Past

Neural networks have shown great success at learning function approximators between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}. In many problems arising in the physical and engineering sciences it is desirable to generalize this setting to learn operators between spaces of functions X and Y. The talk will overview recent work in this context.

Then the talk will focus on work aimed at addressing the problem of learning operators which define the constitutive model characterizing the macroscopic behaviour of multiscale materials arising in material modeling. Mathematically this corresponds to using machine learning to determine appropriate homogenized equations, using data generated at the microscopic scale. Applications to visco-elasticity and crystal-plasticity are given.

In the '80s, Gromov proved that sequences of Riemannian manifold with a lower bound on the Ricci curvature and an upper bound on the dimension are precompact in the measured Gromov--Hausdorff topology (mGH for short). Since then, much attention has been given to the limits of such sequences, called Ricci limit spaces. A way to study these limits is to introduce a synthetic definition of Ricci curvature lower bounds and dimension upper bounds. A synthetic definition should not rely on an underlying smooth structure and should be stable when passing to the limit in the mGH topology. In this talk, I will briefly introduce CD spaces, which are a generalization of Ricci limit spaces.

Function approximation, as a goal in itself or as an ingredient in scientific computing, typically relies on having a basis. However, in many cases of interest an obvious basis is not known or is not easily found. Even if it is, alternative representations may exist with much fewer degrees of freedom, perhaps by mimicking certain features of the solution into the “basis functions" such as known singularities or phases of oscillation. Unfortunately, such expert knowledge typically doesn’t match well with the mathematical properties of a basis: it leads instead to representations which are either incomplete or overcomplete. In turn, this makes a problem potentially unsolvable or ill-conditioned. We intend to show that overcomplete representations, in spite of inherent ill-conditioning, often work wonderfully well in numerical practice. We explore a theoretical foundation for this phenomenon, use it to devise ground rules for practitioners, and illustrate how the theory and its ramifications manifest themselves in a number of applications.

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

A well-known conjecture of Ivanov states that mapping class groups of surfaces with genus at least 3 virtually do not surject onto the integers. Putman and Wieland reformulated this conjecture in terms of higher Prym representations of finite-index subgroups of mapping class groups. We show that the Putman-Wieland conjecture holds for geometrically uniform subgroups. Along the way we construct a cover S of the genus 2 surface such that the lifts of simple closed curves do not generate the rational homology of S. This is joint work with Markovic.

This talk aims to be a rigorous introduction to Social Choice Theory, a sub-branch of Game Theory with natural applications to economics, sociology and politics that tries to understand how to determine, based on the personal opinions of all individuals, the collective opinion of society. The goal is to prove the three famous and pessimistic impossibility theorems: Arrow's theorem, Gibbard's theorem and Balinski-Young's theorem. Our blunt conclusion will be that, unfortunately, there are no ideally fair social choice systems. Is there any hope yet?

In recent years, our understanding of the asymptotic behavior of characteristic polynomials of random matrices has seen much progression. A key paradigm in this area is that the asymptotic behavior is often captured by an appropriate family of Gaussian multiplicative chaos (GMC) measures (defined heuristically as the normalized exponential of log-correlated random fields). Indeed, such results have been shown for Harr distributed matrices for U(N), O(N), and Sp(2N), as well as for one-cut Hermitian invariant ensembles (and in particular, GUE(N)). In this talk we explain an extension of these results to GOE(2N) and GSE(N). The key tool is a new asymptotic relation between the moments of the characteristic polynomials of all three classical ensembles. 

Given a fixed poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-free if it does not contain an (induced) copy of $\mathcal P$. And we say that $F$ is $\mathcal P$-saturated if it is maximal $\mathcal P$-free. How small can a $\mathcal P$-saturated family be? The smallest such size is the induced saturation number of $\mathcal P$, $\text{sat}^*(n, \mathcal P)$. Even for very small posets, the question of the growth speed of $\text{sat}^*(n,\mathcal P)$ seems to be hard. We present background on this problem and some recent results.

In the first part of the seminar I will introduce shallow neural-networks from a statistical-mechanics perspective, focusing on simple cases and on a naive scenario where information to be learnt is structureless. Then, inspired by biological information processing, I will enrich this framework by accounting for structured datasets and by making the network able to perform challenging tasks like generalization or even "taking a nap”. Results presented are both analytical and numerical.

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.

In 1989, Bryant and Salamon constructed the first Riemannian manifolds with holonomy group $\Spin(7)$. Since a crucial aspect in the study of manifolds with exceptional holonomy regards fibrations through calibrated submanifolds, it is natural to consider such objects on the Bryant-Salamon manifolds.

