#### Joint Problem Session for Floer Theory and Low-dimensional Topology; Investigating the L-space Conjecture; and Knots, Surfaces, and 4-manifolds

You can download the pdf here!

Problem Session Write-Up

Categories

You can download the pdf here!

Problem Session Write-Up

Categories

Dear NCNGT participants,

We are very excited to be starting up in just three days- we have 80+ fantastic talks recorded across 8 topic groups, a variety of live office hours, coffee breaks, and problem sessions scheduled, and what we hope will be an active mathematical conversation ongoing via the comment section! More details below:

**Introductions: **

In our experience, one of the most valuable aspects of the conference experience, especially for graduate students, is meeting other people interested in the same ideas. To facilitate this in our online setting, we encourage all participants to briefly introduce themselves here and say a little bit about their mathematical interests.

**Talks:**

We are primarily asychronous, so you can watch our pre-recorded talks whenever you like during the conference! This talks page is the central reference for all our talks. Clicking on the title of a mini-session will take you to its page. These currently just have titles and abstracts and as of Monday will also have videos. In addition, each of these pages has a comment sections which the speakers and organizers will be checking regularly, and which will serve as the central discussion/ questions location for the conference. Commenting is restricted to registered users (like you!), though they can be viewed by anyone- please remember to be professional, friendly, and welcoming. If at any point you see inappropriate behavior or experience harassment, please get in contact with either the relevant topic group organizers (listed on the main page) or the conference organizers (Martin Bobb and Allison Miller), and we will intervene as soon as possible.

**Live events:**

We have a variety of live events scheduled (see here for a calendar and listing by group), all of which will be run via Zoom. We will email the links and passwords out on a daily basis, so e.g. Monday's three events will be sent out Sunday night.

**Office hours:**

These live events, hosted by our speakers, are intended for graduate students to get face time with the speakers and dig into some of the details of their talks. If you are not a graduate student, you are welcome to attend these office hours, as long as you give priority to any students who are present. We also encourage all conference attendees to reach out to other participants to discuss research and learn new things.

“See” you soon!

Best,

The NCNGT organizers (aka Allison N. Miller and Martin Bobb)

P.S. Please add our email (organizers@ncngt.org) to your address book to ensure you receive future emails!

Categories

This is a space for all conference participants to introduce themselves and hopefully meet others with overlapping mathematical interests!

As well as name, career stage, home institution, and research focus (as specifically as you like), consider giving a short description of a math paper you read recently and think is really great!

**Codimension-1 Flats in Convex Projective Geometry- Martin Bobb (Michigan)
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Abstract: Convex projective manifolds generalize hyperbolic manifolds while allowing for some similarities to non-positively curved spaces, and some interesting deformation theory. In this lecture we will discuss the structure of codimension-1 flats in compact convex projective manifolds.

**Rank one phenomena in convex projective geometry- Mitul Islam (Michigan)**

Abstract: The goal of this talk is to develop analogies between rank one non-positive curvature/ CAT(0) and convex projective geometry. We will introduce the notion of rank one automorphisms of properly convex domains and characterize them as contracting group elements. We will prove that a discrete rank one automorphism group is either virtually cyclic or acylindrically hyperbolic. This leads to some applications like computation of space of quasimorphisms, counting of closed geodesics, and genericity results.

**Moduli space of unmarked convex projective surfaces- Zhe Sun (Luxembourg)**

Abstract: Mirzakhani found a beautiful recursive formula to compute the volume of the moduli space of Riemann surfaces. We discuss the possible similar recursive formula where the Riemann surfaces are replaced by the convex projective surfaces. We investigate the boundedness of projective invariants, area, and many other notions that are uniformly related to each other and we show one of these bounded subsets has polynomially bounded Goldman symplectic volume.

**Proper actions on symplectic groups and their Lie algebras- Jean-Philippe Burelle (Sherbrooke)**

Abstract: Danciger-Guéritaud-Kassel developed a theory of proper actions on PSL(2,R) (anti-de Sitter space) and its Lie algebra sl(2,R) (Minkowski space) using length contraction/expansion properties. They applied this machinery to obtain several results on the structure of anti-de Sitter and flat Lorentzian manifolds in dimension 3, including proofs of the tameness of Margulis space times and of the crooked plane conjecture. I will show how part of this theory generalizes to symplectic groups Sp(2n,R) and their Lie algebras sp(2n,R). This is joint work with Fanny Kassel.

