## The Washington University in St. Louis Geometry and Topology Seminar

Organizer: Michael Landry
Time: 4-5 pm in 199 Cupples I, unless otherwise specified.

Spring 2020:
Date Speaker Affiliation Title and abstract
1/17 Thomas Koberda University of Virginia Arithmetic groups, thin groups, and commensurators
Arithmetic groups are an important class of lattices in Lie groups which are of interest from a dynamical, geometric, and number theoretic perspective. These groups were characterized among lattices in a purely intrinsically algebraic way, by a famous result of Margulis. I will survey some of the ideas surrounding arithmetic groups and Margulis' theorem, and then move on to a discussion of thin groups. Thin groups are certain discrete subgroups of Lie groups which occur naturally in many contexts in mathematics, from number theory and spectral theory to quantum computing. Thin groups have much less structure than lattices, though they seem to follow some organizational principles analogous to Margulis' theorem. I will survey some recent results in this direction.
1/24 MurphyKate Montee University of Chicago Cubulating random groups at density d<3/14
For random groups in the Gromov density model at d<3/14, we construct walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities d<1/5 and d<5/24, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density d<1/4.
1/31 James Farre Yale University Flat and hyperbolic geometry of surfaces
The uniformization theorem tells us that the deformation space of constant curvature metrics on a surface also describes the moduli space of its complex structures. Mirzakhani proved that two flows, namely Thurston’s earthquake flow from hyperbolic geometry and Teichmüller horocycle flow on the principal stratum of quadratic differentials from complex analysis, are measurably isomorphic. We will spend some time reviewing some constructions coming from hyperbolic geometry and some coming from complex analysis to explain Mirzakhani’s correspondence. Then, we will introduce a new coordinate system for Teichmüller space that allows us to extend Mirzakhani’s conjugacy to other strata of quadratic differentials, answering a question of Alex Wright. This is joint work (in progress) with Aaron Calderon.
2/7 Gong Cheng WUSTL Bilinear Systems on Lie Groups: An Example of Geometric Control Theory
The study of bilinear system is one of the central topics in nonlinear control and geometric control theory. Investigating fundamental properties of such systems has been a prosperous and continuing research topic for decades, but still remains challenging. In this talk, I'll introduce the general theory of geometric control, its connection to differential geometry, and discuss our recent work on the controllability of bilinear systems on $SO(n)$. This is joint work with Wei Zhang and Jr-Shin Li.
2/14 Jintao Deng Texas A&M University The Novikov conjecture and group extensions
The Novikov conjecture is an important problem in higher dimensional topology. It claims that the higher signatures of a compact smooth manifold are invariant under orientation preserving homotopy equivalences. The Novikov conjecture is a consequence of the strong Novikov conjecture in the computation of the K-theory of group C^*-algebras. In this talk, I will talk about the Novikov conjecture for groups which are extensions of coarsely embeddable groups.
2/21 Rafael Potrie CMAT
2/28 Alex Rasmussen Yale University
3/6
3/20 Tyrone Crisp University of Maine
3/27 Aaron Calderon Yale University
4/3 Shintaro Nishikawa Pennsylvania State University
4/10 Henry Segerman Oklahoma State University
4/17
4/24 Daniel Cristofaro-Gardiner IAS/UC Santa Cruz