In this talk, I will describe the construction and the geometry of (possibly singular) Cayley fibrations on each Bryant-Salamon manifold. These will arise from a natural family of structure-preserving $\SU(2)$ actions. The fibres will provide new examples of Cayley submanifolds.

Volumetric X-ray tomography is used in many areas, including applications in medical imaging, many fields of scientific investigation as well as several industrial settings. Yet complex X-ray physics and the significant size of individual x-ray tomography data-sets poses a range of data-science challenges from the development of efficient computational methods, the modelling of complex non-linear relationships, the effective analysis of large volumetric images as well as the inversion of several ill conditioned inverse problems, all of which prevent the application of these techniques in many advanced imaging settings of interest. This talk will highlight several applications were specific data-science issues arise and showcase a range of approaches developed recently at the University of Southampton to overcome many of these obstacles.

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.

This event will take place in L1 and on Teams. A link will be available 30 minutes before the session begins. 

It's hard to believe: I've spent nearly 40 years in STEM. In that time, much changed: we changed from typewriters to PCs, from low performance to high  performance computing, from data-supported research to data-driven research, from traditional languages such as Fortran to a plethora of programming environments. And the rate of change seems to increase constantly. Some things have stayed more or less the same, such as the (lack of) diversity of the STEM community, the level of stress and the struggles we all experience (and the joys!). In this talk, I will reflect on those years, on lessons learned and not learned or unlearned, on things I wish I understood 40 years ago, and on things I still don't understand.

Margot is a professor at Stanford University in the Department of Energy Resources Engineering (ERE) and the Institute of Computational & Mathematical Engineering (ICME). Margot was born and raised in the Netherlands. Her STEM education started in 1982. In 1990 she received a MSc in applied mathematics at Delft University and then left her home country to search for sunnier and hillier places. She moved to Colorado and a year later to California to join the PhD program in Scientific Computing and Computational Mathematics at Stanford. During her PhD, Margot spent several quarters at Oxford University (with very good memories). Before returning to Stanford as faculty member in ERE, Margot spent 5 years as lecturer at the University of Auckland, New Zealand. From 2010-2018, Margot was the director of ICME. During this directorship, she founded the Women in Data Science initiative, which is now a global organization in over 70 countries. From 2015-2020, Margot was also the Senior Associate Dean of Educational Affairs at Stanford's school of Earth, Energy & Environmental Sciences. Currently, Margot still co-directs WiDS and is the Chair of the Board of SIAM. She has since moved back to the mountains (still sunny too) and now lives in Bend, Oregon.

Supersymmetric gauge theories in five dimensions, although power counting non-renormalizable, are known to be in some cases UV completed by a superconformal field theory. Many tools, such as M-theory compactification and pq-web constructions, were used in recent years in order to deepen our understanding of these theories. This framework gives us a concrete way in which we can try to search for additional IR conformal field theory via deformations of these well-known superconformal fixed points. Recently, the authors of 2001.00023 proposed a supersymmetry breaking mass deformation of the E_1theory which, at weak gauge coupling, leads to pure SU(2) Yang-Mills and which was conjectured to lead to an interacting CFT at strong coupling. During this talk, I will provide an explicit geometric construction of the deformation using brane-web techniques and show that for large enough gauge coupling a global symmetry is spontaneously broken and the theory enters a new phase which, at infinite coupling, displays an instability. The Yang-Mills and the symmetry broken phases are separated by a phase transition. Quantum corrections to this analysis are discussed, as well as possible outlooks. Based on arXiv: 2109.02662.

Acyclic partial matchings on simplicial complexes play an important role in topological data analysis by facilitating efficient computation of (persistent) homology groups. Here we describe probabilistic properties of critical simplex counts for such matchings on clique complexes of Bernoulli random graphs. In order to accomplish this goal, we generalise the notion of a dissociated sum to a multivariate setting and prove an abstract multivariate central limit theorem using Stein's method. As a consequence of this general result, we are able to extract central limit theorems not only for critical simplex counts, but also for generalised U-statistics (and hence for clique counts in Bernoulli random graphs) as well as simplex counts in the link of a fixed simplex in a random clique complex.

