Office 209, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, CANADA

rayan (at) math.usask.ca | Dept of Math & Stats | U of S

I am an Assistant Professor and NSERC-supported researcher in the **Department of Mathematics & Statistics** at the **University of Saskatchewan**. I am a faculty member of both the **College of Arts & Science** and the **College of Graduate & Postdoctoral Studies**. My primary areas of research are:

**algebraic geometry and topology of moduli spaces****geometric and topological techniques in mathematical physics**

While I believe that geometry and topology are, in and of themselves, vastly interesting and exquisitely beautiful, I am always keen to find new applications of geometry to physics and other areas. Currently, I am investigating applications of algebraic geometry and topology to new developments in condensed matter physics centered around quantum materials. As such, I belong to the **Centre for Quantum Topology and Its Applications (quanTA)**, of which I am Founding Director.

You can learn more about my research by clicking on the appropriate tab above, or you can go directly to my **papers**.

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 Schaposnkik (University of Illinois at Chicago) 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."

I am also a member of the **NSF GEAR Network** and **Geometry Labs United**.

I am currently co-editing a special issue of the journal **SIGMA** on **Integrability, Geometry, Moduli** around the work of Motohico Mulase. If you have an article relevant to these themes, please consider submitting it to us.

I am the **Chair of the Mathematical Physics BScH Program** at the University of Saskatchewan. This is an exciting interdisciplinary undergraduate program that combines the best of two worlds. The program provides a rigorous foundation in both mathematics and physics and upper-year students are rewarded with special topics courses ranging from algebraic geometry to mathematical foundations of quantum mechanics to string theory. Students in the program tend to be successful in attaining paid summer research positions and have an excellent track record of being admitted into graduate schools. If you are performing well in first-year and second-year mathematics and physics courses and are considering the possibility of applying for Mathematical Physics, please schedule an appointment with me.

I am the **Graduate Chair** in the Department of Mathematics & Statistics. If you have a strong record of undergraduate and/or postgraduate academic achievement in the mathematical sciences; are keen to pursue Master's or PhD-level level studies in pure mathematics, applied mathematics, discrete mathematics, or statistics; and are interested in the research of one or more of our faculty members, then I encourage you to **apply for admission** to one of our graduate programs.

**Papers**

- January 2020: A new preprint,
**Moduli spaces of generalized hyperpolygons**, written with L. Schaposnik, is online. - 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*. - December 2019: My recent paper,
**Principal co-Higgs bundles on**, written with I. Biswas, O. García-Prada, and J. Hurtubise, has been accepted by**P**^{1}*Proceedings of the Edinburgh Mathematical Society*. - October 2019: A new preprint,
**Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces**, written with I. Biswas, is online - August 2019: My recent paper,
**Twisted cyclic quiver varieties on curves**, written with my student E. Sundbo, has been accepted by*European Journal of Mathematics*. - June 2019: My recent paper,
**Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices**, written with P. Crooks, now appears in*Mathematical Research Letters*. - April 2019: My recent paper,
**The Calogero-Françoise integrable system: algebraic geometry, Higgs fields, and the inverse problem**, written with T. Stanley and J. Szmigielski, has been accepted for*Integrable Systems and Algebraic Geometry*, a refereed LMS volume in honour of Emma Previato. - March 2019: My recent paper,
**Nonabelian Hodge theory and vector-valued modular forms**, written with C. Franc, has been accepted for*AMS Contemporary Mathematics*(special issue on*Vertex Operator Algebras, Number Theory, and Related Topics*in honour of Geoffrey Mason). - December 2018: My article,
**Aspects of the topology and combinatorics of Higgs bundle moduli spaces**, now appears in SIGMA. - December 2018: A new preprint,
**Spectral curves for the triple reduced product of coadjoint orbits for SU(3)**, written with J. Hurtubise, L. Jeffrey, P. Selick, and J. Weitsman, now appears in*Geometry and Physics*, a refereed festschrift in honour of Nigel Hitchin. - June 2018: My recent paper,
**Geodesics on a Kerr-Newman-(anti-)de Sitter instanton**, written with my student A. Lindberg, now appears in*Journal of Geometry and Physics*. - May 2018: My recent paper,
**Twisted argyle quivers and Higgs bundles**, written with my student E. Sundbo, now appears in*Bulletin des Sciences Mathématiques*. - May 2017: A recent paper,
**A vanishing theorem for co-Higgs bundles on the moduli space of bundles**, written with I. Biswas, now appears in*Geometriae Dedicata*. - March 2017: My recent paper,
**The quiver at the bottom of the twisted nilpotent cone on**, now appears in**P**^{1}*European Journal of Mathematics*.

