- , Speaker: Sekh Kiraj Ajij, TIFR Mumbai
3:15 PM IST
Classification of transitive Anosov flows
Anosov flows are generalization of geodesic flow in hyperbolic manifolds. It has been a long standing problem to classify Anosov flows in 3-dimensions up to orbit equivalence. In this talk we will try to give a sketch of orbit equivalence of R-covered Anosov flows based on a recent work of Thomas Barthelmé and Kathryn Mann.
We will also introduce hyperbolic-like actions on R and show how it can be used for some classifications. Additionally we will discuss some topological restrictions on manifold in admitting an Anosov flow. If time permits we will discuss some applications.
- , Speaker: Rakesh Halder, TIFR Mumbai
3:15 PM IST
Surjectivity of the Cannon-Thurston map in metric bundle
Consider a hyperbolic metric bundle X over (a hyperbolic base) B such that fibers are uniformly hyperbolic and possess uniformly coarsely surjective barycenter maps. Mj and Sardar showed that the inclusion of a(ny) fiber into X extends continuously to their (Gromov) boundaries. (Such maps known as Cannon-Thurston (CT) maps.) Later, Bowditch proved that if fibers are isometric to hyperbolic plane and base is a geodesic ray then this CT map is surjective. (It is a result of Krishna and Sardar that it suffices to prove the surjectivity of the CT map when base is a geodesic ray to get surjectivity when base is arbitrary hyperbolic space.)
Motivated by this result, in a paper by Lazarovich, Margolis and Mj posed the question whether surjectivity holds when fibers are one-ended, uniformly hyperbolic and barycenter maps are uniformly coarsely surjective.
In this talk, we provide a positive answer to this question. If time permits, we will discuss a discretized version of the result, where one-ended condition on the fiber can be removed. This is ongoing work.
- , Speaker: Indranil Bhattacharyya, TIFR Mumbai
3:15 PM IST
Existence of train tracks for free group automorphisms
Train tracks were initially introduced by Thurston in the classification of surface automorphisms. Later, the idea was generalized in case of free groups by Bestvina and Handel to keep track of the direction of the edges in a graph. In this talk, we will prove the following theorem of Bestvina - "Every irreducible automorphism of the free group can be realized topologically by a train track map". Bers classified surface automorphisms using Teichmuller metric. In the same analogy, we will introduce a metric space called Outer space and define an asymmetric metric on it. Then we shall classify elliptic, parabolic and hyperbolic automorphisms to prove the theorem. The existence of a train track map helps us to understand the growth of the conjugacy class of a certain group element under a free group automorphism.
- , Speaker: Kuwari Mahanta, TIFR Mumbai
3:30 PM IST
Uniform hyperbolicity of curve graphs
The curve graph of a compact orientable surface is a 1-dimensional simplical complex which was introduced in 1978 as a tool to study the Teichmüller spaces of Riemann surfaces. Since then it has also been used to study the hyperbolic structures of 3-manifolds and the mapping class group of surfaces. In 1999, Masur and Minsky showed that the curve graph is a δ-hyperbolic space. Following this many authors have independently showed that there is an uniform δ for connected curve graphs.
In this talk, we will look at a short and self-contained proof of the fact that curve graphs are uniformly hyperbolic. This proof was given by Hensel, Przytycki and Webb in [1]. One of the main tools that they use are a special kind of paths in curve graphs known as unicorn paths.
[1] Hensel, S., Przytycki, P., and WEBB, R. C. H. 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs. J. Eur. Math. Soc. (JEMS) 17 (2015), 755–762.
- , Speaker: Rashmita Hore, TIFR Mumbai
3:15 PM IST
Conformal mating by Schwarz reflection map of the deltoid
Inspired by the Fatou-Sullivan dictionary between Kleinian groups and complex dynamics, it is natural to think of iterations of anti-holomorphic rational maps on the Riemann sphere as the complex dynamics counterpart of actions of Kleinian reflection groups. Schwarz reflections associated with (disjoint unions of) quadrature domains generate a class of anti-holomorphic dynamical systems. In certain cases, such systems provide a framework for conformally combining (mating) the dynamics of rational maps and that of reflection groups in the same dynamical plane. In this talk, we will demonstrate conformal matings between quadratic anti-holomorphic polynomials and reflection groups using Schwarz reflection maps, for the particular example of the deltoid reflection.
