It is a truth universally acknowledged, that a local system on a connected topological manifold is completely determined by its attached monodromy representation of the fundamental group. Similarly, lisse ℓ-adic sheaves on a connected variety determine and are determined by representations of the profinite étale fundamental group. Now if one wants to classify constructible sheaves by representations in a similar manner, new invariants arise. In the topological category, this is the exit path category of Robert MacPherson (and its elaborations by David Treumann and Jacob Lurie), and since these paths do not ‘run around once’ but ‘run out’, we coined the term exodromy representation. In the algebraic category, we define a profinite ∞-category – the étale fundamental ∞-category – whose representations determine and are determined by constructible (étale) sheaves. We describe the étale fundamental ∞-category and its connection to ramification theory, and we summarise joint work with Saul Glasman and Peter Haine.

One of the fundamental themes of geometric group theory is to
view finitely generated groups as geometric objects in their own right,
and to then understand to what extent the geometry of a group determines
its algebra. A theorem of Stallings says that a finitely generated group
has more than one end if and only if it splits over a finite subgroup.
In this talk, I will explain an analogous geometric characterisation of
when a group admits a splitting over certain classes of infinite subgroups.

The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of S^3 along a knot, defined by Lipshitz, Hendricks, and Sarkar, 
is an isomorphism class of F_2[\theta]-modules. In this talk, we show that this invariant is natural, and is functorial under based cobordisms. 
Given a transverse knot K in the standard contact 3-sphere, we define an element of the Z/2-equivariant Heegaard Floer cohomology 
that depends only on the tranverse isotopy class of K, and is functorial under certain symplectic cobordisms.

Rips filtrations over a finite metric space and their corresponding persistent homology are prominent methods in Topological Data Analysis to summarize the ``shape'' of data. For finite metric space $X$ and distance $r$  the traditional Rips complex with parameter $r$ is the flag complex whose vertices are the points in $X$ and whose edges are $\{[x,y]: d(x,y)\leq r\}$. From considering how the homology of these complexes evolves we can create persistence modules (and their associated barcodes and persistence diagrams). Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric space then the bottleneck distance between persistence modules constructed by the Rips filtration is bounded by $2d_{GH}(X,Y)$ (where $d_{GH}$ is the Gromov-Hausdorff distance). Using the asymmetry of the distance function we construct four different constructions analogous to the persistent homology of the Rips filtration and show they also are stable with respect to the Gromov-Hausdorff distance. These different constructions involve ordered-tuple homology, symmetric functions of the distance function, strongly connected components and poset topology.

McKean-Vlasov SDEs are SDEs where  the coefficients depend on the law of the solution to the SDE. Their interest is in the links with nonlinear PDEs on one side (the SDE-related Fokker-Planck equation is nonlinear) and with interacting particles on the other side: the McKean-Vlasov SDE be approximated by a system of weakly coupled SDEs. In this talk we consider McKean-Vlasov SDEs with irregular drift: though well-posedness for this SDE is not known, we show a large deviation principle for the corresponding interacting particle system. This implies, in particular, that any limit point of the particle system solves the McKean-Vlasov SDE. The proof combines rough paths techniques and an extended Vanrdhan lemma.

This is a joint work with Thomas Holding.

 Consider dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation norm, 1<= p < 2, and f is a Lipchitz continuous function. Following the geometric solution construction of Levy-driven stochastic differential equations in (Williams 2001), we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both the geometric solution Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the uniform metric on compact time intervals with probability 1. Moreover, the users can adaptively choose epsilon’ < epsilon, so that Y_epsilon’ can be constructed conditional on Y_epsilon. This tolerance-enforcement feature allows us to easily combine our algorithm with Multilevel Monte Carlo for efficient estimation of expectations, and adding as a benefit a straightforward analysis of rates of convergence. This is joint with Jose Blanchet, Fei He and Offer Kella.

We present a support theorem for subcritical parabolic stochastic partial differential equations (SPDEs) driven by Gaussian noises. In the spirit of the classical theorem by Stroock and Varadhan for ordinary stochastic differential equations, we identify the support of the solution to singular SPDEs with the closure of the union of the support of solutions to approximate and renormalized equations. We implement our approach in the setting of regularity structures and obtain a general result covering a range of singular SPDEs (including $\Phi^4_3$, $\Phi^d_2$, KPZ, PAM (2D+3D), SHE, ...). As a Corollary to our result we obtain the uniqueness of invariant measures for various interesting SPDEs. This is a joint work with Martin Hairer.

We present a numerical investigation of stochastic transport for the damped and driven incompressible 2D Euler fluid flows. According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles of transformation theory and multi-time homogenisation, respectively, imply a physically meaningful, data-driven approach for decomposing the fluid transport velocity into its drift and stochastic parts, for a certain class of fluid flows. We develop a new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretisation. We show our numerical method is consistent.
Numerically, we perform the following analyses on this velocity decomposition. We first perform uncertainty quantification tests on the Lagrangian trajectories by comparing an ensemble of realisations of Lagrangian trajectories driven by the stochastic differential equation, and the Lagrangian trajectory driven by the ordinary differential equation. We then perform uncertainty quantification tests on the resulting stochastic partial differential equation by comparing the coarse-grid realisations of solutions of the stochastic partial differential equation with the ``true solutions'' of the deterministic fluid partial differential equation, computed on a refined grid. In these experiments, we also investigate the effect of varying the ensemble size and the number of prescribed stochastic terms. Further experiments are done to show the uncertainty quantification results "converge" to the truth, as the spatial resolution of the coarse grid is refined, implying our methodology is consistent. The uncertainty quantification tests are supplemented by analysing the L2 distance between the SPDE solution ensemble and the PDE solution. Statistical tests are also done on the distribution of the solutions of the stochastic partial differential equation. The numerical results confirm the suitability of the new methodology for decomposing the fluid transport velocity into its drift and stochastic parts, in the case of damped and driven incompressible 2D Euler fluid flows. This is the first step of a larger data assimilation project which we are embarking on. This is joint work with Colin Cotter, Dan Crisan, Darryl Holm and Igor Shevchenko.

 

I will talk about recent work that uses recent ideas from stochastic analysis to develop robust and non-parametric statistical tests for stochastic processes.