General Profile

Mulase

Motohico Mulase

Professor
Complex algebraic geometry and analysis
Ph.D. and D.Sc., 1985, Kyoto University

Web Page: http://www.math.ucdavis.edu/~mulase/
Email: momulase@ucdavis.edu
Office: MSB 3103
Additional Information: Distinguished Professor of Mathematics

Publications:
Preferred listing: https://www.math.ucdavis.edu/~mulase/publication.html
Math arXiv
Google Scholar

Research, Teaching, and Administration

Motohico Mulase is a creative mathematician whose research has influenced numerous different areas of mathematics, such as algebraic geometry of moduli spaces of Riemann surfaces and Higgs bundles, nonlinear integrable systems such as KP and KdV equations, Gromov-Witten theory, and topological recursion. At the same time, through organizing many conferences and workshops in emerging frontiers of mathematics, such as the 2016 AMS von Neumann Symposium, Mulase has also helped forming communities of young mathematicians on new research subjects.

Mulase's work on the unique solvability of infinite-dimensional integrable systems, such as the KP equations, in terms of Birkhoff decomposition has become a fundamental theorem. Its application includes his construction of the Riemann surface from its period matrix, by identifying a commutative ring of ordinary differential operators from the Riemann period matrix through the Lax equation formalism. The projective scheme of this ring is the desired algebraic curve. Here, analysis of Lax equations is integrated into scheme theory of Grothendieck to show the power of cross fertilization beyond the boundary. This idea also led him to a solution of the Schottky problem.

His work on ribbon graphs (Grothendieck's dessins d'enfants), Strebel differentials, and Belyi morphisms with his former student Penkava is also well cited. This work relates arithmetic geometry (curves defined over the field of algebraic numbers), complex analysis, combinatorics, and the orbifold topology of the moduli spaces of pointed algebraic curves.

The way Mulase asks his question is unique. Witten-Kontsevich theorem says that the generating function of cotangent class intersection numbers on the moduli space of stable curves satisfies the KdV equations. Since KdV is a time-evolution equation, he asked: Where do the cotangent class intersection numbers evolve through the KdV equations? The surprising answer he discovered with his student Safnuk is: They evolve into Mirzakhani's volume function of the moduli space of bordered hyperbolic surfaces.

When topological recursion of Eynard-Orantin has caught attention of pure mathematicians, an important open problem was the conjecture of physicists Bouchard and Mariño on Hurwitz numbers. After hundreds of pages of hand calculations and computer experiments, Mulase discovered that the conjectural formula is exactly the Laplace transform of the cut-and-join equation of Ravi Vakil and combinatorialists Goulden and Jackson. He then noticed that this particular Laplace transform also gives a simple and short proof of the Witten-Kontsevich theorem and the λg-formula of Faber and Pandharipande.

More recently, Mulase with his collaborators Dumitrescu, Fredrickson, Kydonakis, Mazzeo and Neitzke, solved a conjecture of Gaiotto on constructing opers via conformal limit of the non-abelian Hodge correspondence. This procedure connects the concept of quantum curves, opers, and moduli spaces of Higgs bundles, and has triggered further study by Collier, Wentworth, and others on the stratification of Hitchin moduli spaces.

As an instructor, Mulase is a Distinguished Teaching Award winner at UC Davis. He has served over a decade on the administration in various capacities and chair of Academic Senate Committees, including Department Chair (1998--2001, 2004--2007), Chair of the Faculty Personnel Committee 3 in College of L&S (2010--2013), and Associate Dean of the Faculty for Mathematical and Physical Sciences (2017--2019).

