STEVEN RAYAN

Assistant Professor, Department of Mathematics & Statistics, University of Saskatchewan
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
I am a pure mathematician influenced in large part by physics. By training I am a geometer and my work is concerned chiefly with moduli spaces and their fascinating features. Such features include special metrics, algebraic structures, and integrable systems. Moduli spaces arising from physics — such as the moduli space of Higgs bundles on an algebraic curve — as well as moduli spaces whose total spaces are Fano or Calabi-Yau are especially interesting to me. In the broadest of strokes, I seek to reduce questions of physics to geometric questions. Then, wherever possible I attempt to reduce geometric and topological questions to combinatorial ones. One example of this process that is foundational to my interests and work is the passage from physics in the form of the Yang-Mills equations to geometry in the form of moduli of Higgs bundles, and then finally to combinatorics in the form of representations of quivers.

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.



Undergraduate Studies


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.

 

Graduate Studies


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.

 


Recent Activity


Papers

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.