Past

Most Ricci flow theory takes the short-time existence of solutions as a starting point and ends up concerned with understanding the long-time limiting behaviour and the structure of any finite-time singularities that may develop along the way. In this talk I will look at what you can think of as singularities at time zero. I will describe some of the situations in which one would like to start a  Ricci flow with a space that is rougher than a smooth bounded curvature Riemannian manifold, and some of the situations in which one considers smooth initial data that is only achieved in a non-smooth way. A particularly interesting and useful case is the problem of starting a Ricci flow on a Riemann surface equipped with a measure. I will not be assuming expertise in Ricci flow theory. Parts of the talk are joint with either Hao Yin (USTC) or ManChun Lee (CUHK).

We consider the four-point correlator of the stress-energy tensor in N=4 SYM, to leading order in inverse powers of the central charge, but including all order corrections in 1/lambda. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small alpha'/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude, and determine all coefficients in the low energy expansion in terms of the CFT data for heavy string operators, in principle available from integrability. For the first set of corrections to the flat space amplitude we find a unique solution consistent with the results from integrability and localisation.

A few years back, Smale spaces were shown to exhibit noncommutative Poincaré duality (with Jerry Kaminker and Ian Putnam). The fundamental class was represented as an extension by the compacts. In current work we describe a Fredholm module representation of the fundamental class. The proof uses delicate approximations of the Smale space arising from a refining sequence of (open) Markov partition covers. I hope to explain all these notions in an elementary manner. This is joint work with Dimitris Gerontogiannis and Joachim Zacharias.

In this talk, I will introduce our investigations on how to make deep learning easier to optimize, faster to train, and more robust to out-of-distribution prediction. To be specific, we design a group-invariant optimization framework for ReLU neural networks; we compensate the gradient delay in asynchronized distributed training; and we improve the out-of-distribution prediction by incorporating “causal” invariance.

Dr Oliver Sheridan-Methven from Citadel Securities, (an InFoMM and MScMCF alumni), will be talking about his experiences from studying at the Mathematical Institute, interviewing for jobs, to working in finance. Now in Zurich, Oliver is a quantitative developer in the advanced scientific computing team at Citadel Securities, a world leading market maker. Citadel Securities specialises in ultra high frequency trading, low latency execution, and their researchers tackle cutting edge machine learning and data science problems on colossal data sets with humongous computational resources. Oliver will be talking about his own experiences, and also how mathematicians are naturally great fits for a huge number of roles at Citadel Securities.

We discuss linear convolutional neural networks (LCNs) and their critical points. We observe that the function space (that is, the set of functions represented by LCNs) can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the function space.

For instance, for LCNs with one-dimensional convolutions having stride one and arbitrary filter sizes, we provide a full description of the boundary of the function space. We further study the optimization of an objective function over such LCNs: We characterize the relations between critical points in function space and in parameter space and show that there do exist spurious critical points. We compute an upper bound on the number of critical points in function space using Euclidean distance degrees and describe dynamical invariants for gradient descent.

This talk is based on joint work with Thomas Merkh, Guido Montúfar, and Matthew Trager.

The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.

In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.

This is based on ongoing joint work with Adeel Khan and David Rydh.

We consider close-packed tiling models of geometric objects -- a mixture of hardcore dimers and plaquettes -- as a generalisation of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a z-link or a plaquettein the x-y plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type `higher-rank electrostatics', which can exhibit a plaquette-dimer liquid and an ordered phase. We analyse this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Random walks are a common model for the exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? In this talk we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is noisy, exposure theory produces a highly accurate prediction of aggregate exploration statistics. As a specific application, we extend exposure theory to better understand human learning with its typical mental errors, and thus account for distortions of learned networks.

This talk is based on https://arxiv.org/abs/2202.11262

A classical theorem of Molloy and Johansson states that if a graph is triangle free and has maximum degree at most $\Delta$, then it has chromatic number at most $\frac{\Delta}{\log \Delta}$. This was extended to graphs with few edges in their neighbourhoods by Alon-Krivelevich and Sudakov, and to list chromatic number by Vu. I will give a full and self-contained proof of these results that relies only on induction and the first moment method.

The on-shell superspace formulation of N=4 SYM allows the writing of all possible scattering processes in one compact object called the super-amplitude.   Famously, the super-amplitude integrand can be extracted from generalized polyhedra called the amplituhedron. In this talk, I will review this construction and present a natural generalization of the amplituhedron that we proved at tree level and conjectured at loop level to correspond to the product of two parity conjugate superamplitudes. The sum of all parity conjugate amplitudes corresponds to a particular limit of the supercorrelator through the Wilson Loop/Amplitude duality.  I will conclude by discussing this connection from a geometrical point of view. This talk is based on the reference arXiv:2106.09372 .

TBC

As biological data has become more accessible and available in biology and healthcare, we find increasing opportunities to leverage artificial intelligence to help with data integration and picking out patterns in complex data. In this short talk, we will provide glimpses of what we do at the Novo Nordisk Research Centre Oxford towards understanding patient journeys, and in the use of knowledge graphs to draw insights from diverse biomedical data streams

Spectral sequences are computational tools to find the (co-)homology of mathematical objects and are used across various fields. In this talk I will focus on the LHS spectral sequence, which we associate to an extension of groups to compute group cohomology. The first part of the talk will serve as introduction to both group cohomology and general spectral sequences, where I hope to provide and intuition and some reduced formalism. As main example, and core of this talk, we will look at the LHS spectral sequence associated to the group extension $(\mathbb{Z}/3\mathbb{Z})^3 \rightarrow S \rightarrow \mathbb{Z}/3\mathbb{Z}$, where $S$ is a Sylow-3-subgroup of $SL_2(\mathbb{Z}/9\mathbb{Z})$. In particular I will present arguments that all differentials on the $E^2$ page vanish.

The presence of glaciers and ice sheets leaves a significant imprint
on Earth's surface.  We steadily improve our physical understanding of
the involved processes, from the erosion of kilometer-deep fjords in
crystalline bedrock to the broad ice-marginal deposition of sediments.
This talk will highlight observations of landforms, sedimentary deposits,
laboratory experiments, and models that aim to capture the interplay
between ice and substratum.  I show how the interplay may play a role in
the future evolution of the West Antarctic Ice Sheet in a warming climate.

In the 1980s, gauge theory was used to provide new invariants (up to
diffeomorphism) of orientable four dimensional manifolds, by counting
solutions of certain equations up to to a choice of gauge. More
recently, similar techniques have been used to study manifolds of
different dimensions, most notably on Spin(7) and G_2 manifolds. Using
dimensional reduction, one can find candidates for gauge theoretic
equations on manifolds of lower dimension. The talk will give an
overview of gauge theory in the 4 and 8 dimensional cases, and how
gauge theory on Spin(7) manifolds could be used to develop a gauge
theory on 5 dimensional manifolds.