MIT Geometry and Topology Seminar

The seminar meets on Mondays 4:30-5:30 in room 2-449.

Please contact Josh (joshuaxw@mit.edu) for seminar-related matters.
Organizers: Hokuto Konno, Tom Mrowka, Josh Wang, and Jonathan Zung.

Spring 2024

Abstracts

Feb 12

Speaker: Karola Mészáros (Cornell University)
Title: Log-concavity of the Alexander polynomial
Abstract: Almost a century after the introduction of the Alexander polynomial, it still presents us with tantalizing questions, such as Fox's conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial Delta_L(t) of an alternating link L are unimodal. Fox's conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus 2 alternating knots, among others. We settle Fox's conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of Delta_L(t), where L is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. This talk is based on joint work with Elena Hafner and Alexander Vidinas.

Feb 26

Speaker: Luke Conners (UNC Chapel Hill)
Title: Row-Column Mirror Symmetry for Colored Torus Knot Homology
Abstract: The colored HOMFLYPT polynomial is a 2-variable invariant of links colored by Young diagrams generalizing the celebrated Jones polynomial and other Type A quantum link polynomials. Constructions of this invariant from a physical perspective reveal a natural symmetry describing its behavior under exchanging each Young diagram coloring a fixed link with its transpose.

One categorical level up, Khovanov and Rozansky constructed a triply-graded homological link invariant that recovers the (trivially colored) HOMFLYPT polynomial upon taking Euler characteristic. Various authors have constructed colored analogues of this invariant and, in special cases, conjectured that their constructions satisfy a categorical lift of this decategorified symmetry. In this talk, we will precisely formulate a version of this conjecture applying to all colorings and outline a recent proof in the special case of a positive torus knot colored by a single row or column of arbitrary length.

Mar 4

Speaker: Abhishek Mallick (Rutgers University)
Title: Detecting corks
Abstract: Corks are fundamental to the study of exotic smooth structures on 4-manifolds. In this talk we will discuss how to use Heegaard Floer homology to detect corks. We will then describe some new examples of corks which partially address a question posed by Gompf. This is joint with Irving Dai and Ian Zemke.

Mar 11

Speaker: Juan Muñoz-Echániz (Simons Center for Geometry and Physics)
Title: Dehn twists on 4-manifolds with Seifert-fibered boundary
Abstract: If X is a smooth 4-manifold with an embedding of a 3-manifold Y with circle action, then one can perform a "Dehn twist" diffeomorphism of X along Y. Such diffeomorphisms naturally arise, for instance, as the monodromy of isolated singularities. By results of Orson--Powell, these Dehn twists are topologically trivial in several instances, but a systematic way of detecting whether they are smoothly trivial or not is not known.

I will discuss the following result: for any symplectic filling with b^+ > 0 of either of the Brieskorn spheres Sigma(2,3,7) or Sigma(2,3,11), the Dehn twist along the boundary is an exotic diffeomorphism with infinite order in the mapping class group. The proof relies on the Mrowka--Ozsváth--Yu description of the Seiberg--Witen moduli spaces on Seifert-fibered spaces, together with Francesco Lin's Morse--Bott model for the monopole Floer homology groups. This is based on joint work in progress with Hokuto Konno, Jianfeng Lin and Anubhav Mukherjee.

Mar 18

Speaker: Cameron Gordon (UT Austin)
Title: Cyclic branched covers of links and the L-space conjecture
Abstract: The L-space conjecture asserts that for a closed, connected, orientable, prime 3-manifold M, the following are equivalent: (1) M is not a Heegaard Floer L-space, (2) pi_1(M) is left-orderable, and (3) M supports a co-orientable taut foliation. We will discuss these properties for cyclic branched covers of prime knots and links. This is joint work with Steve Boyer and Ying Hu.

Apr 1

Speaker: Inanc Baykur (UMass Amherst)
Title: A dichotomy in dimension four
Abstract: Does every four-manifold admit either no smooth structure or infinitely many of them? I'll report on recent work, joint with A. Stipsicz and Z. Szabó, addressing this question for four-manifolds with finite cyclic fundamental groups. A bonus discussion may feature fake projective planes in the context of yet another dichotomy.

Apr 16

Room change: 4-231
Speaker: Ian Agol (UC Berkeley)
Title: Ribbon concordance of knots is a partial order
Abstract: We will discuss the result in the title, answering a question of Gordon. The proof makes use of representation varieties of the knot group to SO(N). If there is time, we will discuss the prospect of answering some other questions from Gordon's paper about ribbon concordance.

