"In the category of debts from many years ago, I should also like to thank the people at Harvard who fostered an environment of interest in the topics of representations of fundamental groups and harmonic maps. These include Y. Siu and his students, N. Boston and his fellow students of number theory, and many others. It is remarkable that, unbeknown to any of us, we were all working on the same things."
- Carlos Simpson, "Higgs Bundles and Local Systems"
The Triceratop Seminar is a learning seminar for graduate students interested in gauge theory, low-dimensional topology, and related areas. This semester it has been hijacked by number theorists :P
In the Spring 2026 semester, the topic is the Simpson-Corlette correspondence, and we meet weekly on Fridays from 10:30 am to 11:30 am (overflow time to 12 pm). We will be meeting in different rooms on different days; see the schedule below for which room we assigned.
Please email joyec@mit.edu or franklu@math.harvard.edu if you'd like to be added to the mailing list, if you'd like to give a talk, or if you have other questions!
Expand out each talk to see the abstract and speaker notes.
Notes typed by Frank (errors are likely due to Frank and not the speaker)
02/13
Title: The non-abelian Hodge theorem: a historical view
Speaker: Ollie Thakar
Room: Sever Hall 213
Abstract: Definition of Higgs bundles and flat bundles. The (smooth) Riemann-Hilbert correspondence. Definitions of stability for Higgs bundles, flat bundles, and representations of fundamental group. Rough statement of Simpson-Corlette correspondence (both the stable version and the general version), statement of Narasimhan-Seshadri theorem.
References:
Notes:
02/20
Title: Bundles from the Differential Geometric Perspective
Speaker: Daniel Hu
Room: Sever Hall 213
Abstract: Definitions of curvature, metrics, connections, and compatiblity between connections and metrics. Statement of Chern's theorem on connections on holomorphic vector bundles, Newlander-Nirenberg theorem. Definition of Chern classes as cohomology classes of differential forms (a la Chern-Weil theory), flat bundles have vanishing Chern classes. Statement of the Corlette and Uhlembeck-Yau results and relation to the Simpson correspondence. If time: moduli space of flat/Higgs bundles, extension of Hodge decomposition to cohomology of general bundles.
References: For the last two points, see Simpson, Higgs Bundles and Local Systems, pp. 19-26.
Notes:
02/27
Title: GIT Quotients
Speaker: Anne Larsen
Room: Emerson Hall 104
Abstract: Brief discussion of affine schemes as commutative rings. Definition of GIT quotient for affine schemes and projective schemes, examples. Notion of categorical/geometric quotient, GIT quotient as categorical/geometric quotient. If time: stacks as another notion of quotient in the scheme-theoretic world.
References: Thomas, A Gentle Introduction to the Non-Abelian Hodge Correspondence, pp. 23-25. Fogarty and Mumford, Geometric Invariant Theory, Sections 0.1, 1.2, 1.4. Alper, Stacks and Moduli, 3.5 (for Stacks), 8.1-8.2 (for GIT).
Notes:
03/06
Title: Stability and the Kempf-Ness theorem
Speaker: Elias Sink
Room: Science Center 304
Abstract: Definitions of stable/semistable/etc. in general for GIT. Relation to the notion of stability (via rank and degree) of vector bundles (details depending on time). The Kempf-Ness theorem.
References: Thomas, A Gentle Introduction to the Non-Abelian Hodge Correspondence, pp. 26-27. Fogarty and Mumford, Geometric Invariant Theory, Section 1.4. Alper, Stacks and Moduli, 9.5. Kempf and Ness, The Length of Vectors in Representation Spaces.
Notes:
03/13
Title: An Introduction to Harmonic Heat Flow
Speaker: Joye Chen
Room: Science Center 228
Abstract: TBA
References: Donaldson, "A new proof of a theorem of Narasimhan and Seshadri", Donaldson, "Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles", Jost, Harmonic Mappings between Riemannian manifolds, Ch. 3
Notes:
03/20
Title: Canonical Metrics on Flat Bundles I
Speaker: Frank Lu
Room: Science Center 228
Abstract: Definition of the moment map in the setting of Corlette. Recalling definition of stability in flat bundle side. Definition of harmonic metrics and relation to Simpson correspondence. Analogy with Kempf-Ness.
References: Corlette, Flat G-Bundles with Canonical Metrics, sections 2-3.
Notes:
04/03
Title: Canonical Metrics on Flat Bundles II
Speaker: Daniel Santiago-Alvarez
Room: Science Center 228
Abstract: Proof of Theorem 3.3 of Corlette, Sec. 4. Set-up of non-linear heat equation. Short-time existence, global existence (Weitzenbock formula for the D^+ Laplacian, Uhlenbeck compactness), global convergence.
References: Corlette, Flat G-Bundles with Canonical Metrics, section 4.
Notes:
04/10
Title: An Overview of Hermitian-Yang-Mills connections
Speaker: TBA
Room: Northwest Building B103
Abstract: TBA
References: Uhlenbeck-Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles.
Notes:
04/17
Title: The Simpson-Corlette Correspondence, at last
Speaker: TBA
Room: Science Center 228
Abstract: TBA
References:
Notes:
04/24
Title:
Speaker: TBA
Room: Science Center 228
Abstract: TBA
References:
Notes:
05/01
Title:
Speaker: TBA
Room: Science Center 228
Abstract: TBA
References:
Notes: