L4

We study 5d N=1 theories engineered by (p,q) five-brane webs and their 3d N=4 magnetic quivers, focusing on the N + SU(N)^(k-1) + N gauge theory. We establish an exact parameter dictionary mapping the 5d flavour masses and instanton couplings to the FI parameters of the strong-coupling magnetic quiver. We test this dictionary against mass deformations, weak-coupling limits, and moment map VEVs, implemented via the 5d superconformal index and the 3d squashed-sphere partition function. We analyze these deformations on the magnetic-quiver side using a new framework that exactly tracks Higgsing patterns, decoupled sectors, and infrared theories. This promotes magnetic quivers to quantitative tools for studying how the 5d superconformal index responds to physical deformations
We show that large 5d moment map VEVs render the superconformal index distributional, mirroring bad theories in lower dimensions. We also develop a brane-web interpretation of these results, identifying the Hanany–Witten moves associated with the cascade of VEVs and relating inequivalent frames to distinct tessellations of the generalized toric diagram. Furthermore, we demonstrate that magnetic quivers remain powerful predictive tools even when the theory enters the bad regime.

For subcritical singular SPDEs, solutions are typically exhibited as the limit of solutions of regularised and renormalised PDEs as the regularisation scale goes to 0. Existing general solution theories typically yield a very small exponent for the rate of convergence. This suggests the question of identifying the sharpest possible rate of convergence that can be achieved in some space of distributions. 

In this talk, I will discuss recent work with Máté Gerencsér and Yueh-Sheng Hsu in which we identify the sharp rate of convergence in the case of the KPZ equation.

Finitely presented multiparameter persistence modules are a natural target category for encoding homological invariants, but comparing such modules requires choosing a notion of distance. Our goal is to develop a zoo of distances on multiparameter persistence modules, capturing both the algebraic features of the representation category as well as the geometry of the parameter space. The parameters of this construction make the comparison flexible by capturing different notions of noise and similarity. This is joint work with Jens Agerberg and Wojciech Chachólski.

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

We extend the construction of BPHZ renormalised models in regularity structures via the spectral gap inequality to the fractional heat operator. Its Green’s function fails to be smooth across the entire time-zero hyperplane, violating the classical assumption that kernels are smooth away from the origin. The proof requires a delicate balance between the gain in regularity provided by the spectral gap approach and the limited spatial decay of the kernel. 

Joint work with L. Broux and R. Steele.

TBA