The SGGTC seminar meets on Fridays
in Math
520 from 10:30-11:30am and in **Math 407** from 1-2pm, unless noted otherwise (in red).

Previous semesters: Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.

Our e-mail list.

# Abstracts

#### February 1st, 2019: Florent Schaffhauser (Universidad de Los Andes and Université de Strasbourg) " Higher Teichmüller spaces for orbifolds "

*Abstract:*
The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group of X into PGL(2,R). It is a contractible space. For closed orientable surfaces, "Higher analogues" of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of $\pi_1(X)$ that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yield new examples of Higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.

#### February 8th, 2019: Siddhi Krishna (Boston College) " Taut Foliations, Positive 3-Braids, and the L-Space Conjecture "

*Abstract:*
The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll build taut foliations for manifolds obtained by surgery on positive 3-braid closures. As an example, we'll construct taut foliations in *every* non-L-space obtained by surgery along the P(-2,3,7) pretzel knot. No background in Heegaard-Floer or foliation theories will be assumed.

#### February 15th, 2019: Marco Castronovo (Rutgers) " Counting branes on the Gelfand-Cetlin Lagrangian "

*Abstract:*
Starting from Kontsevich's homological mirror symmetry conjecture, derived equivalences between symplectic and algebraic geometry have been intensely investigated, even beyond the Calabi-Yau setting. In the Fano case, Marsh-Rietsch proposed Landau-Ginzburg mirrors for the complex Grassmannians Gr(k,n) that fit in a general picture relating mirror symmetry and Langlands duality. We show that a Lagrangian torus fiber of the Gelfand-Cetlin integrable system on Gr(k,n) always supports nonzero objects of the Fukaya category, and for n prime these suffice to generate and establish an equivalence with the category of singularities of the mirror. The arithmetic restriction is curiously related to the values of Schur polynomials at roots of unity.

#### Februrary 22nd, 2019: Nate Bottman (IAS) " Family polyfolds and an $\mathcal{A}_\infty$ structure on the bar construction "

*Abstract:*
I will describe two areas of recent progress in the
construction and study of Symp, the symplectic
$(\mathcal{A}_\infty,2)$-category.
First, I will explain how associating operations to fiber products of
2-associahedra gives rise to a coherent algebraic structure, which in
particular endows the bar construction $TA[1]$ with an $\mathcal{A}_{\infty}$
structure. This is one way to circumvent the fact that strata of
2-associahedra involve fiber products of lower-dimensional
2-associahedra. I will also mention a related construction, which
exhibits $QC^*$ as a module over a decategorification of Symp.
Second, I will present work-in-progress with Katrin Wehrheim in which
we aim to construct a notion of "family polyfolds". This will be a
framework for setting up Fredholm problems that involve adiabatic
limits. Our motivating example is the strip-shrinking degeneration,
which is a key feature of the moduli spaces of pseudoholomorphic
quilts involved in the definition of Symp.

#### February 22th, 2019: Barney Bramham (IAS and Bochum) " Orbit travel and symbolic dynamics for 3D Reeb flows using finite energy foliations "

*Abstract:*
Suppose a non-degenerate Reeb flow of a closed contact 3-manifold admits a finite energy foliation F. The rigid leaves in F divide the space into various regions A, B, C etc. A typical Reeb trajectory then spits out an infinite sequence of letters which record which regions it visits and in which order. It is natural to investigate the converse. Indeed, it turns out one needs surprisingly little information to determine, for a given sequence of letters, whether there exists a Reeb trajectory realising this sequence as its
itineraries. I will explain this mechanism with pictures and discuss some applications. This is joint work with Umberto Hryniewicz and
Gerhard Knieper.

#### March 1st, 2019: Du Pei (QGM and Caltech) " Modular tensor categories from the Coulomb branch "

*Abstract:*
In this talk, I will review an emerging link between the geometry of moduli spaces and the representation theory of vertex operator algebras. The construction goes through a class of four-dimensional quantum field theories that are said to satisfy "property F". Each such theory gives rise to a family of modular tensor categories, whose algebraic structures are encoded in the geometry of the Coulomb branch.

