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: A complex of singular Bott—Samelson bimodules with many symmetries
by Josh Wang
We review singular Bott—Samelson bimodules through the lens of equivariant cohomology of generalized Bott—Samelson varieties. The homotopy category of these bimodules is relevant to low-dimensional topology. For example, Rouquier and Chuang—Rouquier defined faithful embeddings of the braid group into this homotopy category. The space of maps between the objects assigned to two braids is an invariant of the link obtained by pairing the braids together via braid closure.
We report the discovery of a highly structured complex of singular Bott—Samelson bimodules. We present many symmetries and conjectural properties of the complex, including its relationship with the complexes assigned to braids. We explain their expected relevance to computing the colored triply-graded homology of torus knots, with an eye towards the refined exponential growth conjecture of Gorsky—Gukov—Stosic.
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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