**Growth of quadratic forms under Anosov subgroups- León Carvajales (Universidad de la República & Sorbonne)**

Abstract: For positive integers p and q we define a counting problem in the (pseudo-Riemannian symmetric) space of quadratic forms of signature (p,q) on R^{p+q}. This is done by associating to each quadratic form o a totally geodesic copy of the Riemannian symmetric space of PSO(o) inside the Riemannian symmetric space of PSL_{p+q}(R), and by looking at the orbit of this geodesic copy under the action of a discrete subgroup of PSL_{p+q}(R). We then present some contributions to the study of this counting problem for Anosov subgroups of PSL_{p+q}(R).

**Topological dynamics of the Weyl chamber flow- Nguyen-Thi Dang (Heidelberg)**

Abstract: Let G be a connected, real linear, semi-simple Lie group without compact factors and D be a discrete subgroup. Let K be a maximal compact subgroup, A a Cartan subgroup for which Cartan decomposition holds. Consider an action D\G on the right by a one-parameter subgroup of A. As a family of topological dynamical systems, when do all orbits diverge ? What are the interesting sets for the dynamics ? Are there dense orbits in those sets ? Is there topological mixing ? When D is a cocompact lattice, there are no diverging orbits and every flow is exponentially mixing by a result of Howe-Moore. Assume now that D is only Zariski dense in G. For SO(n,1)⁰, the corresponding flow is the frame flow which factors over the geodesic flow of a hyperbolic manifold. Mixing properties of such flows in convex-cocompact hyperbolic manifolds were obtained by Winter in 2016 and improved recently by Winter-Sarkar. In 2017, Maucourant-Schapira proved topological mixing when D is Zariski dense. In this talk, I'll focus in the case where G is a higher rank split simple Lie group and D a Zariski dense subgroup. The existence of Weyl chambers in A allows to define regular Weyl chamber flows as those parametrised by an element of the interior a Weyl chamber. The Benoist cone, which contains all the information about the spectrum of D will then give us a necessary condition for the existence of non-diverging orbits. Finally I'll explain a topological mixing criteria for regular flows.

**Counting meanders and square-tilings on surfaces- Eduard Duryev (Institut de mathématiques de Jussieu in Paris)**

Abstract: Meander is a homotopy class of a pair of transversal simple closed curves on a sphere. They appear, in particular, as natural enumeration of polymer foldings. Statistics of meanders as the number of intersection grows is one question about meanders people have been interested in. We will show how meanders are related to a particular class of square tilings on surfaces and explain an approach to this more general counting problem that uses ribbon graphs and intersection theory on moduli space.

**Part 1/2: **

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**Extremal length systole of the Bolza surface- Didac Martinez-Granado (Indiana University Bloomington)**

Abstract: In Bers-Teichmueller theory there is a rich interaction between the hyperbolic geometry and the conformal geometry, as a result of the uniformization theorem. If the length notion in the hyperbolic side of this dichotomy is the hyperbolic length, on the conformal side the counterpart is extremal length, a conformal invariant notion of homotopy classes of curves. For a choice of hyperbolic structure, there is an essential curve of minimal extremal length and one of minimal hyperbolic length. We call both the curve and its length, respectively, the extremal length and hyperbolic length systoles. These are functions on moduli space. Bounds for maximal values of the hyperbolic length systole for some geni are known, and for genus 2, its absolute maximum is the Bolza surface, a triangle surface. In this talk we show that the Bolza surface realizes a local maximum of the extremal length systole, and compute its value: square root of 2. This is work in progress with Maxime Fortier Bourque and Franco Vargas Pallete.

**Video: **

**Gaps of saddle connection directions for some branched covers of tori- Anthony Sanchez (University of Washington Seattle)**

Abstract: Translation surfaces given by gluing two identical tori along a slit have genus two and two cone-type singularities of angle $4\pi$. There is a distinguished set of trajectories called saddle connections that are the straight lines trajectories between cone points. We can associate a holonomy vector in the plane to each saddle connection whose components are the horizontal and vertical displacement of the saddle connection. How random is the planar set of holonomy of saddle connections? We study this question by computing the gap distribution for slopes of saddle connections for these and other related classes of translation surfaces.

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**The topology and geometry of random square-tiled surfaces- Sunrose Thapa Shrestha (Tufts University)**

Abstract: A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk, we will consider a randomizing model for STSs based on permutation pairs and see how to use it to compute the genus distribution. We will also look at holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. We will obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

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**Periodic billiard paths on regular polygons- Diana Davis (Swarthmore College)**

Abstract: Mathematicians have understood periodic billiards on the square for hundreds of years, and my collaborator Samuel Lelièvre and I have understood them on the regular pentagon for about five years now. During the COVID-19 pandemic, I have been in France, working with Samuel to extend our understanding to all regular polygons with an odd number of sides. In this talk, I'll briefly explain results and techniques for the square and pentagon, and then show lots of nice pictures of billiards on polygons with more than 5 sides, that we have created recently.

