Hyperbolic Geometry Seminar
Location
Starting in Spring 2023 the seminar will go back to in-person talks. We will also attempt to run the seminar as hybrid using zoom. Zoom doors will open a few minutes before the seminar 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 2:45pm–3:45pm, room 5382.
Organizers
Ara Basmajian (CUNY, Graduate Center and Hunter College)
Email: abasmajian@gc.cuny.edu
Dragomir Saric (CUNY, Graduate Center and Queens College)
Email: dragomir.saric@qc.cuny.edu
Nick Vlamis (CUNY, Graduate Center and Queens College)
Email: nvlamis@gc.cuny.edu
Spring 2024 Schedule (room 5382)
February 6: Meenakshy Jyothis (Binghampton U.)
Title: Automorphisms of geodesic currents preserve intersection form
Abstract: We will discuss progress in proving Ivanov’s meta conjecture in the context of geodesic currents. Ivanov’s meta conjecture says that every object naturally associated with a surface and having a ’sufficiently rich’ structure has the mapping class group as its group of automorphisms. The conjecture has been proven for various combinatorial objects associated with a surface as well as for the Teichmüller space of a surface. The space of geodesic currents contains many of these structures, such as the set of closed curves up to homotopy and the Teichmüller space. We discuss progress in showing Ivanov’s meta conjecture for a natural group of automorphisms of currents.
February 13: Chaitanya Tappu (Cornell University) [Cancelled due to inclement weather]
Title: A Moduli Space of Marked Hyperbolic Structures for Big Surfaces
Abstract: We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichmüller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.
February 20: (No seminar)
Colloquium talk 2:00-3:00pm by Alexander Gamburd (Graduate Center).
February 27: Pat Hooper (City College and Graduate Center)
Title: Geodesic representatives on surfaces without metrics
Abstract: A translation surface is a singular geometric structure on a surface modeled on the plane where transition maps are translations. Some recent research has focused on extending results known for translation surfaces to dilation surfaces, where we broaden allowable transition maps to include dilations of the plane. Such surfaces do not have natural metrics; however, one can ask: “Are natural analogs of geodesic representatives in this context?” Relatedly, translation surfaces which are not closed (e.g., infinite genus surfaces) may or may not have geodesic representatives in every homotopy class. We will describe conditions on surfaces that guarantee that canonical representatives of homotopy classes of curves exist. In doing so, we realize that even less structure is needed: we describe a class of geometric structures on surfaces that are not modeled on the plane at all, but still have canonical curve representatives. This is joint work with Ferrán Valdez and Barak Weiss.
March 5: Dragomir Saric (Queens College and Graduate Center)
Title: A Dichotomy for Geodesic Flows and finite-area Holomorphic Quadratic Differentials
Abstract: Let $X$ be an infinite Riemann surface with a conformally hyperbolic metric. E. Hopf proved that almost every geodesic is recurrent or almost every geodesic is transient. In the first case, the geodesic flow is ergodic; in the second, it is not. Hopf posed the question of finding geometric conditions on $X$ to detect which of the two conditions holds for $X$. Hopf-Tsuji-Sullivan theorem states that the geodesic flow is ergodic iff the Poincare series is divergent iff the Brownian motion is recurrent, and many other equivalent conditions are given in the literature. We added an equivalent condition: the Brownian motion on $X$ is recurrent iff almost every leaf of every finite-area holomorphic quadratic differential is recurrent.
From now on, assume that $X$ is equipped with a geodesic pants decomposition whose cuffs are bounded. We parametrize the space of finite-area holomorphic quadratic differentials on $X$. Namely, a holomorphic quadratic differential is determined by the intersection numbers of its horizontal foliation with the cuffs (and “adjoint cuffs”). We establish that finite-area holomorphic quadratic differentials are in one-to-one correspondence with the intersection numbers that are square summable. Using this parametrization, we establish that the Brownian motion on $X$ is recurrent iff the random walk on the pants graph is recurrent.
March 12: Brandis Whitfield (Temple U.)
Title: Short curves of end-periodic mapping tori
Abstract: Let $S$ be a boundaryless infinite-type surface with finitely many ends and consider an end-periodic homeomorphism $f$ of S. The end-periodicity of $f$ ensures that $M_f$, its associated mapping torus, has a compactification as a $3$-manifold with boundary; and further, if $f$ is atoroidal, then $M_f$ admits a hyperbolic metric.
As an end-periodic analogy to work of Minsky in the finite-type setting, we show that given a subsurface $Y\subset S$, the subsurface projections between the ``positive" and ``negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of $Y$ as it resides in $M_f$.
In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic $3$-manifolds, and how these techniques may be used in the infinite-type setting.
March 19: Jing Tao (U. of Oklahoma)
Title: Taming tame maps of surfaces of infinite type
Abstract: A cornerstone in low-dimensional topology is the Nielsen-Thurston Classification Theorem, which provides a blueprint for understanding homeomorphisms of compact surfaces up to homotopy. However, extending this theory to non-compact surfaces of infinite type remains an elusive goal. The complexity arises from the behavior of curves on surfaces with infinite type, which can become increasingly intricate with each iteration of a homeomorphism. To address some of the challenges, we introduce the notion of tame maps, a class of homeomorphisms that exhibit non-mixing dynamics. In this talk, I will present some recent progress on extending the classification theory to such maps. This is joint work with Mladen Bestvina and Federica Fanoni.
March 26: (No Seminar)
April 2: Jane Gilman (Rutgers University)
Title: Palindromes in Teichmuller Theory and New Orders on the Rational Numbers
Abstract: It is well known that every primitive word in a rank two free group is conjugate to either a palindrome or the product of two palindromes. We present an enumeration scheme that gives each primitive as the unique palindrome in its conjugacy class or as a product of two unique palindromes that have already appeared in the enumeration scheme. We use this result to obtain some necessary and sufficient geometric discreteness conditions in PS L(2, C) (equivalently I som+ (H3 )). We define new orders on the positive rational numbers and pose some open questions about the new orders. This is joint work with L. Keen.
April 9: Chaitanya Tappu (Cornell University)
Title: A Moduli Space of Marked Hyperbolic Structures for Big Surfaces
Abstract: We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichmüller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.
April 16: (No seminar)
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April 23: (No seminar-Spring recess)
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April 30: (No seminar-Spring recess)
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May 7: Bill Goldman (U. of Maryland)
Double header with Einstein Chair Seminar. This talk will not be zoomed. Note the change in time and place.
1:30-2:30 (Part 1) Room 6417 GC.
2:45-3:45 (Part 2) Room 6417 GC.
Title: Dynamical systems arising from classification of geometric structures on manifolds
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May 14: Hrant Hakobyan (Kansas State U.)
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Related seminars:
History of the hyperbolic geometry seminar