Apr 29

Speaker: John Baldwin (Boston College)
Title: Bordered 3-manifolds and surgery triads
Abstract: We prove that for any closed, oriented, connected surface S, the set of bordered 3-manifolds with boundary S is generated through surgery triads by a finite set of bordered handlebodies; furthermore, gluing together any two handlebodies in this set yields a connected sum of (S^1 x S^2)s. This result is a key topological ingredient for connected sum formulae in monopole Floer homology, and in Culler's proof of Blokhuis-Brouwer's conjecture on spanning sets for binary symplectic dual polar space. It is also a crucial ingredient in our proposed construction of bordered monopole Floer homology. I'll focus on the topological result but will survey some of the applications. This is joint work with Jon Bloom.

Fall 2023

Abstracts

Sep 18

Speaker: Qiuyu Ren (UC Berkeley)
Title: Lee filtration structure of torus links, adjunction inequality, and applications
Abstract: We state our recent result determining the quantum filtration structure of the Lee homology of torus links. This implies a relative adjunction-type inequality for s-invariants of two links related by a link cobordism in kCP^2, originally conjectured by Manolescu-Marengon-Sarkar-Willis. As two applications of this inequality, we show the existence of knots with arbitrarily large CP^2 genus, and we sketch a potential approach to produce and detect exotic kCP^2's.

Sep 25

Speaker: William Ballinger (Harvard University)
Title: The SO(8) invariant of trivalent graphs and its categorification
Abstract: The SO(2n) Reshetikhin-Turaev invariant of a link in the vector representation admits a conjectural categorification due to Khovanov and Rozansky - they prove that their construction is invariant under the first two Reidemeister moves, but not the third. I will present a proof that this is in fact a link invariant in the SO(8) case, based on extending the construction to give an invariant for knotted trivalent graphs, colored by the vector and two spinor representations. Essential simplifications are provided by the triality automorphisms, which allow the three colors to be treated symmetrically. I will also discuss some skein theory for the decategorified invariant.

Oct 2

Speaker: Thomas Barthelmé (Queen's University)
Title: Group actions on bifoliated planes and classification of (pseudo)-Anosov flows in dimension 3
Abstract: An old problem in dynamical systems is to try to classify Anosov flows up to orbit-equivalence. This question is particularly interesting in dimension 3 where we both have lots of examples and a rich, but still poorly understood, relationships between the dynamics of the flow and the topology of the manifold. By a result of T. Barbot, classifying Anosov flows (or more general pseudo-Anosov flows) in dimension 3 up to orbit equivalence restricts to classifying, up to conjugacy, certain actions of \pi_1(M) on the orbit space, a topological plane with two transverse foliations. In this talk, I will recall the above and discuss a new complete invariant for transitive (pseudo)-Anosov flows which often reduces to just knowing which conjugacy classes in \pi_1(M) are represented by periodic orbits of the flow. If time permits, I'll talk about some applications with link to contact geometry. This is all joint work with Kathryn Mann and Steven Frankel.

Oct 16

Speaker: Joe Boninger (Boston College)
Title: Twisted knots and the perturbed Alexander invariant
Abstract: The perturbed Alexander invariant, defined by Bar-Natan and van der Veen, is an infinite family of polynomial invariants of knots in the three-sphere. The first invariant in the family, rho_1, is quick to compute and often superior to Khovanov homology at distinguishing knots. We will discuss the perturbed Alexander invariant and properties of the rho_1 invariant in particular. We will then explore the behavior of the rho_1 invariant and the Alexander polynomial under the operation of applying full twists to a knot. Our arguments use a model of random walks on knot diagrams.

Oct 23

Speaker: Oliver Edtmair (UC Berkeley)
Title: The subleading asymptotics of symplectic Weyl laws
Abstract: Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics and C^0 symplectic geometry. For example, it plays a key role in the proof of the smooth closing lemma for three-dimensional Reeb flows and area preserving surface diffeomorphisms, and in the proof of the simplicity conjecture. In this talk I will report on work in progress concerning the subleading asymptotics of these Weyl laws and the connection to symplectic packing problems.

