This is the website for the weekly Geometry and Topology seminar at MIT. The seminar meets on Mondays at 3:00-4:00PM, in room 2-449. If you have trouble getting through a certain locked door, contact Jonathan Zung (jzung@mit.edu) and he might help you out.
Spring 2023
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Feb 13: Stabilizations, Satellites, and Exotic Surfaces
by Gary Guth
A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. One naturally asks, "how many stabilizations are necessary, and is one always enough?" By considering certain satellite operations, we provide an answer to this question in the case of exotic surfaces with boundary. (This draws on work-in-progress with Hayden-Kang-Park.)
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Feb 27: Homological instability for moduli spaces of 4-manifolds
by Hokuto Konno
We prove that homological stability with respect to connected sums of S^2×S^2 fails for moduli spaces BDiff(X) of simply-connected closed 4-manifolds X. This makes a striking contrast with other dimensions: in all even dimensions except for 4, analogous homological stability for moduli spaces has been established by work of Harer and of Galatius and Randal-Williams. The proof of the above result is based on a characteristic class constructed using the Seiberg-Witten equations. This is joint work with Jianfeng Lin.
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Mar 6: Up and Down the Bow Construction
by Sergei Cherkis
Quivers play prominent role in geometric representation theory and in quantum gauge theory. Bows give a first step in a series of generalizations of quivers. The study of Yang-Mills instantons on hyperkaehler spaces provides a reliable guide in finding these generalizations.
The construction of Kronheimer and Nakajima of instantons on Asymptotically Locally Euclidean spaces is formulated naturally in terms of quivers. We present a construction of instantons on Asymptotically Locally Flat spaces in terms of bows, identify the instanton topological class in terms of the bow representation, and prove that our construction is complete. These results are obtained in collaboration with Andres Larrain-Hubach and Mark Stern.
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Mar 13: Lattice cohomology and q-series invariants of 3-manifolds
by Ross Akhmechet
I will discuss joint work with Peter Johnson and Slava Krushkal which introduces an invariant of negative definite plumbed 3-manifolds that unifies and extends two theories with quite different origins. The first is lattice cohomology, due to Némethi, whose degree zero part is described by a certain graph and is isomorphic to Heegaard Floer homology for a large subclass of such 3-manifolds. The second theory is the Z-hat q-series of Gukov-Pei-Putrov-Vafa, a power series which conjecturally recovers SU(2) quantum invariants at roots of unity. I will explain lattice cohomology, Z-hat, and our unification of these theories. I will also point out some key features of the new invariant and mention work in progress with Peter Johnson and Sunghyuk Park on extending the construction to knot complements.
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Mar 20: Homology cobordism & knot concordance
by Sally Collins
We can associate to a knot K in the 3-sphere a 3-manifold, called the zero-surgery of K, via performing zero-framed Dehn surgery on the knot. It is a natural question to ask: if two knots have zero-surgeries which are homology cobordant via a cobordism that preserves the homology class of the two positively oriented knot meridians, does that imply that the knots must be smoothly concordant? We review the history and motivation of this question and questions like it and describe our own result, ending with a brief discussion of a proof technique of obstructing torsion in the smooth concordance group.
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Apr 3: Concordance of surfaces
by Mark Powell
I will prove that topologically concordant surfaces are smoothly concordant, and compare with the situation in other dimensions.
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Apr 10: Classifying plane curves and symplectic 4-manifolds using braid groups: The symplectic isotopy conjecture in \(CP^2\)
by Amitesh Datta
The question of which symplectic 4-manifolds are complex projective surfaces reduces in principle (via branched covering constructions) to the question of which symplectic curves in the complex projective plane \(CP^2\) are isotopic to algebraic curves - the latter is known as the symplectic isotopy problem.
The longstanding symplectic isotopy conjecture posits that every smooth symplectic curve in \(CP^2\) is isotopic to an algebraic curve. In this talk, I will describe a new algebraic theory I have developed on the braid groups in order to prove that all degree three symplectic curves in \(CP^2\) with only \(A_n\)-singularities (an \(A_n\)-singularity is locally modelled by \(w^2 = z^n\) and includes nodes and cusps) are isotopic to algebraic curves. The proof is independent of Gromov's theory of pseudoholomorphic curves, and the theory also addresses the symplectic isotopy conjecture in full generality in upcoming work.
I will review the necessary background from scratch, and along the way, we will discuss beautiful ideas from algebraic geometry, symplectic geometry, monodromy theory and geometric group theory and how they unite in the study of plane curves and 4-manifolds.
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Apr 24: Asymptotic behavior of invariants of homology spheres
by Miriam Kuzbary
As shown by Morita, every integral homology 3-sphere Y has a decomposition into two simple pieces (called a Heegaard splitting) glued along a surface diffeomorphism which acts trivially on the homology of the surface. These diffeomorphisms form the Torelli subgroup of the mapping class group of the surface, and the Torelli group is finitely generated for surfaces of genus 3 or higher. Though the group of integral homology spheres is infinitely generated, by fixing the genus of a Heegaard splitting we can use finite generation in the surface setting to better understand how invariants like the Rokhlin and Casson invariant change. This perspective has led to important results in the study of the Torelli group and the Casson invariant in work of Birman-Craggs-Johnson, Morita, and Broaddus-Farb-Putman. In work in progress with Santana Afton and Tye Lidman, we show that the d-invariant from Heegaard Floer homology of an integral homology sphere is bounded above by a linear function of the word length of a corresponding gluing map in the Torelli group. Moreover, we show the d-invariant is bounded for homology spheres corresponding to various large families of mapping classes. If time permits, we will discuss the case of rational homology spheres.
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May 15: Arborealization and the construction of Floer homotopy types for Weinstein manifolds
by Abigail Ward
One of the aims of Floer homotopy theory is to produce refined Floer-theoretic invariants, called Floer homotopy types, of symplectic manifolds; for example, in 2007 Cohen showed that one can recover the suspension spectrum of the free loop space of a manifold M from the Hamiltonian Floer theory of \(T^*M\), strengthening Viterbo's result that \(H^*(LM)\) and \(HF^*(T^*M)\) are isomorphic. Weinstein manifolds form a class of open symplectic manifolds which generalize cotangent bundles, and it is a natural question to ask when Floer homotopy types exist for such manifolds. Arborealizable Weinstein manifolds are a subclass of Weinstein manifolds which retract onto Lagrangian skeleta with particularly simple singularities that can be described combinatorially. We explore the relationship between the arborealizability of a Weinstein manifold X and the topological conditions that allow one to construct a Floer homotopy type for X. This is joint work in progress with Daniel Álvarez-Gavela and Tim Large.
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May 22: Exotic contact structures on \(\mathbb{R}^n\)
by Joj Helfer
Contact homology is a Floer-type invariant for contact manifolds, and is a part of Symplectic Field Theory. One of its first applications was the existence of exotic contact structures on spheres.
Originally, contact homology was defined only for closed contact manifolds. We will describe how to extend it to open contact manifolds that are "convex". As an application, we prove the existence of (infinitely many) exotic contact structures on \(\mathbb{R}^{2n+1}\) for all \(n>1\).
This is joint work with François-Simon Fauteux-Chapleau.