Mazur observed that in many cases where an elliptic curve E has a non-trivial element C in its Tate-Shafarevich group, one can find another elliptic curve E' such that ExE' admits an isogeny that kills C. For elements of order 2 and 3 one can prove that such an E' always exists. However, for order 4 this leads to a question about rational points on certain K3-surfaces. We show how to explicitly construct these surfaces and give some results on their rational points.

This is joint work with Tom Fisher.
 

I will describe how to recast perturbative quantum gravity using non-perturbative techniques from conformal field theory, focussing on the case of N=4 super Yang-Mills theory. By resolving the degeneracy among double trace operators at large N we are able to bootstrap one-loop supergravity corrections from the OPE of the CFT.
 

A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle.  In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, where we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon.  This is consequence of a technical result about Liouville currents associated to nonpositively curved Euclidean cone metrics on surfaces.  In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of how one determines the shape of the polygon from its bounce spectrum.

Based on the notion of paracontrolled distributions, existence and uniqueness results are presented for rough convolution equations. In particular, this wide class of equations includes rough differential equations with possible delay, stochastic Volterra equations, and moving average equations driven by Lévy processes. The talk is based on a joint work with Mathias Trabs.

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.