STEVEN RAYAN

Associate 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 | USask

 

In the broadest of strokes, I am attempting to pursue a deeper understanding of problems at the interface of complex algebraic geometry and mathematical physics.


Research interests broadly

  • complex algebraic geometry
  • symplectic geometry
  • representation theory
  • mathematical / theoretical physics
  • gauge theory
  • integrable systems
  • quantum geometry and topology

Research interests less broadly

  • moduli spaces of vector bundles, sheaves, and Higgs bundles on complex varieties
  • Nakajima quiver varieties (and quiver varieties in other categories)
  • nonabelian Hodge theory
  • hyperkähler geometry
  • mirror symmetry, especially with regards to Higgs bundles and integrable systems
  • topological recursion
  • applications of modern geometry and topology to quantum condensed matter physics
  • applications of geometry outside mathematics and physics

Research interests even less broadly


Moduli spaces

My work begins and ends with moduli spaces, coming either from pure mathematics or from the equations of theoretical physics. The moduli space that motivates much of my work is that of Higgs bundles on a complex variety, which arises as a gauge-theoretic quotient via the Hitchin equations (a reduction of the self-dual Yang-Mills equations) and also as a geometric-invariant theory quotient. The link between the two points of view is provided by a complex reductive analogue of the Hitchin-Kobayashi correspondence, which can be regarded as an infinite-dimensional version of the Kempf-Ness Theorem. The numerous geometric features of the Hitchin fibration, which is a fibration of the moduli space of Higgs bundles on a curve by compact (possibly degenerate) tori, are especially fascinating to me. While the moduli space of Higgs bundles is typically considered on an algebraic curve of genus 2 or larger, I have investigated twisted versions of the moduli space over the projective line and elliptic curves as well as instances of the moduli space over higher-dimensional varieties, including toric varieties of arbitrary dimension.

Topological questions

My earliest work on moduli spaces revolved around global topological questions. In particular, I have tackled the problem of computing topological invariants (e.g. Betti numbers) of certain moduli spaces using tools from representation theory, analysis, and combinatorics. One key result arising from this work is a proof that the genus zero limit of the ADHM formula of Chuang, Diaconescu, and Pan yields correct Betti numbers for moduli spaces of twisted Higgs bundles for ranks at least up to a certain bound. I have also characterized the geometry of the global minimum of L2-norm on the moduli space of twisted Higgs bundles on the projective line for any rank, degree, and twist. My work in this direction also applies to hyperpolygon space, which is a Nakajima quiver variety for the star-shaped quiver. In joint work with Jonathan Fisher, we produced a recursion relation that yields the rational Betti numbers of hyperpolygon space for any rank and any number of vertices. My work on the topology of both Higgs bundle and hyperpolygon moduli spaces has involved studying the geometry of spaces of quiver representations, both in the order linear sense and in the category of sheaves on a fixed variety. This has led to two papers with my student Evan Sundbo on quiver varieties in twisted categories. Woven through all of this work is a common thread of reducing geometry and topology to representation theory and, finally, to combinatorics. I am currently working with Laura Fredrickson to compute the Poincaré series of the moduli space of twisted wild Higgs bundles at genus 0. This moduli space, which consists of the Higgs bundles with a pole of arbitrary order at infinity, is closely related to N=2 supersymmetric field theory and the theory of vertex algebras.

Integrable systems

Given the origins of the moduli spaces mentioned above, it is no coincidence that they also admit the structures of completely integrable systems. This observation has led me to investigate a number of problems concerning the algebraic geometry of integrable systems including: the explicit construction of the Liouville fibration for the Guilleman-Sternberg integrable system on the triple reduced product of coadjoint orbits for SU(3) (accomplished across two articles with various collaborators); the construction of noncommutative integrable systems on hyperkähler varieties from Slodowy slices (with Peter Crooks); the application of (twisted) Higgs bundles and their spectral curves to understanding the Camassa-Holm and Calogero-Françoise systems (with Thomas Stanley and Jacek Szmigielski); and a proof that the complete and minimal hyperpolygon spaces of any rank and with any number of vertices admit the structure of Gelfand-Cetlin-type integrable systems (starting in work with Jonathan Fisher and then with Laura Schaposnik). My work in integrable systems has an additional motivation, which is that of mirror symmetry and symplectic duality. My work on the topology of the Hitchin fibration also engages with this theme. My PhD student Christopher Mahadeo and I are combining all of these directions — Higgs bundles, integrable systems, and mirror symmetry — in his thesis work on generalizations of topological recursion.

Quantum geometry and topology

My research has branched out naturally into applications of pure mathematical techniques to the search for hidden features of the equations of theoretical physics. This is already observed in my work on the Calogero-Françoise system, yielding an algebro-geometric means for understanding and resolving collisions in that system. In a sense, this brings my work full circle, from having used equations from physics to generate interesting moduli spaces to using tools originating in the study to moduli spaces to probe the equations themselves. My work with Chouchkov, Ercolani, and Sigal on the Ginzburg-Landau equations on higher-genus Riemann surfaces is another example of this idea. Here, we use ideas analogous to those arising in nonabelian Hodge theory (which relates the holomorphic geometry of Higgs bundles to the smooth geometry of flat connections) to study the nature of solutions to the equations, which generalize Abrikosov vortex lattices. As the Ginzburg-Landau equations lie at the heart of the macroscopic theory of superconductivity, it is natural to ask whether algebro-geometric techniques can be applied even more broadly to understanding phenomena in condensed matter. A prime phenomenon to explore is quantum condensed matter — including topological phases of matter — given the spectacular successes that have been achieved here through clever applications of algebraic topology. Is there a role in quantum matter for algebraic geometry? I am part of an interdisciplinary collaboration that is making significant progress on this question. A recent milestone of this collaboration is a paper that I wrote jointly with a colleague in condensed matter physics that establishes, for the first time, an electronic band theory for periodic hyperbolic geometries. This discovery paves the way for many wholly new examples of quantum materials, where the underlying structure resembles a hyperbolic crystal as opposed to a standard Euclidean crystal. Given the many potential real-world applications of quantum matter, this work has the potential to create new pathways between algebraic geometry and the development of next-generation technologies.




You can find my papers here or on the arXiv, INSPIRE-HEP, Google Scholar, and ORCiD.