Some of the most studied examples of conformal field theories
include
the Wess-Zumino-Witten models. These are conformal field theories exhibiting
affine Lie algebra symmetry at non-negative integers levels. In this talk I
will
discuss conformal field theories exhibiting affine Lie algebra symmetry at
certain rational (hence fractional) levels whose structure is arguably even
more intricate than the structure of the non-negative integer levels,
provided
one is prepared to look beyond highest weight modules.

Hardy's axiomatic approach to quantum theory revealed that just one axiom
distinguishes quantum theory from classical probability theory: there should
be continuous reversible transformations between any pair of pure states. It
is the single word `continuous' that gives rise to quantum theory. This
raises the question: Does there exist a finite theory of quantum physics
(FTQP) which can replicate the tested predictions of quantum theory to
experimental accuracy? Here we show that an FTQP based on complex Hilbert
vectors with rational squared amplitudes and rational phase angles is
possible providing the metric of state space is based on p-adic rather than
Euclidean distance. A key number-theoretic result that accounts for the
Uncertainty Principle in this FTQP is the general incommensurateness between
rational $\phi$ and rational $\cos \phi$. As such, what is often referred to
as quantum `weirdness' is simply a manifestation of such number-theoretic
incommensurateness. By contrast, we mostly perceive the world as classical
because such incommensurateness plays no role in day-to-day physics, and
hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum
variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger
equations based on the rational differential calculus: one for rational
$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation
on the complex continuum that has led to such problems over the years. Based
on this splitting of the Schr\"{o}dinger equation, the measurement problem
is trivially solved in terms of a nonlinear clustering of states on $I_U$.
Overall these results suggest we should consider the universe as a causal
deterministic system evolving on a finite fractal-like invariant set $I_U$
in state space, and that the laws of physics in space-time derive from the
geometry of $I_U$. It is claimed that such a  deterministic causal FTQP will
be much easier to synthesise with general relativity theory than is quantum
theory.

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.