Simplicial SL(2,R) Chern-Simons theory and Boltzmann entropy on 3-manifolds

We show that real solutions of Thurston's equation on triangulated 3-manifolds are maximizers of the Boltzmann entropy.

Comparing Invariants of 3-Manifolds Derived from Hopf Algebras

In this talk, we will compare two different quantum 3-manifold invariants, both of which are given using a finite dimensional Hopf Algebra H. One is the Hennings invariant, given by an algorithm involving the link surgery presentation of a 3-manifold and the Drinfeld double D(H); the other is the Kuperberg invariant, which is computed using a Heegaard diagram of the 3-manifold and the same H. We show that when H is semi-simple, these two invariants are equal. The proof is entirely in Hopf algebraic terms and does not rely on the representation theory of H or general results involving categorical invariants.

An introduction to the rational genus of a knot

What is the "simplest" knot in a given three-manifold Y? We know that the answer is the unknot when Y=S^3, as the unknot happens to be the only knot in the three-sphere with the smallest genus (=0). In this talk, we will discuss the more general notion of the rational genus of knots. In particular, we will show that the simple knots are really the "simplest" knots in the lens spaces in the sense of being a genus minimizer in its homology class. This is a joint work with Yi Ni.

Organizing principles of Discrete Differential Geometry

We describe the recent development of the part of Discrete Differential Geometry related to the theory of integrable systems. Elementary geometrical roots of some complicated differential geometric notions will be explained. The notion of multidimensional consistency as the definition of integrability, as well as a sort of discrete Erlangen program will be discussed in detail.

Diameter controls and smooth convergence away from large singular sets

Given a sequence of Riemannian metrics on a compact manifold, one can talk about different notions of convergence (or limits) of these Riemannian manifolds. We will consider the two most important notions of convergence namely, the Gromov-Hausdorff convergence (limit) and the Smooth Convergence (away from singularities). We will provide conditions under which these two limits agree.

Horospherical limit points

I'll begin by defining the Bieri-Neumann-Strebel invariant of a group, something introduced in the 1980's. With that as a jumping-off point, I'll describe a generalization involving an isometric action of a group G on a proper CAT(0) space M (e.g. a non-positively curved simply connected manifold or a euclidean building). Given a finitely generated ZG-module, there is a way of seeing it as sitting equivariantly "over" M, and one defines the set of horospherical limit points of this situation. There are connections with arithmetic groups, with groups acting on hyperbolic space, and with tropical geometry. This is joint work with Robert Bieri.

The finiteness obstruction

The talk will discuss occurences of K₀ obstructions in topology, including Wall's finiteness obstruction and Siebenmann's end obstruction. A survey article on this topic may be downloaded at The Finiteness Obstruction.

Computer assisted proof for normally hyperbolic manifolds

In the talk we present a topological proof of existence of normally hyperbolic invariant manifolds for maps. The proof is constructive and our theorems are formulated so the their assumptions are verifiable using interval-arithmetic based, computer assisted estimates. Throughout the majority of the talk we focus on a two dimensional, driven logistic map. On this examples we demonstrate the main features of the construction. The example is of interest since standard computer simulations suggest that the map has a chaotic attractor. Using our method we can prove that the numerical evidence is false, and that the map has a globally attracting smooth invariant curve. This is joint work with Carles Simo.

Motivic topology is a way of doing homotopy theory for schemes over a fixed base field. Topological techniques, e.g., Adams spectral sequences, are heavily used to inform computations of motivic invariants. The talk will survey such techniques and computations. Our main examples of base fields are finite and p-adic fields,the complex, real and rational numbers. The arithmetic of the base field enters in our computations.

Knot homologies and ideal quotients

A knot homology theory assigns a (bi- or triply-)graded chain complex to a knot such that the complex's chain homotopy type is an invariant of the knot and the complex's graded Euler characteristic is a classical knot polynomial. This talk will describe work in progress that aims to relate Khovanov and Rozansky's HOMFLY-PT homology (which lifts the HOMFLY-PT polynomial) to Ozsv\'ath-Szab\'o and Rasmussen's knot Floer homology (which lifts the Alexander polynomial). We will first define the two theories in apparently similar, purely algebraic terms. The main result is a relationship between the two algebraic constructions expressed in terms of ideal quotients. We will sketch a proof for this result, then discuss its consequences in the context of a spectral sequence that conjecturally relates the two homology theories.

In this talk I will revisit the algebraic interpretation of persistent homology introduced by Carlsson and Zomorodian and discuss how this interpretation can give a more unified algorithmic picture for all of the variants of persistence. This will include known variants as well as other well-known constructions which can be computed. I will also show how this can be used to parallelize certain computations and certain natural extensions of persistence which come up as a result.

Statistics for Teichmuller geodesics

We describe two ways of picking a geodesic "at random" in a space, one coming from the standard Lebesgue measure on the visual sphere, and the other coming from random walks. The spaces we're interested in arehyperbolic space and Teichmuller space, together with some discrete group action on the space. We investigate the growth rate of word length as you move along the geodesic, and we show these growth rates are different depending on how you choose the geodesic. This is joint work with Vaibhav Gadre and Giulio Tiozzi.

Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Given a punctured surface S, its Ptolemy groupoid is a natural object associated to ideal triangulations on the surface. The action of the mapping class group on ideal triangulations extends to a natural homomorphism to this groupoid. Using hyperbolic geometry, in our contextshear coordinates on Teichmuller space, this can be used to construct representations of the mapping class group in terms of rational functions. This was described first by R. Penner using the closely related lambda-length coordinates. In this talk we will describe how this construction behaves when pinching simple closed curves on S. This has combinatorial implications, with the construction of ideal triangulations on pinched surfaces and the effect on the Ptolemy groupoid, and geometrical, with a natural extension of shear coordinates to the augmented Teichmuller space. In both cases we explain how this applies to the action of the mapping class group. If time permits we will describe some applications to the study of quantum Teichmuller theory.

Thurston's gluing equations for PGL(n,C)

Thurston's gluing equations are polynomial equations invented by Thurston to explicitly compute hyperbolic structures or, more generally, representations in PGL(2,C). This is done via so called shape coordinates. We generalize the shape coordinates to obtain a parametrization of representations in PGL(n,C). We give applications to quantum topology, and discuss an intriguing duality between the shape coordinates and the Ptolemy coordinates of Garoufalidis-Thurston-Zickert. The shape coordinates and Ptolemy coordinates can be viewed as 3-dimensional analogues of the X- and A-coordinates on higher Teichmuller spaces due to Fock and Goncharov.

Structure and rigidity of toatally periodic pseudo-Anosov flows in graph manifolds

This is joint work with Thierry Barbot. A graph manifold is an irreducible manifold so that all pieces of the torus decomposition are Seifert fibered. We consider pseudo-Anosov flows in graph manifolds so that all pieces are periodic. We consider pseudo-Anosov flows in graph manifolds so that each Seifert fibered piece is periodic. This means that a regular fiber is freely homotopic to a closed orbit of the flow. We show that these flow are rigid, that is, they are completely determined by the dynamics and topological structure of a dynamical spine associated to the flow.

Quantifying Residual Finiteness and LERF-ness in Terms of Geometric Data

This talk will begin by defining residual finiteness (RF) and locally extended residual finiteness (LERF) for groups, followed by a brief history of the results that study the connection between these algebraic properties and the fundamental groups of surfaces and 3-manifolds. We will then describe what it means to quantify these group properties and present the results that quantify RF-ness and LERF-ness of hyperbolic surface groups in terms of geometric data. If time permits, we will conclude with an overview of similar techniques used to quantify residual finiteness for particular hyperbolic 3-manifold groups.

Hyperbolic cone metrics on 3-manifolds with boundary

We prove that a hyperbolic cone metric on an ideally triangulated compact $3$-manifold with boundary consisting of surfaces of negative Euler characteristic is determined by its combinatorial curvature. The proof uses a convex extension of the Legendre transformation of the volume function. Several related results on maximum volume angle structures are obtained. This is a joint work with Feng Luo.

Skein Algebras and the Decorated Teichmuller Space

The Kauffman bracket skein module K(M) of a 3-manifold M is defined by Przytycki and Turaev as an invariant for framed links in M satisfying the Kauffman skein relation. For a compact oriented surface S, it is shown by Bullock-Frohman-Kania-Bartoszynska and Przytycki-Sikora that K(Sx[0,1]) is a quantization of the SL_2 C -characters of the fundamental group of S with respect to the Goldman-Weil-Petersson Poisson bracket.

In the joint work with Julien Roger that I will be talking about, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in Sx[0,1] satisfying the skein relations of crossings both in the surface and at punctures. This algebra quantizes a Poisson algebra of loops and arcs on S in the sense of deformation of Poisson structures. This construction provides a tool to quantize the decorated Teichmuller space; and the key ingredient in this construction is a collection of geodesic lengths identities in hyperbolic geometry which generalizes/is inspired by Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula.

Collapsing of Kahler metrics and analytic space

In this talk, we study the relationship of Kahler metrics along a degeneration of projective manifolds and the solutions of the non-archimedean Monge-Ampere equation. For a degeneration of projective manifolds, the solutions of the non-archimedean Monge-Ampere equation were obtained on the corresponding Berkovich analytic space by Bouckson, Favre and Jonsson. In this talk, we show that the solution can be approximated by a family of Kahler metrics on the original family of projective manifolds in the potential function sense. This result generalizes a theorem of Haase and Zharkov for Calabi-Yau hypersurfaces in toric varieties.

A Transcendental Invariant of Pseudo-Anosov Maps

For each pseudo-Anosov map phi on surface S, we will associate it with a Q-submodule of R, denoted by A(S,phi). A(S,phi) is defined by interaction between Thurston norm and dilatation of pseudo-Anosov map. We will develop a few nice properties of A(S,phi) and give a few examples to show that A(S,phi) is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen: the minimal point of the restriction of the dilatation function on a fibered face need not be a rational point.