#### March 1st, 2019: Yaron Ostrover (IAS and Tel Aviv University) " Computational aspects of symplectic measurements "

*Abstract:*
We will discuss certain computational aspects of symplectic capacities for convex domains in the classical phase space. In particular, I will describe the state-of-the-art of some related open questions and conjectures, and explain how probability theory
and asymptotic geometric analysis can be used in the framework of symplectic geometry.

#### March 8th, 2019: Biji Wong (CIRGET, UQaM) " Three-orbifolds, bordered Floer theory, and twisted coefficients "

*Abstract:*
Using bordered Floer theory, we associate to any 3-orbifold Y^orb with singular set a knot an invariant HFO(Y^orb) that generalizes the hat flavor of Heegaard Floer homology for 3-manifolds. If Y denotes the 3-manifold underlying the 3-orbifold Y^orb, we relate the invariant HFO(Y^orb) to HF-hat of Y twisted by certain coefficients. We then use this to show that when Y is integer surgery on a knot K in S^3, twisted homology HF-hat of Y admits a description in terms of the epsilon invariant of K and the untwisted homology HF-hat of Y.

#### March 8th, 2019: Matt Hedden (Michigan State) " Satellites of Infinite Rank in the Smooth Concordance Group "

*Abstract:*
I'll discuss the way satellite operations act on the concordance group, and raise some questions and conjectures. In particular, I'll conjecture that satellite operations are either constant or have infinite rank, and reduce this to the difficult case of winding number zero satellites. I'll then talk about how to use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of smooth concordance, and use this to address the winding zero case. This is joint work with Juanita Pinzon-Caicedo.

#### March 15th, 2019: Ran Tessler (Weizmann Institute) " Open r-spin intersection theory: construction, integrable hierachy and mirror symmetry "

*Abstract:*
I will start by reviewing Witten's r-spin conjecture (now Faber-Shadrin-Zvonkine's theorem) which states that the partition function of the r-spin intersection theory is a Gelfand-Dickey tau function.
I will then describe a construction of the open analog in genus 0 (disks), and relate it to the Gelfand-Dickey wave function (based on a joint work with Buryak and Clader).
If time permits I will describe the mirror symmetry interpretation of the new disk invariants (based on a joint work to appear with Gross and Kelly) or an extension of the theory which includes boundary descendents.

#### March 29th, 2019: Beibei Liu (UC Davis) " Some geometric applications of the link Floer homology "

*Abstract:*
For links in the three sphere, there are two geometric questions: determining the Thurston polytope and 4-genus of links with vanishing pairwise linking numbers. I will explain how to use the Heegaard Floer homology introduced by Ozsvath and Szabo to determine the Thurston polytope, and give some bounds on the 4-genus in terms of the so-called h-function. In particular, for L-space links, the h-function can be computed explicitly by Alexander polynomials of the links and sublinks, and for L-space links with two components, the Thurston polytope is determined by the Alexander polynomials in a combinatorial way. I will also show some examples for both of the questions.

#### March 29th, 2019: Zhengyi Zhou (IAS) " Gysin sequences and cohomology ring of symplectic fillings "

*Abstract:*
It is conjectured that contact manifolds admitting flexible fillings have unique exact fillings. In this talk, I will show that exact fillings (with vanishing first Chern class) of a flexibly fillable contact (2n-1)-manifold share the same product structure on cohomology if one of the multipliers is of even degree smaller than n-1. The main argument uses Gysin sequences from symplectic cohomology twisted by sphere bundles.

#### April 5th, 2019: Aleksander Doan (Stony Brook) " Monopoles and Fueter sections on three-manifolds "

*Abstract:*
Over the last thirty years, the study of the Seiberg--Witten equations has led to spectacular advances in low-dimensional topology. In my talk, I will explain a generalization of the Seiberg--Witten equations and its conjectural relationship to Fueter sections, solutions of a non-linear version of the Dirac equation. I will then state some results, obtained in collaboration with Thomas Walpuski, which make progress towards proving this conjectural relationship. While generalized Seiberg--Witten equations are unlikely to lead to new invariants of low-dimensional manifolds, they have an intriguing connection to the geometry of higher-dimensional Riemannian manifolds with special holonomy.