Fall 2019:
Date Speaker Affiliation Title and abstract
9/13 Michael Landry WUSTL Taut surfaces in hyperbolic 3-manifolds
Abstract: Let M be a closed hyperbolic three-manifold with nonzero first Betti number. The second homology group of M with real coefficients has an interesting norm that people now call the Thurston norm. I will introduce this norm and discuss what it can tell us about so-called taut surfaces embedded in M and their interaction with certain dynamical systems in M called pseudo-Anosov flows. Time permitting, we will discuss some new results which are joint work with Samuel Taylor and Yair Minsky. My goal is for this to be a self-contained talk providing a friendly introduction to the kinds of things I think about for people in the department. I am happy to cover less material than planned if the audience has a lot of questions.
9/20
9/27 Didac Martinez-Granado Indiana University, Bloomington From curves to currents
Abstract: Geodesic currents are measures introduced by Bonahon in 1986 that realize a suitable closure of the space of closed curves on a surface. They are analogous to measured laminations for simple closed curves. Many geometric structures on surfaces, such as hyperbolic structures or half-translation structures can be realized as geodesic currents. Bonahon proved that the notion of hyperbolic length for curves extends to geodesic currents. Since then, many other functions defined on the space of curves have been proven to extend to currents, such as negatively curved lengths, lengths from singular flat structures or stable lengths for surface groups. In this talk, we explain how a function defined on the space of curves satisfying some simple conditions can be extended continuously to geodesic currents. The most important of these is that the function decreases under smoothing of essential crossings. Our theorem subsumes previous extension results. Furthermore, it extends functions that had not been considered before, such as extremal length. This is joint work with Dylan Thurston.
10/4 Eric Samperton UIUC Coloring invariants of knots and links are often intractable
Abstract: I’ll give an overview of my result with Greg Kuperberg concerning the computational complexity of G-coloring invariants of knots, where G is a finite, simple group. We have a similar theorem for closed 3-manifolds. I’ll try to give a sense of the commonalities of the two proofs (e.g. “reversible computing with a combinatorial TQFT”), as well as where they differ (there’s some interesting algebraic topology that needed developing in the knot case).
10/11 Jorge Plazas Pontificia Universidad Javeriana Automorphic functions via noncommutative geometry
In this talk we discuss noncommutative spaces arising naturally in number theory. Considering elliptic curves together with their torsion structure leads to the notion of Q-lattice introduced by Connes and Marcolli. The relation of commensurability among Q-lattices gives rise to a quantum statistical mechanical system with a rich structure encoding the arithmetic of automorphic functions. Looking at automorphic functions from the vantage point of noncommutative geometry provides new tools and insights for their study. We will discuss the Connes-Marcolli system in relation to the "big picture", a combinatorial gadget introduced by Conway for the study of congruence groups. We will also explore the relevance of this setting for the study of principal moduli.
10/18 **Special time/location** Paul Baum Penn State Note: 3-4p in room 6
Index theory on odd-dimensional manifolds
On a closed (i.e. compact, no boundary) odd-dimensional C^{\infty} manifold, the index of any elliptic differential operator is zero. Thus, on odd-dimensional manifolds, interesting examples can only be obtained by dropping either elliptic" or differential". This talk will explain the two cases. If differential" is dropped, then the examples are Toeplitz operators (which are elliptic pseudo-differential operators of order zero). A corollary is the result of Louis Boutet de Monvel about Toeplitz operators associated to the boundary of a strictly pseudo-convex domain. If elliptic" is dropped, then examples are the differential operators on contact manifolds studied by E. van Erp and Baum-van Erp.
10/18 Lvzhou Chen University of Chicago Spectral gap of stable commutator length in graphs of groups and 3-manifolds
The stable commutator length (scl) is a relative version of the Gromov-Thurston norm. It is a group invariant sensitive to geometric and dynamical properties. Many groups that act on non-positively curved spaces are known to have a spectral gap in scl, which is a homological analog of margulis constants for negatively curved manifolds. We will discuss sharp estimates of spectral gaps for groups acting on trees using a topological method. This yields a gap for any 3-manifold group and a new proof for the optimal gap 1/2 by Heuer on all right-angled Artin groups. This is joint work with Nicolaus Heuer.
10/23 **Special time/location** Nikhil Savale Universität zu Köln Note: 3-4p in room 6
Spectrum and abnormals in sub-Riemannian geometry: the 4D quasi-contact case
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry.
10/25 Hao Guo Texas A&M University Coarse Geometry and Callias Quantization
I will report on some recent joint work with Peter Hochs and Mathai Varghese at the University of Adelaide on index theory of a class of elliptic operators on non-compact manifolds that are invertible outside of a small set, with respect to some group of symmetries on the manifold. The index we study takes values in the maximal C*-algebra of this group. Our approach uses a maximal version of the localised Roe algebra. The advantage of using the maximal version of the index is that it admits a natural trace defined by integration. As a motivating application, I will mention a version of Guillemin and Sternberg’s quantization commutes with reduction principle for equivariant indices of spin-c Callias-type operators.
11/1 Ying Hu University of Nebraska at Omaha Euler class of taut foliations and Dehn filling
We will discuss the Euler class of co-orientable taut foliations on rational homology spheres. Given a rational homology solid torus X, we give a necessary and sufficient condition for the Euler class of taut foliations on Dehn fillings of X that are transverse to the core of the filling solid torus to vanish, from which restrictions on the range of the filling slopes are derived. We will also discuss more specific examples of taut foliations as well as the implications of our results regarding the L-space conjecture.
11/8 Peter Hochs University of Adelaide An equivariant Atiyah-Patodi-Singer index theorem for proper actions
Abstract: The Atiyah-Singer index theorem is an equality between an analytic quantity on a compact manifold without boundary, and a topological one. This has various implications in mathematics and physics. Atiyah, Patodi and Singer (APS) generalised this to compact manifolds with boundary. This motivated their invention of the eta-invariant, which has been used in many places since then. An interesting feature of the APS index theorem is that the topological side consists of two terms, coming from the interior and the boundary of the manifold, that depend on the local geometry individually, but become topological invariants when they are added together. This talk is about work with Bai-Ling Wang and Hang Wang on an equivariant extension of the APS index theorem, for possibly noncompact groups acting on possibly noncompact manifolds with boundary.
11/15 Jianchao Wu Texas A&M University The rational strong Novikov conjecture, groups of diffeomorphisms, and symmetric Hilbert-Hadamard spaces
The rational strong Novikov conjecture is a deep problem in noncommutative geometry. It implies important conjectures in manifold topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that the rational strong Novikov conjecture holds for any discrete group admitting an isometric and proper action on an admissible Hilbert-Hadamard space, which is a (typically infinite-dimensional) generalization of complete simply connected nonpositively curved Riemannian manifolds. In particular, a prominent example of an admissible Hilbert-Hadamard space is the space of L^2-Riemannian metrics on a smooth manifold with a fixed density. This space can be viewed as an infinite-dimensional symmetric space. As a result, our result implies the rational strong Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This is joint work with Sherry Gong and Guoliang Yu.
11/22
12/6 Michael Deutsch Universidade Federal do Rio de Janeiro Equivariant differential operators and contact geometry
Given a group G of automorphisms of a complex manifold M, effective methods to construct meromorphic functions on M which are invariant under the action of G are classical, dating back to the 19th century in the case of complex curves. In this talk we describe a method to produce a more general class of functions, those which are *equivariant* with respect to a projective representation of G. In the 1-dimensional case, we use hyperbolic geometry to extend an idea of Doyle and McMullen and obtain an equivariant analogue of the classical Schwarzian derivative. This construction leads via contact geometry to a kind of Cayley transform parametrizing the full monoid of equivariant differential operators, the action of which generates arbitrary equivariant functions from a given one. This in turn generalizes to algebra-valued functions in arbitrary dimension, yielding what appear to be the first examples of such functions in the multi-variable case when the image of G is non-compact.