Non-compactness of stationary harmonic maps in weak sense

Let M and N be two compact Riemannian manifolds. Given a sequence of stationary harmonic maps from M to N with bounded energies, we may assume that it converges weakly to a weakly harmonic map in W^{1,2}(M, N). One can ask whether the limit map could be also stationary or not. In this talk, some basic definitions and results about harmonic maps will be reviewed firstly. Then we will construct a concrete example to show that the limit map may not be stationary.

Taming wild homotopies

We prove that if G is a group with finite asymptotic dimension, then the integral Novikov conjecture is true for G. In the course of the proof, we prove that if G is such a group, then the Higson compactification of EG is mod p acyclic for all p. This is joint work with Dranishnikov and Weinberger.

Algebras Counting Intersections and Self-Intersections of Loops on a Surface

Goldman and Turaev defined a Lie bialgebra structure on the vector space generated by nontrivial free homotopy classes of loops on an oriented surface. The Goldman bracket and Turaev cobracket give lower bounds on the minimal intersection and self-intersection numbers of loops in given free homotopy classes, respectively. Chas showed that these bounds are not equalities in general. Andersen, Mattes, and Reshetikhin defined a Lie bracket that generalizes Goldman's. We show that their bracket gives a formula for the minimal intersection number. We also define an operation that generalizes Turaev's cobracket in the same way that the Andersen-Mattes-Reshetikhin bracket generalizes Goldman's bracket, and show this operation gives a formula for the minimal self-intersection number. Some of this work is joint with Vladimir Chernov.

A topological algorithm for computing the Conley index of Poincare maps in time-periodic differential equations

Conley index is a topological invariant of dynamical systems used to prove the existence of invariant sets. The use of the Conley index in the study of recurrent dynamics of a flow is not straightforward. The Conley index for flows is not very useful and usually the Conley index of a Poincare map is studied. Unfortunately, the computation of the index is not easy, because in general no explicit formula for the Poincare map is available. An alternative is to compute the index numerically. The standard algorithmic technique requires computing outer enclosures of the Poincare map. This requires long time integration along the trajectories of the flow which is computationally expensive and often prohibits successful computations. We will discuss some earlier strategies to overcome the problem as well as a recent topological approach which does not require long time integration. This is research in progress, joint work with R. Srzednicki and F. Weilandt.

Anosov representations along a geodesic lamination

Let λ ⊂ S be a geodesic lamination, where S is a connected, closed, oriented surface of genus g ≥ 2. We consider homomorphisms   ρ: π₁(S)→ PSL_n(R) that satisfy a certain Anosov property along the leaves of  λ. We discuss various geometric properties of these so-called λ-Anosov representations; in particular, we define invariants for these representations that generalize Thurston's length function, and give a characterization of the λ-Anosov property. We then introduce cataclysm deformations, and obtain, under some additional genericity condition, a parametrization of the subset of  λ-Anosov representations via transverse n-twisted cocycles for the geodesic lamination λ.

Prescribing the behavior of Weil-Petersson geodesics

The Weil-Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of Riemann surfaces with negative sectional curvatures which are not bounded away from 0. Brock, Masur and Minsky introduced a notion of "ending lamination" for WP geodesic rays which is an analogue of the vertical foliations of Teichm\"{u}ller geodesics. In this talk we show that these laminations and the associated subsurface coefficients can be used to determine the itinerary of a class of WP geodesics in the moduli space. As a result we give examples of closed WP geodesics staying in the thin part of of the moduli space, geodesic rays recurrent to the thick part of the moduli space and diverging geodesic rays. These results can be considered as a kind of symbolic coding for WP geodesics.

Combinatorial Roundness of Grains in Cellular Microstructures and Random Discrete Morse Theory for Cell Complexes

Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Some of the occurring grain types are substantially more frequent than others. We will observe that the frequent types are ``combinatorially round'' -- which gives us a new starting point for the microstructure analysis of steel.

Computational homology packages such as CHomP or RedHom allow to obtain homological information for large complexes from material sciences or other. These packages extensively use (NP-hard) discrete Morse theory as a (fast) preprocessing step to avoid (slow, polynomial time) Smith Normal Form computations. In fact, it is surprisingly hard to construct ``complicated'' examples on which homology calculations perform poorly. We propose a new library of complicated triangulations and we introduce Random Discrete Morse Theory as a computational scheme to measure the complicatedness of a triangulation. An interesting infinite series of complicated triangulations is based on the Akbulut-Kirby spheres related to the Andrews-Curtis conjecture.

Reflection groups, non-negative curvature and Tits geometry

(i) A complete classification of reflection groups and the equivariant structures of complete non negatively curved manifolds.

(ii) A complete classification of positively curved polar manifolds of cohomogeneity at least 2, which is achieved partially based on Tits geometry.