The talk will be based on this paper.
- , Speaker: Viswanathan S, ICTS-TIFR
3:00 PM IST
Combination theorems in complex dynamics
In this talk we will see a few instances of combining (or mating) complex dynamical systems. Such theorems are used to
(a) prove certain facts about the iterating functions, and
(b) extract some geometric insights on the parameter spaces of these iterating functions.
If time permits, we will introduce algebraic correspondences, and see why they provide the right framework for matings. The talk will be based on this paper.
- , Speaker: Amartya Muthal, TIFR Mumbai
3:15 PM IST
Fenchel-Nielsen coordinates on the Teichmüller space of genus g
A marked surface S of genus g is an orientable surface of genus g along with a specified collection of 2g curves that generate the fundamental group of S. The Teichmüller space of genus g, denoted by Tg, is the space of hyperbolic metrics on marked surfaces of genus g. A hyperbolic surface of genus g can be cut along geodesic curves into simple pieces called "pairs of pants", the resulting decomposition is called the "pants decomposition" of the surface. The hyperbolic structure of the surface is completely determined by the hyperbolic structure of each piece, and the way these pieces are glued together. This gives rise to a global system of coordinates on Tg called Fenchel-Nielsen coordinates. In this talk I'll describe Fenchel-Nielsen coordinates for Tg, and if time permits, I'll discuss the construction of a compactification of Tg due to Thurston.
- , Speaker: Jacob (Yankl) Mazor, TIFR Mumbai
3:00 PM IST
"Geodesic flows" for iterated rational maps
There are a number of similarities between the dynamics of Kleinian groups on the sphere and the dynamics of iterating rational maps. But is there an analog to hyperbolic 3-manifolds on the iterated maps side? I do not know, but in the mid 1990s M. Lyubich and Y. Minsky constructed spaces admitting flows which are analogous to the geodesic flow on the unit tangent bundles of hyperbolic manifolds.
I will describe what these "hyperbolic 3-laminations" are, and how they are analogous to geodesic flow for Kleinian groups. I will then give some of the main ideas and outline the technicalities in their construction. Lastly, if time permits, I will show how Lyubich and Minsky adapted the ideas in the proof of Mostow's rigidity theorem to give an alternate proof of a theorem of W. Thurston's.
This talk is based primarily on: M. Lyubich and Y. Minsky. "Laminations in holomorphic dynamics." Journal of Differential Geometry (1997).
- , Speaker: Aratrika Pandey, IIT Bombay
2:00 PM IST
An overview of the Dimension Drop Conjecture on the space of lattices
Let G be a Lie group and Γ be a lattice in G and X=G/Γ. The Dimension Drop Conjecture states the set of points in X whose gt trajectory misses some open set U is not of full Hausdorff dimension. We will first discuss the setup and the history of the problem for G=SLn(R). Then, we will discuss a technique that has been used to prove the conjecture in this specific case. This talk is based on this paper.
- , Speaker: Anindya Chanda, TIFR Mumbai
3:00 PM IST
Self-orbit equivalences of Anosov flows
Study of Anosov Flows is a central theme in hyperbolic dynamics. These flows are extensively studied due to their rich geometric and dynamical properties. An orbit equivalence is a map which determines when two Anosov flows are 'same'. In recent years, self-orbit equivalence maps of Anosov flows got much traction due to their presence in the progress of two big open problems: classifications of Anosov flows and classifications of partially hyperbolic maps on 3-manifolds. In this talk we will first introduce orbit equivalence maps and their importance. In the second part, we will discuss an example of a hyperbolic manifold whose mapping class group can be represented by self-orbit equivalence maps only. This last part is based on a recent preprint by Bin Yu: https://arxiv.org/html/2312.13177v2.