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. Wherever possible, I seek to reduce geometric and topological questions to combinatorial ones. For moduli spaces of Higgs bundles, this can be achieved by viewing Higgs bundles as 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.

I am also a member of the NSF GEAR Network.

 

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.



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

  • 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 for 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.



   

 

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