Abstracts

Week of April 14, 2024

April 15, 2024
Group Actions, Geometry and Dynamics Closed geodesics and stability of negatively curved metrics 4:00pm -
KT205

The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.
Geometry, Symmetry and Physics Advances in flat space holography 4:30pm -
KT 217

I will review the highlights of Strominger’s program of celestial holography, focusing on emerging connections to twisted holography, topological strings and twistor theory. A running example of the talk will be holography for certain self-dual theories placed on an asymptotically flat, scalar-flat Kahler geometry known as Burns space. This is based on work done in collaboration with Kevin Costello and Natalie M. Paquette.
April 16, 2024
Geometry & Topology Short curves of end-periodic mapping tori 4:00pm -
KT 205

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.
April 17, 2024
Applied Mathematics Efficient Convergent Boundary Integral Methods for Slender Bodies 3:00pm -
LOM 214

The dynamics of active and passive filaments in viscous fluids is frequently used as a model for many complex fluids in biological systems such as: microtubules which are involved in intracellular transport and cell division; flagella and cilia which aid in locomotion. The numerical simulation of such systems is generally based on slender-body theory which give asymptotic approximations of the solution. However, these methods are low-order and cannot enforce no-slip boundary conditions to high-accuracy, uniformly over the boundary. Boundary-integral equation methods which completely resolve the fiber surface have so far been impractical due to the prohibitive cost of current layer-potential quadratures for such high aspect-ratio geometries. In this talk, I will present new quadrature schemes which make such computations possible and new integral equation formulations which lead to well-conditioned linear systems upon discretization. I will present numerical results to show the efficiency of our methods.
Colloquium Ghost polygons, Poisson bracket and convexity 4:00pm -
KT 207

The moduli space of Anosov representations of a surface group in a semisimple group admits many more natural functions than the regular functions including length functions and correlation functions. We consider the Atiyah-Bott/Goldman Poisson bracket for length functions and correlation functions and give a formula that computes their Poisson bracket. This is done by introducing a new combinatorial framework including ghost polygons and a ghost bracket encoded in a formal algebra called the ghost algebra. As a consequence, we show that the set of length and correlation functions is stable under the Poisson bracket and give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichmüller space, secondly we exhibit subalgebras of commuting functions associated to laminations. This is joint with François Labourie.
April 18, 2024
Geometry, Symmetry and Physics Classical Deformations of Celestial Symmetries 2:30pm -
KT801

This talk is based on arXiv:2305.09451, arXiv:2403.18011 and work in progress. I will discuss several deformations of algebras which are closely related to $w_{1+\infty}$ and give bulk interpretations of the respective deformations. Some of these deformations arise naturally from a backreaction in self-dual Einstein gravity analogous to part of the recent top-down construction of Costello, Paquette and Sharma and I will highlight similarities and differences.
Analysis Rigidity of the quintic, nonlinear Schrodinger equation 4:00pm -
KT 201
April 19, 2024
Friday Morning Seminar Solid-On-Solid is liquid (at least when thawed a little) 10:00am -
KT801

The (2+1)D Solid-On-Solid (SOS) model famously exhibits a roughening transition: on an N×N torus with the height at the origin rooted at 0, the variance of h(x), the height at a point x is O(1) when the inverse-temperature β is large, vs O(log |x|) when β is small. The rigidity at large β is believed to fail once the surface is on a slope (tilted boundary conditions), which ought to destabilize it and induce the log-correlated behavior of the small β regine. The only rigorous result on this is by Sheffield (2005): if the slope θ is irrational, then Var(h(x)) diverges with |x| (with no known quantitative bound).

We study this model at a large enough fixed β, on an N×N torus with a nonzero boundary condition slope θ, perturbed by a potential V of strength ε(β) per site (arbitrarily small). Our main result is (a) the measure on the height gradients ∇h has a weak limit μ as N→∞; and (b) the scaling limit of a sample from μ converges to a full plane GFF. In particular, we recover the asymptotics of Var(h(x)). To our knowledge, this is the first example of a random surface of the ∇ɸ family of models, or any perturbation of one, where the scaling limit is recovered at large finite β under tilted boundary conditions.

The proof looks at random monotone surfaces that approximate the SOS surface, and shows that (i) these form a weakly interacting dimer model, and (ii) the renormalization framework of Giuliani, Mastropietro and Toninelli (2017) can be applied to it, leading to the scaling limit.

Joint work with Benoît Laslier.