#### April 5th, 2019: Alex Takeda (UC Berkeley) " Relative stability conditions: recovering the entire stability space for Fukaya categories of surfaces. "

*Abstract:*
In this talk I will present the techniques and results from arXiv:1811.10592, where a new notion of a relative stability condition is presented. This is defined in analogy with compactly supported chains, and using this tool we are able to prove that the stability conditions defined by Haiden, Katzarkov and Kontsevich using quadratic differentials cover the entire stability space of that surface's Fukaya category, in the fully stopped case. This confirms a special case of the general idea that stability on Fukaya categories should be related to the mean Lagrangian flow and special Lagrangians. The definition of these relative versions has only been worked out in the setting of these categories; time allowing I will discuss some expectations and hopes for extending it to a broader context.

#### April 12th, 2019: Yu-Shen Lin (Boston University) " Disc Correspondence theorem for Toric Fano surfaces "

*Abstract:*
In this talk, I will explain how to establish an equivalence of tropical discs counting and open Gromov-Witten invariants for toric Fano surfaces via Lagrangian Floer theory.
On one hand, this leads to a recursive algorithm to compute the bulk-deformed potential which is a canonical deformation of Hori-Vafa potential. On the other hand, we derive a quantum period theorem which provides a way to compute closed descendant Gromov-Witten invariants from open invariants. This is a joint work with Hansol Hong and Jingyu Zhao.

#### April 12th, 2019: Masaki Taniguchi (Tokyo university) " Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds. "

*Abstract:*
For a certain class of pairs of 3- and 4-manifolds, we construct an obstruction in the filtered instanton Floer cohomology to the existence of an embedding with some homological conditions between them. In order to achieve that goal, we study the compactness of the ASD-moduli spaces over 4-manifolds with periodic ends. By using the obstruction, we show that there are several homology 3-spheres which cannot be embedded in any homotopy S^1 × S^3 as a generator of H_3. Moreover, we introduce a real valued homology cobordism invariant r^+ of homology 3-spheres. We see the relation between r^+ and Daemi’s invariants and give some calculations. The study for r^+ is joint work with Nozaki Yuta and Kouki Sato.

#### April 19th, 2019: James Pascaleff (UIUC) " On monoidal structures on Fukaya categories. "

*Abstract:*
Unlike many categories arising in algebra, the Fukaya category
of a symplectic manifold X does not automatically carry a monoidal
structure. It turns out that monoidal structures on the Fukaya category
are related to symplectic groupoid structures on X. This in turn
connects to Poisson geometry, since a symplectic groupoid is an
"integration" of a Poisson manifold. I will outline these connections
and provide some applications to the case of cotangent bundles.

#### April 19th, 2019: Peter Feller (ETH Zurich) " Moebius bands in B^3xS^1 and the square peg problem. "

*Abstract:*
Following an idea of Hugelmeyer, we give a knot theory reproof of the following theorem. Every smooth Jordan curve in the Euclidean plane has an inscribed square. We will comment on possible generalizations to more general Jordan curves.
Our knot theory result, which allows the above application, is the following. For integers $n>1$ that are not squares, the torus knot $T(1,2n)$ in $S^2xS^1$ does not arise as the boundary of a locally-flat Moebius band in $B^3xS^1$. For context, we note that for $n>2$ and the smooth setting, this result follows from a result of Batson about the non-orientable 4-genus of certain torus knots. However, we show that Batson's result does not hold in the locally flat category: the smooth and topological non-orientable 4-genus differ for the $T(9,10)$ torus knot in S^3.
Based on work in progress with Marco Golla.

#### April 26th, 2019: Baptiste Chantraine (Université of Nantes) " Lagrangian cobordisms between Legendrian submanifolds and Lagrangian surgeries. "

*Abstract:*
In this talk we will study Lagrangian cobordisms between Legendrian submanifolds arising from some Lagrangian surgeries. From the Floer theory of those cobordisms we can deduce some geometrical descriptions of certain iterated cones in the Fukaya category. I will then explain how those considerations lead to a proof of the fact that Lagrangian cocores generates the wrapped Fukaya category of a Weinstein manifold. This is joint work with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the "Floer Homology" e-mail list maintained via Google Groups.