We meet Thursdays at 2:45–3:45 pm in Whitney Hall 100E. This semester's organizers are James Hyde and Lorenzo Ruffoni. The seminar has an announcement mailing list open to all.

Topics include: geometric group theory, differential geometry and topology, low-dimensional topology, algebraic topology, and homotopy theory.

• January 29h
  Speaker: Juliet Aygun (Cornell)
  Title: Counting geodesics on prime-order k-differentials

  Abstract: It has been of popular interest over the last several decades to count geodesics with respect to their length on flat surfaces. Asymptotics of these counting functions for generic translation surfaces, which are Riemann surfaces with a holomorphic one form, have been determined by the pioneering work of Eskin-Masur-Zorich. There is a more general type of flat surface called a (1/k)-translation surface, which is a Riemann surface with a k-differential. Equivalently, a (1/k)-translation surface is a collection of polygons in the complex plane with sides identified pairwise by translation and possible rotations of 2pi/k. In this talk, we will discuss asymptotics of these counting functions on generic (1/k)-translation surfaces when k is prime and genus is more than two. The main tools I will discuss are GL+(2,R)-orbit closures and a result of Eskin-Mirzakhani-Mohammadi which relates asymptotics to GL+(2,R)-orbit closures.
  

• February 5th
  Speaker: Barry Minemyer (Commonwealth University - Bloomsburg)
  Title: Negatively Curved Metrics on Complex Hyperbolic Branched Covers

  Abstract: Gromov and Thurston used hyperbolic branched cover manifolds to construct the first known examples of compact manifolds which admit a pinched negatively curved metric, but do not admit a hyperbolic metric. Fine and Premoselli (n=4) and Hamenstadt and Jackel (n > 4) later used these same manifolds to construct the first known examples of negatively curved Einstein metrics (in these respective dimensions) that are not locally symmetric.

  Recently, Stover and Toledo proved that analogous complex hyperbolic branched cover manifolds exist. They also proved that these manifolds do not admit a locally symmetric metric, and a result of Zheng shows that these manifolds are Kahler. In this talk I will present recent work proving the existence of pinched negatively curved metrics, as well as the existence of negatively curved Kahler-Einstein metrics (due to Guenancia and Hamenstadt) on these complex hyperbolic branched cover manifolds. Part of my presented work is joint with Lafont.
  

• February 12th
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• February 19th
  Speaker: Satya Howladar (University of Florida)
  Title: Gromov’s Conjecture for Product of Baumslag-Solitar Groups and some other One relator groups

  Abstract: Gromov introduced macroscopic dimension of metric spaces in order to study large scale properties of manifolds. He conjectured that a closed $n$-manifold which admits Positive Scaler Curvature metric, should have its universal cover to be of macroscopic dimension at most $n-2$, with respect to the pull back metric on it. This conjecture depends a lot on the fundamental group of the base manifold. For $n>4$, closed spin $n$-manifolds $M$, we developed sufficient condition on $\pi_1(M)$, to verify the conjecture. When $\pi_1(M)$ is product of $2$-dimensional groups (i.e. groups with classifying space a $2$-dimensional CW complex), $\mathbb Z_2$-summands in their homology creates a problem for application of our technique. We could resolve this in the case of certain one-relator groups, including Baumslag-Solitar, and certain others, by passing to some finite index subgroup of them not admitting $\mathbb Z_2$-torsion in homology. This is done by the well-known technique of Fox calculus, to analyze boundary maps of cells of finite index covers. I will try to revisit this technique and sketch a proof our result.
  

• February 26th
  Speaker: Lucas Williams (Binghamton University)
  Title: TBA

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• March 5th
  Speaker: Oliver Wang (University of Virginia)
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• March 12th
  Speaker: Sofia Martinez (Bryn Mawr College)
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• March 13th
  PETER HILTON MEMORIAL LECTURE
  SPECIAL TIME AND LOCATION: March 13, 3:30pm, Alumni Lounge at Old O'Connor Hall
  Speaker: Martin Bridson (University of Oxford)
  Title: Chasing finite shadows of infinite groups through geometry

  Abstract: There are many situations in geometry or elsewhere in mathematics where it is natural or convenient to explore infinite groups of symmetries via their actions on finite objects. But how hard is it find these finite manifestations and to what extent does the collection of all such actions determine the infinite group?

  In this colloquium, I will sketch some of the rich history of such problems and then describe some of the great advances in recent years. I'll describe pairs of distinct groups that have the same finite images and I'll sketch the proof of some “profinite rigidity results”, i.e. theorems showing that in certain circumstances one can identify an infinite group if one knows its set of finite images.
  

• March 19th
  Speaker: Francesco Lin (Columbia)
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• March 26th
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• April 2nd

  (Spring break - no seminar)
  

• April 9th
  Speaker: Sanjana Agarwal (Indiana University, Bloomington)
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• April 16th
  Speaker: Josefien Kuijper (University of Toronto)
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• April 23th
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• April 30th
  Speaker: Urshita Pal (University of Michigan)
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