Waseda Special Lectures

Special Lecturers (series of lectures)

Professor Franz Pedit (5 lectures, University of Massachusetts Amherst, USA) 

Theme: Harmonic maps, Higgs bundles, loop groups, and surface geometry 

Abstract: This series of lectures will begin with the geometric and gauge-theoretic framework for harmonic maps from compact Riemann surfaces into symmetric spaces. I will explain the fundamental distinction between compact and non-compact target symmetric spaces. The former leads to the theory of infinite-dimensional integrable systems and loop groups, whereas the latter lies within the realm of non-Abelian Hodge theory (see the lectures by Charles Ouyang). The general framework will be illustrated through a variety of examples drawn from classical surface geometry. I will then discuss how the genus of the domain Riemann surface influences the theory, focusing first on the cases of genus zero and one. The final lectures will address the higher-genus case, where loop Weierstrass representations, symmetries, character varieties, and implicit function techniques can be employed to construct new examples.

Professor Charles Ouyang (4 lectures, Washington University in St. Louis, USA) 

Theme: Surface group representations, Higgs bundles, and applications to surface geometry

Abstract: The non-Abelian Hodge correspondence is a highly transcendental correspondence linking surface group representations into a non-compact Lie group with Higgs bundles over a Riemann surface. This is achieved via harmonic maps into a non-compact symmetric space. The purpose of these four lectures is to explain this beautiful correspondence, starting with a simple case where the group is C*, which will be an abelian version of this correspondence, involving very classical objects such as periods of 1-forms, harmonic forms, and the Jacobian of a Riemann surface. We will then discuss the correspondence in generality for groups such as GL(n,C), and here, harmonic maps and surface geometry will feature prominently. For certain rank 2 groups, minimal surfaces in non-compact symmetric spaces will appear, and we will discuss the role they play here, in particular, applications to higher Teichmüller theory. (This should be compared to the loop group methods discussed in Professor Pedit's lectures, which allow for minimal surfaces into compact symmetric spaces.) Finally, we will discuss a concrete application of the theory for SL(3,R), where applications to SYZ geometry can be made, highlighting the scope and reach of the theory.

Invited Lecturers

Dr. Kento Sakai  (University of Tokyo, JSPS PD)
Title: Harmonic maps and degeneration of Riemann surfaces  

Abstract: In 1989, Wolf constructed a parametrization of Teichmüller space by the space of holomorphic quadratic differentials on a fixed Riemann surface using harmonic maps.  He also proved that the compactification of the Teichmüller space induced by this parametrization agrees with the Thurston compactification, whose boundary is the space of projective measured foliations.  As shown by Wolf's work and subsequent developments, several tools and concepts in classical Teichmüller theory can be understood from the perspective of harmonic maps.  In this talk, we focus on Wolf's compactification theorem and the degeneration of Riemann surfaces arising from his parametrization, and discuss several variations of his result.

Dr. Jun Sasaki  (Institute of Science Tokyo, JSPS DC1)
Title: The moduli space of Higgs pairs 

Abstract: Let E be a smooth complex vector bundle over a compact Riemann surface.  A Higgs pair (D'',s) consists of a structure of Higgs bundle D'' in E  and a smooth section s of E such that D''s=0 holds.  Higgs pairs can be considered as a generalization of holomorphic pairs introduced by Bradlow. Thus, one can define the notion of τ-stability for Higgs pairs,  depending on a real number τ. The notion of a Higgs pair was introduced by M. Mehta in 2003. He proved that a certain open subset of the moduli space of τ-stable Higgs pairs is non-singular.  In this talk, we prove that for a suitable choice of τ, the entire moduli space is non-singular. Moreover, under assumptions that rank E=2 and deg E is odd, we determine the Betti numbers of the moduli space.

Kento Sakai Kento Saka

Related Research Activity

Waseda Young Researchers Meeting on Geometry 2026

organized by Yuichiro Sato
July 8 and 22, 2026

Organizers

Yoshihiro Ohnita (chair, Waseda University & OCAMI)

Yuichiro Sato (Waseda University)

Masashi Yasumoto (Tokushima University)

Katsuhiro Moriya (University of Hyogo) 

Martin Guest (professor emeritus of Waseda University)

Support

- JSPS Grant-in-Aid for Scientific Research (C) No.22K03293 (Principal Investigator: Katsuhiro Moriya)

- JSPS Grant-in-Aid for Scientific Research (A) No.23H00083 (Principal Investigator: Martin Guest)

- JSPS Grant-in-Aid for Scientific Research (A) No.22H00094 (Principal Investigator: Masa-hiko Saito)

- JSPS Grant-in-Aid for Scientific Research (C) No.26K06792 (Principal Investigator: Yoshihiro Ohnita)

- Waseda Institute for Mathematical Science (WIMS) 

- Osaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan University  

Contact

Yoshihiro Ohnita (Waseda U. & OCAMI) ohnita @omu.ac.jp

Links

Waseda Special Lectures 2024
"Integrabilities in Differential Geometry, and their Applications"

Waseda Special Lectures 2025
"Exceptional Symmetric Spaces and Related Topics"

Waseda Special Lectures 2025 in Fall
"Geometric Analysis of Hypersurfaces - Rigidity, Uniqueness and Geometric Inequalities -"

Waseda Special Lectures 2026 in Spring
"Geometry of Quantum Cohomology and Related Topics"