Oct 30

Speaker: Sunghyuk Park (Harvard University)
Title: Quantum UV-IR map and curve counts in skeins
Abstract: Quantum UV-IR map (a.k.a. q-nonabelianization map), introduced by Neitzke and Yan, is a map from UV line defects in a 4d N=2 theory of class S to those of the IR. Mathematically, it can be described as a map between skein modules and is a close cousin of quantum trace map of Bonahon and Wong. In this talk, I will discuss how quantum UV-IR map can be generalized to a map between HOMFLYPT skein modules, using skein-valued curve counts of Ekholm and Shende.

Nov 6

Speaker: Kyle Hayden (Rutgers University-Newark)
Title: An atomic approach to Wall-type stabilization problems
Abstract: In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. I will discuss Wall's stabilization problem and some of its variants, all of which aim to quantify this instability. In particular, I'll outline a simple "atomic" approach to these problems and, as a proof of concept, use it to produce exotic pairs of knotted surfaces (with boundary) in the 4-ball that remain exotic after one "internal stabilization". The key obstruction comes from the universal version of Khovanov homology. Time permitting, I will speculate on some potential connections to Floer homology.

Nov 13

Speaker: Rohil Prasad (UC Berkeley)
Title: Dense existence of proper compact invariant sets for torsion 3D contact forms
Abstract: This is joint work in progress with Dan Cristofaro-Gardiner. The study of periodic orbits of Reeb flows on 3-dimensional contact manifolds has seen major progress over the past two decades. This talk, however, will be about the topological dynamics of Reeb flows beyond periodic orbits. For any contact form for a torsion contact structure on a closed 3-manifold Y, without any genericity conditions, I'll show that any point in Y is arbitrarily close to some proper compact subset of Y which is invariant under the Reeb flow. Stronger results can also be established which parallel classical theorems of Le Calvez-Yoccoz, Franks, and Salazar for homeomorphisms of the two-sphere. The proof relies on new analytical results for holomorphic curves, various properties of embedded contact homology, and the following fun fact: a connected subset of a Hausdorff space with at least two points has no isolated points.

Nov 20

Speaker: Ka Ho Wong (Yale University)
Title: Asymptotics of the relative Reshetikhin-Turaev invariants
Abstract: In a series of joint works with Tian Yang, we made a volume conjecture and an asymptotic expansion conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it. We propose that their asymptotic behavior is related to the volume, the Chern-Simons invariant and the adjoint twisted Reidemeister torsion associated with the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring.

In this talk, I will first discuss how our volume conjecture can be understood as an interpolation between the Kashaev-Murakami-Murakami volume conjecture of the colored Jones polynomials and the Chen-Yang volume conjecture of the Reshetikhin-Turaev invariants. Then I will describe how the adjoint twisted Reidemeister torsion shows up in the asymptotic expansion of the invariants. Especially, we find new explicit formulas for the adjoint twisted Reidemeister torsion of the fundamental shadow link complements and of the 3-manifolds obtained by doing hyperbolic Dehn-filling on those link complements. Those formulas cover a very large class of hyperbolic 3-manifolds and appear naturally in the asymptotic expansion of quantum invariants. Finally, I will discuss some recent progress of the asymptotic expansion conjecture of the fundamental shadow link pairs.

Nov 27

Speaker: Kim Frøyshov (University of Oslo)
Title: Mod 2 instanton homology and the invariant q_2
Abstract: I will discuss how instanton homology with coefficients in Z/2 gives rise to a homomorphism q_2 from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton h-invariant and the Heegaard correction term.

If a homology 3-sphere Y bounds a smooth, compact, negative definite 4-manifold without 2-torsion in its homology then q_2(Y) is non-negative, and positive if the intersection form is non-standard.

Dec 4

Speaker: Sherry Gong (Texas A&M University)
Title: Ribbon concordances and slice obstructions: experiments and examples
Abstract: We will discuss some computations of ribbon concordances between knots and talk about the methods and the results. This is a joint work with Nathan Dunfield.

Dec 11

Speaker: Mike Miller Eismeier (University of Vermont)
Title: Chern-Simons, diffeomorphism invariants, and cosmetic surgery
Abstract: The Chern-Simons filtration on irreducible instanton homology has produced some excellent results about homology cobordism, but naive considerations also give rise to diffeomorphism invariants. These diffeomorphism invariants can be used to distinguish 3-manifolds, even when we can't compute them.

Everything discussed in this talk is joint with Tye Lidman.