The seminar this semester will be meeting online using Zoom. The Zoom doors will open 10 minutes before the seminar at 2:50pm (EST) for chatter and self provided coffee, tea, beer, wine, cocktail, aperitif, digestif, cheese, and snacks depending on your time zone and predilection. If you would like to be put on the email list contact Ara Basmajian (abasmajian@gc.cuny.edu). The zoom link for the upcoming seminar will be included in the weekly mailing to the email list.
Time: Tuesdays 3:00pm - 4:00pm (Talks are 50 minutes long)
Ara Basmajian (CUNY, Graduate Center and Hunter College)
Email: abasmajian@gc.cuny.edu
Nick Vlamis (CUNY, Queens College)
Email: nicholas.vlamis@qc.cuny.edu
Title: Random 3-manifolds with boundary
Abstract: If one randomly glues a finite number of tetrahedra together along their faces, the probability that the resulting complex is a manifold tends to zero as the number of tetrahedra grows. However, the only non-manifold points are the vertices of this complex. So, if we truncate the tetrahedra at their vertices, we obtain a random manifold with boundary. This talk will be about the geometry and topology of that manifold. This is joint work with Jean Raimbault.
Title: A universal Cannon-Thurston map and the surviving complex
Abstract: The fundamental group of a surface (closed or with punctures) acts on the curve complex of the surface with one additional puncture via the Birman Exact Sequence. I will describe a construction of a continuous, equivariant map from a subset of the circle at infinity of the universal cover of the surface onto the Gromov boundary of the curve complex (along the way, explaining what these objects and actions are). This map is universal with respect to all Cannon-Thurston maps coming from type-preserving Kleinian representations without accidental parabolics. In the case of closed surfaces, this map was constructed in joint work with Mj and Schleimer, and in this talk I will talk about an extension to the case of punctured surfaces obtained in joint work with Gultepe and Pho-On. The proof for punctured surfaces involves constructing a continuous equivariant map to the Gromov boundary of a "larger" complex called the surviving complex. I will describe this complex, its Gromov boundary, and the construction of the map.
Title: Cusped Hitchin representations
Title: Parameter spaces for families of transcendental functions.
Abstract: This lecture is based on joint work with Tao Chen, Nuria Fagella and Yunping Jiang. It is part of a more general program in complex dynamics to understand parameter spaces of transcendental maps.
A function {f_x(z), x in X a complex manifold, z in C}, defines a complex dynamical system for each x. The long term behavior of the orbits f_x^n(z) divides the dynamic plane C into regions where this behavior is predictable and where it is chaotic. We also study how the structure of the dynamic plane depends on x. Is there a ``natural embedding'' of X or ``good parameters'' that exhibit this dependence.
For rational functions, the natural embedding is given by the coefficients of the polynomials defining them.
For transcendental functions, it isn't always obvious what a good embedding is. In this talk,
we will look at two examples of reasonably general families of transcendental meromorphic functions where one can find good parameters using some topological and geometric ideas.
We will see at the end how these examples fit into a larger program.
Title: SLE, energy duality, and foliations by Weil-Petersson quasicircles
Abstract: Loewner's equation encodes a Jordan curve into a real-valued driving function. The Loewner energy of a Jordan curve is defined as the Dirichlet energy of its driving function and was first motivated by the study of small-parameter asymptotic behaviors of random fractal curves: Schramm-Loewner evolution (SLE). It was shown that the Loewner energy is finite if and only if the curve is a Weil-Petersson quasicircle, an interesting class of Jordan curves appearing in Teichmuller theory, geometric function theory, and string theory with currently more than 20 equivalent definitions. In this talk, I will show that the large-parameter large deviations of SLE gives rise to a new Loewner-Kufarev energy, which is dual to the Loewner energy via foliations by Weil-Petersson quasicircles and exhibits remarkable features and symmetries.
References:
[1] Large deviations of radial SLE infinity, Morris Ang, Minjae Park, Yilin Wang
Electron. J. Probab., Vol. 25, paper no. 102, 1-13 (2020)
https://arxiv.org/abs/2002.02654 [arxiv.org]
[2] The Loewner-Kufarev energy and foliations by Weil-Petersson quasicircles, Fredrik Viklund, Yilin Wang (2020)
https://arxiv.org/abs/2012.05771 [arxiv.org]
Title: Volume bounds for collections of simple closed curves
Abstract: Consider a surface S with negative Euler characteristic and a filling system of primitive simple closed curve G. If one drills, at different levels, G from the trivial bundle M we obtain a hyperbolic manifold M_G. We will investigate volume bounds for this setup and the case of the projective bundle and canonical lifts. In particular, we will obtain asymptotics relating the volume to formulas involving pants distances of the associated collection G.
