This is the website for the weekly Geometry and Topology seminar at MIT. The seminar meets on Mondays at 3:30-4:30PM, in room 2-449. Please contact Jonathan Zung (jzung@mit.edu) if you'd like to be added to our seminar mailing list.
Fall 2025
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Sep 15: Boundary Currents of Hitchin Components
by Charlie Reid
A hyperbolic structure on a surface is described by a representation of the fundamental group into PSL(2,R). Higher rank Teichmüller theory aims to go beyond hyperbolic geometry by studying moduli spaces of representations into bigger Lie groups, most quintessentially SL(n,R). I will discuss a SL(n,R) version of a classic piece of hyperbolic geometry—Thurston's compactification of Teichmüller space. Boundary points of Thurston's compactification are measured laminations: certain analytic objects generalizing simple closed curves. One can define a compactification of the SL(n,R) Hitchin component in much the same way, whose boundary points are now geodesic currents which generalize closed curves with more intricate restrictions on self-intersection. Dual to these geodesic currents are n-1 dimensional polyhedral spaces which generalize R-trees.
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Oct 6: Dax invariants, light bulbs, and isotopies of symplectic structures
by Boyu Zhang
In this talk, I will present the following two main results. First, we give a classification of the isotopy classes of embeddings of \(\Sigma\) in \(\Sigma\times S^2\) that are geometrically dual to \(\{pt\}\times S^2\), where \(\Sigma\) is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic. This answers a question of Gabai. Second, we show that the space of symplectic forms on an irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components. This gives the first such example among closed 4-manifolds and answers a question of McDuff-Salamon. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. This is joint work with Jianfeng Lin, Weiwei Wu, and Yi Xie.
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Oct 20: Infinitely many Lefschetz pencils on ruled surfaces
by Seraphina Lee
Works of Donaldson and Gompf show that a closed, oriented 4-manifold admits a symplectic structure if and only if it admits the structure of a Lefschetz pencil. However, the question of how many Lefschetz pencils (or fibrations) a given symplectic 4-manifold admits remains open. Works of Park--Yun and Baykur construct 4-manifolds admitting arbitrarily large (but finite) numbers of Lefschetz pencils or fibrations of the same genus. In this talk, we will construct infinitely many non-isomorphic Lefschetz pencils of the same genus on ruled surfaces of negative Euler characteristic. In fact, our construction gives the first example of infinitely many non-isomorphic but diffeomorphic Lefschetz pencils and fibrations of the same genus. This is joint work in progress with Carlos A. Serván.
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Nov 3: TBA
by Josh Wang
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Nov 17: TBA
by Andreas Stavrou
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Nov 24: TBA
by Ali Sadr
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Dec 1: TBA
by Antonio Alfieri
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Dec 8: TBA
by Mike Miller Eismeier