Selected publications

Book Edited

  • Chiu-Chu Melissa Liu and Motohico Mulase, Editors, "Topological Recursion and its Influence in Analysis, Geometry, and Topology,"
     Proceedings of Symposia in Pure Mathematics 100, 549 pps, American Mathematical Society, (2018). [ISBN: 978-1-4704-3541-7]

Selected Journal Papers

  1. Motohico Mulase, Complete integrability of the Kadomtsev-Petviashvili equation, Advances in Mathematics 54, 57--66 (1984).  
  2. Motohico Mulase, Cohomological structure in soliton equations and Jacobian varieties, Journal of Differential Geometry 19, 403–430 (1984).
  3. Motohico Mulase, Solvability of the super KP equation and a generalization of the Birkhoff decomposition, Inventiones Mathematicae 92, 1--46 (1988).
  4. Motohico Mulase, Category of vector bundles on algebraic curves and infinite dimensional Grassmannians, International Journal of Mathematics 1, 293--342 (1990).
  5. Motohico Mulase, A new super KP system and a characterization of the Jacobians of arbitrary algebraic super curves, Journal of Differential Geometry 34, 651--680 (1991).
  6. Motohico Mulase, Algebraic theory of the KP equations, Perspectives in mathematical physics 3, 151--217 (1994).
  7. Motohico Mulase and Michael Penkava, Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over $\overline{Q}$, Asian Journal of Mathematics 2 (4), 875--920 (1998).
  8. Motohico Mulase and Andrew Waldron, Duality of orthogonal and symplectic matrix integrals and quaternionic Feynman graphs, Communications in Mathematical Physics 240 (3), 553--586 (2003).
  9. Motohico Mulase and Josephine T. Yu, Non-commutative matrix integrals and representation varieties of surface groups in a finite group, Annales de l'Institut Fourier 55, 1001--1036 (2005).
  10. Motohico Mulase and Brad Safnuk, Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy, Indian Journal of Mathematics 50 (1), 189--228 (2008).  
  11. Motohico Mulase and Naizhen Zhang, Polynomial recursion formula for linear Hodge integrals, Communications in Number Theory and Physics 4, 267--294 (2010).  
  12. Bertrand Eynard, Motohico Mulase and Brad Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers, Publications of the Research Institute for Mathematical Sciences 47, 629--670 (2011).  
  13. Kevin Chapman, Motohico Mulase and Brad Safnuk, The Kontsevich constants for the volume of the moduli of curves and topological recursion, Communications in Number Theory and Physics 5, 643--698 (2011).
  14. Motohico Mulase and Michael Penkava, Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves, Advances in Mathematics 230, 1322--1339 (2012).
  15. Olivia Dumitrescu, Motohico Mulase, Brad Safnuk, and Adam Sorkin, The spectral curve of the Eynard-Orantin recursion via the Laplace transform, Contemporary Mathematics 593, 263--315 (2013).
  16. Motohico Mulase, Sergey Shadrin and Loek Spitz, The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures, Communications in Number Theory and Physics 7 (1), 125--143 (2013).
  17. Vincent Bouchard, Daniel Hernadez Serrano, Xiaojun Liu, and Motohico Mulase, Mirror symmetry for orbifold Hurwitz numbers, Journal of Differential Geometry, 98, 375--423 (2014).
  18. Olivia Dumitrescu and Motohico Mulase, Quantum curves for Hitchin fibrations and the Eynard-Orantin theory, Letters in Mathematical Physics 104, 635--671 (2014).
  19. Motohico Mulase and Piotr Sulkowski, Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion, Advances in Theoretical and Mathematical Physics 19, 955--1015 (2015).
  20. Petr Dunin-Barkowski, Motohico Mulase, Paul Norbury, Alexandr Popolitov, and Sergey Shadrin, Quantum spectral curve for the Gromov-Witten theory of the complex projective line, Journal für die reine und angewandte Mathematik 2017-726, 267--289 (2017).
  21. Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo, Motohico Mulase, and Andrew Neitzke, From the Hitchin section to opers through nonabelian Hodge, Journal of Differential Geometry 117 (2), 223--253 (2021).

Honors and Awards

Last updated: 2025-09-20