Oxford

Holographic correlation functions are under good analytic control when none of the single trace operators live in long multiplets. This is famously the case for SCFTs with sixteen supercharges but it is also possible to construct examples with eight supercharges by exploiting space filling branes in AdS. In particular, one can study 4d N=2 theories which are related to each other by an S-fold in much the same way that N=3 theories are related to N=4 Super Yang-Mills. I will describe how modern methods provide a window into their correlation functions with an emphasis on anomalous dimensions. To compare the different S-folds we will need to go to one loop, and to go to one loop we will need to account for operator mixing. This provides an example of resolving degeneracy by resolving degeneracy.

I will review aspects of the theory of symmetry-protected topological phases, focusing on the case of one-dimensional quantum chains. Important concepts include the bulk-boundary correspondence, with bulk topological invariants leading to interesting boundary phenomena. I will discuss topological invariants and associated boundary phenomena in the case that the system is gapless and described at low energies by a conformal field theory. Based on work with Ruben Verresen, Ryan Thorngren and Frank Pollmann.

One of the lasting puzzles in quantum gravity is whether the holographic description of a gravitational system is a single quantum mechanical theory or the disorder average of many. In the latter case, multiple copies of boundary observables do not factorize into a product, but rather have higher moments. These correlations are interpreted in the bulk as due to geometries involving spacetime wormholes which connect disjoint boundaries. 

I will talk about the question of factorization and the role of wormholes for supersymmetric observables, specifically the supersymmetric index. Working with the Euclidean gravitational path integral, I will start with a bulk prescription for computing the supersymmetric index, which agrees with the usual boundary definition. Concretely, I will focus on the setting of charged black holes in asymptotically flat four-dimensional N=2 ungauged supergravity. In this case, the gravitational index path integral has an infinite family of Kerr-Newman classical saddles with different angular velocities. However, fermionic zero-mode fluctuations annihilate the contribution of each saddle except for a single BPS one which yields the expected value of the index. I will then turn to non-perturbative corrections involving spacetime wormholes, and show that fermionic zero modes are present for all such geometries, making their contributions vanish. This mechanism works for both single- and multi-boundary path integrals. In particular, only disconnected geometries without wormholes contribute to the index path integral, and the factorization puzzle that plagues the black hole partition function is resolved for the supersymmetric index. I will also present all other single-centered geometries that yield non-perturbative contributions to the gravitational index of each boundary. Finally, I will discuss implications and expectations for factorization and the status of supersymmetric ensembles in AdS/CFT in further generality. Talk based on [2107.09062] with Luca Iliesiu and Joaquin Turiaci.

Over the complex numbers, the Bomolgorov-Tian-Todorev theorem asserts that Calabi-Yau varieties have unobstructed deformations, so any n^{th} order deformation extends to higher order.  We prove an analogue of this statement for the nicest kind of Calabi-Yau varieties in characteristic p, namely ordinary ones, using derived algebraic geometry. In fact, we produce canonical lifts to characteristic zero, thereby generalising results of Serre-Tate, Deligne-Nygaard, Ward, and Achinger-Zdanowic. This is joint work with Taelman.

One core aim of (supervised) machine learning is to approximate an unknown function given a dataset containing examples of input-output pairs. Real-world examples of such functions include the mapping from an image to its label or the mapping from a molecule to its energy. For a variety of such functions, while the precise mapping is unknown, we often have knowledge of its properties. For example, the label of an image may be invariant to rotations of the input image. Generally, such properties formally correspond to the function being equivariant to certain actions on its input and output spaces. This has led to much research on building equivariant function classes (aka neural networks). In this talk, we survey this growing field of equivariance in deep learning for a mathematical audience, motivating the need for equivariance, covering concrete examples of equivariant neural networks, and offering a learning theoretic perspective on the benefits of equivariance. 

The Iwasawa algebra of a compact $p$-adic Lie group is fundamental to the study of the representations of the group. Understanding this representation theory is crucial in progress towards a (mod p) local Langlands correspondence. However, much remains unknown about Iwasawa algebras and their modules.

In this talk we'll aim to measure the size of the Iwasawa algebra and its representations. I'll explain the algebraic tools we use to do this - Krull dimension and canonical dimension - and survey previously known examples. Our main result is a new bound on these dimensions for the group $SL_2(O_F)$, where $F$ is a finite extension of the p-adic numbers. When $F$ is a quadratic extension, we find the Krull dimension is exactly 5, as predicted by a conjecture of Ardakov and Brown.

Computing Donaldson-Thomas partition function of a G2 manifold has been a long standing problem. The key step for the problem is to understand the G2 instanton moduli space. I will discuss a string theory way to study the G2 instanton moduli space and explain how to compute the instanton partition function for a certain G2 manifold. An important insight comes from the twisted M-theory on the G2 manifold. This talk is based on a work with Michele del Zotto and Yehao Zhou.

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. If time allows I will also discuss some applications of this result. (Joint with Chieu-Minh Tran and Ruixiang Zhang)