centre de recherches mathématiques. My talk, which was held online, was a survey of Hitchin systems together with an announcement of my recent results in hyperbolic band theory, emphasizing connections between the two.
• May - July 2020: I was an invited speaker at the Workshop on Higgs Bundles and Related Topics and at the Conference on Lie Theory and Integrable Systems in Symplectic and Poisson Geometry, both of which were held online. At both venues, I spoke on the relationship between generalized hyperpolygons and Higgs bundles on punctured curves and the existence of a Gelfand-Tsetlin-type integrable system on the moduli space of generalized hyperpolygons for certain flag types. I also gave a talk on the algebraic geometry of quantum matter at the SIDE Math Conference and a "state of the lab" address at the Geometry Labs United Conference.