**Talks**

- February 2020: I gave the
**Colloquium**in the Department of Mathematical Sciences at George Mason University. The title of the lecture was "Hitchin Systems in Mathematics and Beyond", like my Ethel Raybould Colloquium at Queensland, although this talk emphasized a somewhat different side of Hitchin systems. - January 2020: I spoke in the
**Geometry, Algebra, and Physics Seminar**and the**Theoretical Physics Institute Seminar**at the University of Alberta. The topics were the relationship of Nakajima quiver varieties to Hitchin systems and an overview of the moduli space of Higgs bundles, respectively. - March - July 2019: During this time, I spoke twice at the Simons Centre for Geometry and Physics in Stony Brook during the
**Thematic Program on Geometry and Physics of Hitchin Systems**, then at the**Workshop on Geometry and Physics of Higgs Bundles**in Oberwolfach, twice at the**CMS Summer Meeting in Regina**, and finally at the Leibniz Universitä Hannover during the**Workshop on Geometric and Analytic Aspects of Moduli Spaces**. The topics of my talks ranged from hyperpolygon spaces to quivers in twisted categories to spectral data for Hitchin fibrations in low genus. - November 2018: I spoke in the
**Geometry & Analysis Seminar**in the University of Oxford's Mathematical Institute. The topic was the hyperkähler geometry of hyperpolygon spaces, including the problem of how to extend the McKay correspondence to certain non-ADE quivers. - June 2018: I gave the
**Ethel Raybould Colloquium**in the School of Mathematics & Physics at the University of Queensland. The talk was titled "Hitchin Systems in Mathematics and Beyond". - May 2018: I gave a talk in the
**Geometry Seminar**at Stanford University and two in the**Geometry and Topology Seminar**at Caltech. The talk at Stanford described a new construction of solutions to twisted**t=0**Kapustin-Witten equations on a compact Kähler surface. The second talk at Caltech was similar, but was preceded by an introductory talk on Higgs bundles and Hitchin systems. - April 2018: I gave a talk in the
**Geometry / Topology Seminar**at Duke University and a talk in the**Analysis and Geometry Seminar**at Central Michigan University. Both were on asymptotics of hyperpolygon spaces. I also gave the**Colloquium**at SNOLab on what I consider to be spectacular interactions between geometry and physics in the 20th and 21st centuries! - February 2018: I gave the
**Colloquium**in the Department of Mathematics at the University of Manitoba on the problem of finding finite-dimensional presentations of infiniite dimensional hyperkähler quotients, with a particular focus on hyperpolygons. - January 2018: I gave a talk in the
**Algebra & Geometry Seminar**at the University of British Columbia on moduli spaces of hyperpolygons. - October 2017: I gave a talk at the
**Singular Geometry and Higgs Bundles in Sring Theory**workshop at AIM in San Jose. The topic was the**C**-action on the global nilpotent cone.^{*} - August 2017: I gave a talk at the
**Analysis of Gauge-Theoretic Moduli Spaces**workshop at BIRS in Banff. The topic was the large-scale geometry of hyperpolygon spaces. - July 2017: I spent a week at the
**Mathematical Congress of the Americas**in Montréal, where I gave talks in two scientific sessions. - June 2017: Together with H. Weiss, I gave a
**Master Class**titled "Asymptotics of Higgs Bundles and Hyperpolygons" at the Centre for the Quantum Geometry of Moduli Spaces in Aarhus, Denmark. We gave six lectures each. - June 2017: I gave the
**Colloquium**in the Department of Mathematics at the Christian-Albrechts Universität in Kiel, Germany. The topics were Higgs bundles and the Hitchin system. - February 2017: I gave the
**Colloquium**at the Center for Mathematical Sciences and Applications at Harvard University. The topics were Higgs bundles and the Hitchin system.