# Geometry and Topology Seminar

Traditionally, the Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.

In Spring of 2016, the seminar will meet on **Thursdays** at 2:25PM in B 231 of Van Vleck.

For more information, contact Tullia Dymarz or Alexandra Kjuchukova.

## Contents

## Spring 2016

In Spring of 2016, the seminar will meet on Thursdays at 2:25PM in B 231 of Van Vleck.

## Spring Abstracts

### Ryan Blair

*Distance and Exceptional Surgeries on Knots*

Distance is a measure of complexity for a bicompressible surface in a 3-manifold which is defined using the curve complex for the surface. Recently, distance has been used to better understand Dehn surgery on knots in 3-manifolds. In particular, I will present results which show that knots with high distance surfaces do not admit non-hyperbolic surgeries or cosmetic surgeries. Applications to the cabling conjecture and the Berge conjecture will also be discussed.

### Brett Kotschwar

*Ricci flow and bounded curvature*

The problem of determining when a given solution to the Ricci flow with initially bounded curvature will continue to have bounded curvature has bearing on both the uniqueness and long-time existence of solutions to the flow. I will discuss two results in this direction which are equally valid in the noncompact setting, the first based on a simple proof of an extension to the standard uniqueness theorem of Hamilton and Chen-Zhu, and the second, joint with Ovidiu Munteanu and Jiaping Wang, based on new explicit local estimates of the curvature under a uniform bound on the Ricci tensor.

### Partha Solapurkar

*Some new surfaces of general type with maximal Picard number*

The Picard number $ \rho(X) $ of a surface $ X $ is the rank of its Neron-Severi group. It is bounded above by the Hodge number $ h^{11}(X) $. We say that a surface has maximal Picard number if it has the largest possible Picard number: $ \rho(X) = h^{11}(X) $. In 1972, Shioda constructed elliptic modular surfaces and among other things, proved that they have maximal Picard number. Our first idea is to take elliptic modular surfaces and replace each elliptic curve with a canonically constructed genus 2 curve. Under nice circumstances the resulting surfaces do indeed have maximal Picard number. There is also a second set of examples that arise as the total space of the moduli space of quaternionic Shimura curves. This is joint work with my advisor Prof. D. Arapura.

### Botong Wang

*A family of Symplectic-Complex Calabi-Yau Manifolds that are NonKahler*

A Kahler manifold is a smooth manifold with compatible complex and symplectic structures. In general, a compact manifold which admits both complex and symplectic structures may not admit any Kahler structure. Hodge theory and hard Lefschetz theorem have very strong implications on the homotopy type of compact Kahler manifolds. We introduce a family of 6-dimensional compact manifolds $M(A)$, which admit both Calabi-Yau symplectic and Calabi-Yau complex structures. They satisfy all the consequences of classical Hodge theory and hard Lefschetz theorem. However, we show that they are not homotopy equivalent to any compact Kahler manifold using a recently developed cohomology jump loci method. This is joint work with Lizhen Qin.

### Mike Cantrell

"Asymptotic shapes for ergodic families of metrics on nilpotent groups"

Let G be a finitely generated virtually nilpotent group. We consider three closely related problems: (i) convergence to a deterministic asymptotic cone for an equivariant ergodic family of inner metrics on G, generalizing Pansu’s theorem; (ii) the asymptotic shape theorem for First Passage Percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of G; (iii) the sub-additive ergodic theorem over a general ergodic G-action. The limiting objects are given in terms of a Carnot-Carathérodory metric on the graded nilpotent group associated to the Mal’cev completion of G.

### Talia Fernos

"The Roller Compactification and CAT(0) Cube Complexes"

The Roller compactification is a beautiful object that ties together the combinatorial and geometric properties that characterize CAT(0) cube complexes. In this talk, we will discuss this compactification in the context of superrigidity, the fixed point Property (FC), and the abundance of regular elements (i.e. automorphisms that are rank-1 in each irreducible factor). These results are collaborations with Caprace, Chatterji, and Iozzi, as well as, Lécureux, and Mathéus.

### Patricia Cahn

"Knots Transverse to a Vector Field"

We study knots transverse to a fixed vector field V on a 3-manifold M up to the corresponding isotopy relation. Such knots are equipped with a natural framing. Motivated by questions in contact topology, it is natural to ask whether two V-transverse knots which are isotopic as framed knots and homotopic through V-transverse immersed curves must be isotopic through V-transverse knots. When M is R^3 and V is the vertical vector field the answer is yes. However, we construct examples which show the answer to this question can be no in other 3-manifolds, specifically S^1-fibrations over surfaces of genus at least 2. We also give a general classification of knots transverse to a vector field in an arbitrary closed oriented 3-manifold M. We show this classification is particularly simple when V is the co-orienting vector field of a tight contact structure, or when M is irreducible and atoroidal. Lastly, we apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges. This work is joint with Vladimir Chernov.