**Video: **

**An invitation to dilation surfaces- Selim Ghazouani (University of Warwick)**

Abstract: In this video talk, I will introduce dilation surfaces, their moduli spaces and related foliations on surfaces and then try to give some motivation for a range of open problems.

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**You can “hear” the shape of a polygonal billiard table- Chandrika Sadanand (University of Illinois Urbana Champaign)**

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge-itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

**Video: **

**Chabauty Limits of the Diagonal Subgroup in SL(n,Q_p)- Arielle Leitner (Weizmann)**

Abstract: A conjugacy limit group is a limit of a sequence of conjugates of the positive diagonal subgroup C in SL(n) in the Chabauty topology. In low dimensions, there are finitely many limits up to conjugacy, and we explain why there are more limits over Q_p than over R. In higher dimensions there are infinitely many limits up to conjugacy. We can understand limits of C by understand how to go to infinity in the building (you won't need to know what a building is for this talk, we'll explain the geometry with low dimensional examples). We use the geometry of the building to classify limits of C. This is joint work with C. Ciobotaru and A. Valette. The hidden agenda of this talk is to convince you that Q_p is friendly, and things that we do over R and C can work over Q_p as well.

**Sequences of Hitchin representations of Tree-Type- Giuseppe Martone (Michigan)**

Abstract: Let S be an oriented surface of genus greater than 1. The Teichmuller space of S can be described as a connected component of the space of representations of the fundamental group of S into the Lie group PSL(2,R). The Hitchin component generalizes this classical picture to the Lie group PSL(d,R). Hitchin representations are a prominent subject of study in the field of Higher Teichmuller theory. Motivated by classical work of Thurston, one wishes to understand the asymptotic behavior of sequences of Hitchin representations. In this talk we describe non-trivial sufficient conditions on a diverging sequence of Hitchin representations so that its limit can be described as an action on a tree. In other words, we single out sequences whose asymptotic behavior is similar to diverging sequences in the Teichmuller space. Our non-trivial conditions are given in terms of Fock-Goncharov coordinates on moduli spaces of positive tuples of flags.

**In the moduli space of Abelian differentials, big invariant subvarieties come from topology!- Paul Apisa (Yale University)**

Abstract: It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus. In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety - a measure of size related to dimension - and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani. No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright.

**Video:**

**Translation surfaces with multiple short saddle connections- Ben Dozier (Stony Brook University)**

Abstract: The SL_2(R) action on strata of translation surfaces allows us to answer many questions about the straight-line flow on individual translation surfaces (and this flow is in turn closely connected to billiards on rational polygons). By the pioneering work of Eskin-Mirzakhani, to understand dynamics on strata one is led to study "affine" measures. It is natural to ask about the interaction between volumes of certain subsets of surfaces and the geometric properties of the surfaces. I will discuss a proof of a bound on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. This is a strengthening of the regularity result proved by Avila-Matheus-Yoccoz. A key tool is the new smooth compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.

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**What does your average translation surface look like?- Anja Randecker (Heidelberg University)**

Abstract: Almost every talk on translation surfaces starts with a double pentagon or with an octagon. But are these n-gons generic examples of translation surfaces? We will look at this question from the point of view of the geometry of the translation surfaces. Specifically, we prove an upper bound for the average diameter (resp. covering radius) in a stratum of translation surfaces of large genus. The first part of my talk assumes only basic knowledge about translation surfaces, the second part assumes more background to give a sketch of the proof. Both are based on joint work with Howard Masur and Kasra Rafi (available at https://arxiv.org/abs/1809.10769).

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**Developing software for skein computations in knot complements- Rachel Marie Harris (Texas Tech University)**

Abstract: Skein manipulations prove to be computationally intensive due to the exponential nature of skein relations. The purpose of this project is to construct an automated tool to generate a library of examples for use in testing new conjectures in Chern-Simons theory.

**Finding Structure in Polynomial Invariants using Data Science- Jesse Levitt (UCLA)**

Abstract: Authors: Pawel Dlotko, Mustafa Hajij, Jesse Levitt (presenter), Radmila Sazdanovic Abstract: Using Principal Component Analysis and Topological Data Analysis we analyze the distributions of the knot polynomials in coefficient space. These tools prove useful for both distinguishing how well different invariants separate the knots into distinct families and for how these families suggest correlations between different knot invariants. We focus on how the Ball Mapper of P. Dlotko, an exploratory data analysis tool that builds graphs from high dimensional clouds of data using just a radius measure, confirms and further illuminates substructure in this data. This includes some specific ways in which the s-invariant and signature differ through the distribution of the Alexander and Jones polynomials.