Past

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

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.

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

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.

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

The recent, rapid advances in modern biology and data science have opened up a whole range of challenging mathematical problems. In this talk I will discuss a class of interacting particle models with anisotropic repulsive-attractive interaction forces. These models are motivated by the simulation of fingerprint databases, which are required in forensic science and biometric applications. In existing models, the forces are isotropic and particle models lead to non-local aggregation PDEs with radially symmetric potentials. The central novelty in the models I consider is an anisotropy induced by an underlying tensor field. This innovation does not only lead to the ability to describe real-world phenomena more accurately, but also renders their analysis significantly harder compared to their isotropic counterparts. I will discuss the role of anisotropic interaction in these models, present a stability analysis of line patterns, and show numerical results for the simulation of fingerprints. I will also outline how very similar models can be used in data classification, where it is desirable to assign labels to points in a point cloud, given that a certain number of points is already correctly labeled.

Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD),  that assigns a so-called “barcode" to any geometric tree (i.e, any finite binary tree embedded in R^3). We showed that the TMD effectively determines  reliable clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the relatively small number of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type, in terms of morpho-electrical properties and  connectivity. We synthesized networks of structurally altered neurons, revealing principles linking branching properties to the structure of large-scale networks.  We have also successfully applied these classification and synthesis techniques to microglia and astrocytes, two other types of cells that populate the brain.

In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties.

This talk is based on work in collaborations led by Lida Kanari at the Blue Brain Project.

Knot projections for knots in the 3-sphere allow us to easily describe knots, compute invariants, enumerate all knots, manipulate them under Reidemister moves and feed them into a computer. One might hope for a similar representation of knots in general 3-manifolds. We will survey properties of knots in general 3-manifolds and discuss a proposed diagram-esque representation of them.

I will discuss a string theory way to study G_2 instanton moduli space and explain how to compute the instanton partition function for a certain G_2 manifold. An important insight comes from the twisted M-theory on the G_2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G_2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on https://arxiv.org/abs/2109.01110 and a series of works in progress with Michele Del Zotto and Yehao Zhou.

A Hausdorff and etale groupoid is said to be C*-simple if its reduced groupoid C*-algebra is simple. Work on C*-simplicity goes back to the work of Kalantar and Kennedy in 2014, who classified the C*-simplicity of discrete groups by associating to the group a dynamical system. Since then, the study of C*-simplicity has received interest from group theorists and operator algebraists alike. More recently, the works of Kawabe and Borys demonstrate that the groupoid case may be tractible to such dynamical characterizations. In this talk, we present the dynamical characterization of when a groupoid is C*-simple and work out some basic examples. This is joint work with Xin Li, Matt Kennedy, Sven Raum, and Dan Ursu. No previous knowledge of groupoids will be assumed.

In this talk we cover recent work in collaboration with Diego Granziol and Steve Roberts where we study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian and derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. 

In this talk we will look at the different ways to define city boundaries, and the relevance to consider socio-demographic and spatial connectivity in urban systems, in particular if interventions are to be considered.

In commutative algebra localizing a ring and its modules is a fundamental technique. In the general case of a Grothendieck abelian category or even a triangulated category with small direct sums this is replaced by forming the quotient category by a localizing subcategory. Therefore the classification of these localizing subcategories becomes an important problem. I will begin by recalling the case of the (derived) module category of a commutative noetherian ring due to Gabriel and Hopkins/Neeman, respectively, in order to give an idea how such a classification can look like.

The case of interest in this talk is the derived category D(G) of smooth representation in characteristic p of a p-adic Lie group G. This is motivated by the emerging p-adic Langlands program. In joint work with C. Heyer we have some modest initial results if G is compact pro-p or abelian. which I will present.