## Fall 2015

## Fall Abstracts

### Hung Tran

*Relative divergence, subgroup distortion, and geodesic divergence*

In my presentation, I introduce three new invariants for pairs $(G;H)$ consisting of a finitely generated group $G$ and a subgroup $H$. The first invariant is the upper relative divergence which generalizes Gersten's notion of divergence. The second invariant is the lower relative divergence which generalizes a definition of Cooper-Mihalik. The third invariant is the lower subgroup distortion which parallels the standard notion of subgroup distortion. We examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups. We answer the question of Behrstock and Drutu about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. More precisely, we show that for each real number $s>2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.

### Tullia Dymarz

*Non-rectifiable Delone sets in amenable groups*

In 1998 Burago-Kleiner and McMullen constructed the first examples of coarsely dense and uniformly discrete subsets of R^n that are not biLipschitz equivalent to the standard lattice Z^n. Similarly we find subsets inside the three dimensional solvable Lie group SOL that are not bilipschitz to any lattice in SOL. The techniques involve combining ideas from Burago-Kleiner with quasi-isometric rigidity results from geometric group theory.

### Jesse Wolfson

*Counting Problems and Homological Stability*

In 1969, Arnold showed that the i^{th} homology of the space of un-ordered configurations of n points in the plane becomes independent of n for n>>i. A decade later, Segal extended Arnold's method to show that the i^{th} homology of the space of degree n holomorphic maps from \mathbb{P}^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems. We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.

### Matthew Cordes

*Morse boundaries of geodesic metric spaces*

I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the ``hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on Morse boundary of the mapping class group and briefly describe joint work with David Hume developing a capacity dimension for the Morse boundary.

### Anton Izosimov

*Stability for the multidimensional rigid body and singular curves*

A classical result of Euler says that the rotation of a torque-free 3-dimensional rigid body about the short or the long axis is stable, while the rotation about the middle axis is unstable. I will present a multidimensional generalization of this result and explain how it can be proved using some basic algebraic geometry of singular curves.

### Jacob Bernstein

*Hypersurfaces of low entropy*

The entropy is a quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the geometric complexity of a hypersurface of Euclidean space. It is closely related to the mean curvature flow. On the one hand, the entropy controls the dynamics of the flow. On the other hand, the mean curvature flow may be used to study the entropy. In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy really are simple.

### Yun Su

*Higher-order degrees of hypersurface complements.*

### Gao Chen

*Classification of gravitational instantons*

A gravitational instanton is a noncompact complete hyperkahler manifold of real dimension 4 with faster than quadratic curvature decay. In this talk, I will discuss the recent work towards the classification of gravitational instantons. This is a joint work with X. X. Chen.

### Dan Cristofaro-Gardiner

*Higher-dimensional symplectic embeddings and the Fibonacci staircase*

McDuff and Schlenk determined when a four dimensional symplectic ellipsoid can be embedded into a ball, and found that when the ellipsoid is close to round, the answer is given by an infinite staircase determined by the odd-index Fibonacci numbers. I will explain joint work with Richard Hind, showing that a generalization of this holds in all even dimensions.

### Danny Ruberman

*Configurations of embedded spheres*

Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane.

### Quinton Westerich

*Harmonic Chern Forms on Polarized Kähler Manifolds*

Abstract: The higher K-energies are functionals whose critical points give Kähler metrics with harmonic Chern forms. In this talk, we relate the higher K-energies to discriminants and use the theory of stable pairs to obtain results on their boundedness and asymptotics.

### Tommy Wong

*Milnor Fiber of Complex Hyperplane Arrangement*

The existence of Milnor fibration creates rooms and provides a platform to discuss the topology of complex algebraic varieties. In this talk, the study of hyperplane arrangements will be specified. Many open questions have been raised subject to the Milnor fiber of the mentioned fibration. For instance,while the homology of the arrangement complement can be described by the Orlik-Soloman Algebra, which is combinatorically determined by the intersection poset, it has been conjectured that the poset also determines the homology of the Milnor fiber. There are active work on this open conjecture, especially in C^3. Several classical results will be mentioned in the talk. A joint work with Su, serving as an improvement of some of the classical work, will also be briefly described.

## Archive of past Geometry seminars

2014-2015: Geometry_and_Topology_Seminar_2014-2015

2013-2014: Geometry_and_Topology_Seminar_2013-2014

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

2010: Fall-2010-Geometry-Topology