Title: Presentations for cusped arithmetic hyperbolic lattices
Abstract: We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice Gamma, applying a classical result of Macbeath to a suitable Gamma-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups PU(2,1,O_d) for d=1,3,7 and the quaternion-hyperbolic lattice PU(2,1,H) with entries in the Hurwitz integer ring H. This is joint work with Alice Mark.
Title: Geodesic flow and the Fenchel-Nielsen coordinates
Abstract: This is a joint work with A. Basmajian and H. Hakobyan.
A Riemann surface is of parabolic type iff the geodesic flow is ergodic iff its Poincare series diverges iff Brownian motion is recurrent iff the limit set of a qc deformation is either a circle or it has Hausdorff dimension greater than 1. These and other equivalent conditions are obtained in the work of many authors.
The type problem (which is notoriously difficult) is to determine when an explicitly given Riemann surface is of parabolic type. Various constructions and criteria were classically considered in the geometric function theory. More recent emphasis was to determine under which conditions regular covers of compact Riemann surfaces are parabolic. This problems was solved by Mori and Rees (in a more general context) in the case of Abelian covers. Lyons, McKean and Sullivan proved that Z^2 covers of a thrice punctured sphere is not parabolic, and Rees gave a full characterization for the Abelian covers of finite punctured surfaces.
We consider the set of all infinite surfaces and pose the question of deciding when a Riemann surface is of parabolic type. We “parametrize" the set of all Riemann surfaces by the use of Fenchel-Nielsen coordinates and consider parabolicity in terms of these coordinates. Our methods are that of modulus of a curve family. We introduce a non-standard collar around cuffs of a pants decomposition used to define the Fenchel-Nielsen. Then we give a sufficient condition for parabolicity in terms of divergence of a series whose terms involve the lengths and twists of the cuffs. Note here that our series has significantly less terms than the Poincare series. The sufficient condition that we obtained is also necessary and (we believe) close to necessary in many cases.
Title: Blown up Jacobians of Riemann surfaces with nodes and twists at the nodes.
Abstract: In joint work with Eran Makover, Bjoern Muetzel and Robert Silhol we study the Jacobians of families of compact Riemann surfaces of genus g > 1 in which a geodesic is pinched to zero. When the geodesic separates then the Jacobians converge to the direct product of the Jacobians of the two limit surfaces whose genera add up to g. When the geodsic does not separate then the limit surface, S, has genus g-1 and its Jacobian is not a limit. However one may understand S as a genus g surface with a node. We shall mark S with a twist parameter at the node and propose a concept of blown up Jacobian for S with this marking.
The approach is via the energy distribution of harmonic 1 forms along thin handles. The Jacobians are represented by Gram period matrices associated to a homology basis of harmonic 1 forms. We shall, however, never see these forms explicitly, just estimates of their energy distributions.
(Mathematische Zeitschrift, 297(3), 1899-1952)
Title: Combinatorial Growth in the Modular Group (Postponed until Fall 2021)
Abstract: Consider the modular surface, that is, the (2,3, \infty) triangle orbifold. A reciprocal geodesic on the modular surface is a closed geodesic that begins and ends at the order two cone point, traversing its image twice. In this talk we consider an exhaustion of the modular surface by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so called low lying reciprocal geodesics) converges to the growth rate of the full set of reciprocal geodesics on the modular surface. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics. This is joint work with Ara Basmajian.
Title: Hyperbolic 3-manifolds with non-integral trace.
Abstract: A basic consequence of Mostow-Prasad Rigidity is that if
M=H^3/G is an orientable hyperbolic 3-manifold of finite volume, then
the traces of the elements in G are algebraic numbers.
Say that M has non-integral trace if G contains an element whose trace
is an algebraic non-integer. This talk will consider manifolds with
non-integral trace and show for example, that there are infinitely
many non-homeomorphic hyperbolic knot complements S^3\ K_i with
non-integral trace. We will also discuss some open problems about such
manifolds.