From contours of 2D shapes to knotted proteins, many areas of statistics and physical sciences are concerned with data in the form of geometric loops - embeddings of the circle into a Euclidean space. It is desirable to find a representation of the geometry of these loops that is invariant under Euclidean symmetries, and indifferent to how the loops are parametrised. In this joint work with Ka Man Yim (Oxford) and Otto Sumray (MPI Dresden), we propose a novel representation of a loop’s geometry by rep resenting the distance matrix of the loop as a Morse function on a Möbius band - the second finite subset space of the circle. By considering the persistent homology of this function, we obtain a representation of the loop’s geometry that has the desired symmetry invariance. We show that this feature map is stable with respect to perturbations, give a geometric interpretation of the features obtained and explore analogues involving higher order finite subset spaces.

TBA

Scale and conformal symmetries play a central role in various areas of theoretical physics, ranging from statistical mechanics and quantum field theory to string theory. A natural question is under what conditions scale invariance implies the larger symmetry of conformal invariance.  In this talk, I will discuss non-perturbative arguments for this enhancement in two and four dimensions, highlighting the assumptions under which they hold. I will first explain how Zamolodchikov's $c$-theorem, under suitable assumptions, can be used to establish the enhancement of scale invariance to local conformal invariance in two dimensions. I will then turn to four dimensions and discuss the argument of Dymarsky, Komargodski, Schwimmer, and Theisen based on dilaton scattering amplitudes and unitarity. I will conclude by commenting on possible generalisations to other dimensions and, time permitting, recent progress on the question.

Calabi-Yau varieties constitute a wide class which (strictly) includes abelian varieties and K3 surfaces. In this talk, I will discuss  Brantner-Taelman's work on the mixed characteristic deformation theory of Calabi-Yau varieties, and discuss my recent work characterising their "canonical lifts" via $p$-adic Hodge theory. Time permitting, I will also discuss the (in general conjectural) modularity of Calabi-Yau varieties over number fields and its relation to the aforementioned lifts.