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

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

I am an Associate Professor and NSERC-supported researcher in the **Department of Mathematics & 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 & Science** and the **College of Graduate & Postdoctoral Studies**. I am also the Director of the **Centre for Quantum Topology and Its Applications (quanTA)** at the University.

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**

You can read an **overview** of my research or you can see 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."

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

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 enjoy special topics courses ranging from algebraic geometry, to mathematical foundations of quantum mechanics, to string theory. Students in the program are very successful in attaining paid summer research positions and have an excellent track record of being admitted into top 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 also 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 our MMath, MSc, or PhD graduate programs.

**Papers**

- 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. - May 2021: A new preprint,
**Crystallography of hyperbolic lattices**, written with I. Boettcher, A. Gorshkov, A. Kollár, J. Maciejko, and R. Thomale, is online. - 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*. - December 2020: A new preprint,
**The Kapustin-Witten equations and nonabelian Hodge theory**, written with C.-C. Liu and Y. Tanaka, is online. - August 2020: A new preprint,
**Hyperbolic band theory**, written with J. Maciejko, is online. - 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). - June 2020: My recent paper
**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, has been accepted for*Expositiones Mathematicae*. - May 2020: My recent paper
**Principal co-Higgs bundles on P**, written with I. Biswas, O. García-Prada, and J. Hurtubise, now appears in^{1}*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**

- 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(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*AMS Notices*article**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 exclusively online (due to the the COVID-19 outbreak). 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**. A number of other talks that I was invited to deliver during this period, including talks in the**Caltech High Energy Theory Seminar**and the**Workshop on Current Trends on Spectral Data for Higgs Bundles VI**in Chicago), were cancelled due to the COVID-19 outbreak. I was also scheduled to give an invited plenary talk at the**Symposium on Physics, Geometry, and Number Theory**at the Chern Institute in Nankai. The symposium has been postoned to next year due to international travel restrictions associated with the pandemic. - April 2020: Two talks that I were to give this month (in Montrél for the
**CRM Mathematical Physics Seminar**and in Chicago for the**UIC Geometry, Topology, and Dynamics Seminar**) were understandably cancelled due to the COVID-19 outbreak. - 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.