We meet Thursdays at 2:50–3:50 pm in Whitney Hall 100E. This semester's organizer is Cary Malkiewich. 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.

• August 31st
  No seminar this week

• September 7th
  No seminar this week (cancellation)

• September 14th
  Speaker: John Rached (Binghamton)
  Title: Quantitative behavior of horocycle flow on the moduli space of genus 2 surfaces

  Abstract: The action of SL(2,R) on moduli space exhibits measure rigidity, analogously to Ratner’s theorems for unipotent flows on homogeneous spaces, due to the seminal work of Eskin-Mirzakhani. Similar results cannot hold for the horocycle flow on moduli space, but for special subvarieties of strata (eigenform loci), some key tools from homogeneous dynamics have an incarnation in this inhomogeneous setting. A version of Ratner’s theorem holds for eigenform loci, and a flurry of recent work on quantitative results for actions on homogeneous spaces begs a natural question - can one effectivize arguments for the horocycle flow on eigenform loci? We give some support for a positive answer to this question, and make some conjectures.
  

• September 21st
  No seminar this week (cancellation)

• September 28th
  Speaker: Maxine Calle (University of Pennsylvania)
  Title: Nested cobordisms and TQFTs

  Abstract: The folk theorem identifying 2-dimensional TFQTs with Frobenius algebras is a starting point for a lot of interesting mathematics, from mathematical physics to homotopy theory to higher category theory. In this talk, we will explore what happens if we replace the cobordism category with a category of nested cobordisms, where 2-dimensional surfaces may have embedded 1-dimensional submanifolds, and what kind of algebraic structure the corresponding nested TQFTs pick out. This is based on ongoing work joint with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.
  

• October 5th
  Speaker: Cary Malkiewich (Binghamton)
  Title: Higher scissors congruence

  Abstract: Hilbert's Third Problem asks for sufficient conditions that determine when two polyhedra in three-dimensional Euclidean space are scissors congruent. Classically, the attempts to solve this problem (in this and other geometries) lead into group homology and algebraic K-theory, in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that the presence of K-theory here is not ad-hoc, but is integral to the definition of scissors congruence itself. This leads to a natural notion of higher scissors congruence groups, in which the 0th group is the classical one that determines the answer to Hilbert's Third Problem.

  In this talk, I'll describe a surprising recent result that these higher groups arise from a Thom spectrum. Its base space is the homotopy orbit space of a Tits complex, and the vector bundle is the negative tangent bundle of the underlying geometry. Using this result, we can explicitly compute the higher scissors congruence groups for the one-dimensional geometries, and give exact sequences that express them for the two-dimensional geometries. Much of this is joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.
  

• October 12th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• October 19th
  No seminar this week (Fall break)

• October 26th
  Speaker: Brenda Johnson (Union College)
  Title: TBA

  Abstract: TBA
  

• November 2nd
  Speaker: Paul Apisa (University of Wisconsin)
  Title: Hurwitz Spaces, Hecke Actions, and Orbit Closures in Moduli Space

  Abstract: The moduli space of Riemann surfaces is a space whose points correspond to the ways to endow a surface with a hyperbolic metric or, equivalently, complex structure. Geodesic flow on moduli space can be used to generate an action of GL(2, R) on its cotangent bundle. While work of Eskin, Mirzakhani, Mohammadi, and Filip implies that GL(2, R) orbit closures are varieties, the question of which ones occur is wide open. Aside from two well-understood constructions (taking loci of branched covers and subloci of rank two orbit closures) there are only 3 known families of orbit closures: the Bouw-Moller curves, the Eskin-McMullen-Mukamel-Wright (EMMW) examples, and 2 sporadic examples. Building on ideas of Delecroix-Rueth-Wright, I will describe work showing that the Bouw-Moller and EMMW examples can be constructed using just the representation theory of finite groups. The main idea is to connect these examples to Hurwitz spaces of G-regular covers of the sphere for an appropriate finite group G. In the end, I will describe a construction that inputs a finite group G and a set of generators satisfying a combinatorial condition and outputs a GL(2, R) orbit closure in moduli space.
  

• November 9th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• November 16th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• November 23rd
  No seminar this week (Thanksgiving)

• November 30th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• December 7th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• January 18th
  Organizational meeting

• January 25th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• February 1st
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• February 8th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• February 15th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• February 22nd
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• February 29th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• March 7th
  No seminar this week (spring break)

• March 14th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• March 21st
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• March 28th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• April 4th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• April 11th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• April 18th
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA
  

• April 25th
  No seminar this week (Tuesday classes meet)

• May 2nd
  Speaker: TBA (Institution)
  Title: TBA

  Abstract: TBA