This is the website for the weekly Geometry and Topology seminar at MIT. The seminar meets on Mondays at 4:30-5:30PM, in room 2-449. Please contact Jonathan Zung (jzung@mit.edu) if you'd like to be added to our seminar mailing list.
Spring 2025
-
Feb 10: Minimal submanifolds, higher expanders, and waists of
locally symmetric spaces
by Ben Lowe
Gromov initiated a program to prove statements of the
following form: Suppose we are given two simplicial complexes X
and Y, where X is “complicated”a and Y is lower dimensional. Then
any map f: X-> Y must have at least one ”complicated” fiber. In
this talk I will describe various results of this kind for compact
locally symmetric spaces, that are proved by bringing new tools
into the picture from minimal surface theory and representation
theory. Much of the talk will be focused on octonionic hyperbolic
manifolds, the case where our approach seems to work best. If time
permits I will also discuss some applications to systolic geometry,
global fixed point statements for actions of lattices on contractible
CAT(0) simplicial complexes, and/or non-abelian higher expansion
and branched cover stability. Based on joint work with Mikolaj
Fraczyk.
-
Feb 24: Zippers!
by Ino Loukidou
If M is a hyperbolic 3-manifold fibering over the circle, the fundamental group of M acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (eg taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles - i.e. a circle with a faithful pi_1(M) action preserving a pair of invariant laminations - and these play a key role in relating the dynamical structure to the geometry of M. In this talk we introduce the idea of *Zippers*, which give a new and direct way to construct universal circles, streamlining the known construction in some cases, and giving a host of new constructions in others. This is joint work with Danny Calegari.
-
Mar 10: TBA
by Junzhi Huang
-
Mar 17: TBA
by Konstantinos Varvarezos
-
Mar 31: TBA
by Chi Cheuk Tsang
-
Apr 7: TBA
by Autumn Kent
-
Apr 28: TBA
by Ben Knudsen
Fall 2024
-
Sep 9: Floer homology and square pegs
by Josh Greene
The Toeplitz square peg problem (1911) asks whether every Jordan curve in the plane contains the vertices of a square. I will describe a construction in symplectic geometry aimed at proving the existence of such inscribed squares and rectangles in a Jordan curve. The construction is a variation on Lagrangian Floer homology, and its associated spectral invariants give some information about the sizes of rectangles in a smooth Jordan curve. The main application I will describe is that in a rectifiable (a.k.a. finite length a.k.a. Lipschitz continuous) Jordan curve, one can find a large family of inscribed rectangles. Joint work with Andrew Lobb.
-
Oct 7: Unknotting nonorientable surfaces
by Anthony Conway
This talk will describe joint work with Mark Powell and Patrick Orson in which we prove that most closed, nonorientable surfaces in S^4 with knot group Z/2 are topologically unknotted.
-
Oct 21: Torus links and colored Heegaard Floer homology
by Beibei Liu
Link Floer homology is a filtered version of the Heegaard Floer homology defined for links in 3-manifolds. In this talk, we will introduce an algorithm to compute the link Floer homology of algebraic links from its Alexander polynomials. In particular, we give explicit descriptions of link Floer homology of torus link T(n, mn). As an application, we compute the limit of the link Floer homology when m goes to infinity, using certain cobordism maps, which can be used to define colored link Floer homology. This talk includes joint work with Borodzik, Zemke, and with Alishahi, Gorsky.
-
Oct 26: Covers of surfaces
by Ian Biringer
When does one (always orientable) surface cover another surface? For finite type surfaces, there’s an answer in terms of Euler characteristic and the number of cusps. But what if we look at infinite infinite type surfaces? We’ll first show that if S is a surface with nonabelian fundamental group, it is covered by every noncompact surface. The situation changes when we impose conditions on the type of covering. For instance, a cover is `characteristic’ if it corresponds to a subgroup of pi_1 S that is invariant under all automorphisms. We will show that infinite degree characteristic covers of (finite type, say) surfaces are always homeomorphic to one of four surfaces: the disc, the Loch Ness Monster, the flute surface, or the spotted Loch Ness monster. This is a joint project with Chandran, Cremaschi, Tao, Vlamis, Wang, Whitfield.
-
Nov 18: Maximal rotation and stable commutator length
by Lvzhou Chen
The rotation number rot(h) of an orientation-preserving homeomorphism h of the circle measures how fast h rotates, and it captures the dynamical behavior of h. For a free group acting on the circle and a word w in the commutator subgroup, rot(w) is a well-defined real number. The maximal rotation number R(w) is the supremum of rot(w) over all free group actions on the circle. When w is the standard relator of a surface group, this is well understood in Milnor-Wood inequalities. In general, there is an upper bound via Bavard's duality in terms of scl(w), the stable commutator length of w, which is a relative Gromov norm and roughly speaking measures the minimal complexity of surfaces bounding w. Is this inequality sharp? Is R(w) rational (scl(w) is rational in the free group by a theorem of Calegari)? We will discuss these unsolved problems in relation to questions and results about stable commutator length and linear programming. This involves joint works with Nicolaus Heuer and Geoffrey Baring.
-
Dec 2: Spinal open books and symplectic fillings with exotic fibers
by Agniva Roy
The question of understanding symplectic fillings of contact 3-manifolds has a rich history and deep connections to various aspects of low-dimensional topology, such as exotic 4-manifolds, mapping class groups of surfaces, complex geometry, algebraic singularities, among others. The technique of spinal open books, recently introduced by Lisi - Van Horn-Morris - Wendl, describes strong symplectic fillings of planar spinal manifolds in terms of foliations by pseudoholomorphic curves. These foliation descriptions, in the broadest generality, contain singular curves, which are Lefschetz-type singularities, and also a new phenomenon called exotic curves. In the Lefschetz-amenable setting, exotic curves disappear, and fillings can be classified as Lefschetz fibrations, depending on the number of singular curves. In joint work with Hyunki Min and Luya Wang, we give a topological description of the exotic curves in terms of identifying them with a local model, give a count of exotic fibers in any filling, and use these to classify symplectic fillings of certain planar spinal open books that are not Lefschetz-amenable.