Gastrulation is a critical event in vertebrate morphogenesis, characterized by coordinated large-scale multi-cellular movements. One grand challenge in modern biology is understanding how spatio-temporal morphological structures emerge from cellular processes in a developing organism and vary across vertebrates. We derive a theoretical framework that couples tissue flows, stress-dependent myosin activity, and actomyosin cable orientation. Our model, consisting of a set of nonlinear coupled PDEs, predicts the onset and development of observed experimental patterns of wild-type and perturbations of chick gastrulation as a spontaneous instability of a uniform state. We use analysis and numerics to show how our model recapitulates the phase space of gastrulation morphologies seen across vertebrates, consistent with experiments. Altogether, this suggests that early embryonic self-organization follows from a minimal predictive theory of active mechano-sensitive flows. 

 https://www.biorxiv.org/content/10.1101/2021.10.03.462928v2 

We use energy estimates to derive new bounds on the eigenvalues of a generic form of double saddle-point matrices, with and without regularization terms. Results related to inertia and algebraic multiplicity of eigenvalues are also presented. The analysis includes eigenvalue bounds for preconditioned matrices based on block-diagonal Schur complement-based preconditioners, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. The analytical observations are linked to a few multiphysics problems of interest. This is joint work with Susanne Bradley.

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals

Jane Coons

Gaussian graphical models are multivariate Gaussian statistical models in which a graph encodes conditional independence relations among the random variables. Adding colors to this graph allows us to describe situations where some entries in the concentration matrices in the model are assumed to be equal. In this talk, we focus on RCOP models, in which this coloring is obtained from the orbits of a subgroup of the automorphism group of the underlying graph. We show that when the underlying block graph is a one-clique-sum of complete graphs, the Zariski closure of the set of concentration matrices of an RCOP model on this graph is a toric variety. We also give a Markov basis for the vanishing ideal of this variety in these cases.

Topological persistence for multi-scale terrain profiling and feature detection in drylands hydrology

Gillian Grindstaff

With the growing availability of remote sensing products and computational resources, an increasing amount of landscape data is available, and with it, increasing demand for automated feature detection and useful morphological summaries. Topological data analysis, and in particular, persistent homology, has been applied successfully to detect landslides and characterize soil pores, but its application to hydrology is currently still limited. We demonstrate how persistent homology of a real-valued function on a two-dimensional domain can be used to summarize critical points and shape in a landscape simultaneously across all scales, and how that data can be used to automatically detect features of hydrological interest, such as: experimental conditions in a rainfall simulator, boundary conditions of landscape evolution models, and earthen berms and stock ponds, placed historically to alter natural runoff patterns in the American southwest.

We say that a graph G is Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in G is covered by G. Examples include the Euclidean building of PSLn(Qp). We show that the rigidity of the building goes further by proving that a reconstruction is possible from only a partial local information, called “print”. We use this to prove the rigidity of graphs quasi-isometric to the building among which are the torsion-free lattices of PSLn(Qp).

I will present a study of integrable structures and quantum chaos in a class of infinite-dimensional though computationally tractable models, called quantum resonant systems. These models, together with their classical counterparts, emerge in various areas of physics, such as nonlinear dynamics in anti-de Sitter spacetime, but also in Bose-Einstein condensate physics. The class of classical models displays a wide range of integrable properties, such as the existence of Lax pairs, partial solvability or generic chaotic dynamics. This opens a window to investigate these properties from the perspective of the corresponding quantum theory by effectively diagonalising finite-sized matrices and exploring level spacing statistics. We will furthermore analyse the implications of the symmetries for the spectrum of resonant models with partial solvability and discuss how the rich integrable structures can be exploited to constructed novel quantum coherent states that effectively capture sophisticated nonlinear solutions in the classical theory.

TBA

Many applications involve non-Euclidean data, where exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard nonlinear programming approaches. This observation has resulted in an increasing interest in Riemannian methods in the optimization and machine learning community. In this talk, we consider the problem of optimizing a function on a Riemannian manifold subject to convex constraints. We introduce Riemannian Frank-Wolfe (RFW) methods, a class of projection-free algorithms for constrained geodesically convex optimization. To understand the algorithm’s efficiency, we discuss (1) its iteration complexity, (2) the complexity of computing the Riemannian gradient and (3) the complexity of the Riemannian “linear” oracle (RLO), a crucial subroutine at the heart of the algorithm. We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods. Joint work with Suvrit Sra (MIT).

Identifying novel drug-target interactions (DTI) is a critical and rate limiting step in drug discovery. While deep learning models have been proposed to accelerate the identification process, we show that state-of-the-art models fail to generalize to novel (i.e., never-before-seen) structures. We first unveil the mechanisms responsible for this shortcoming, demonstrating how models rely on shortcuts that leverage the topology of the protein-ligand bipartite network, rather than learning the node features. Then, we introduce AI-Bind, a pipeline that combines network-based sampling strategies with unsupervised pre-training, allowing us to limit the annotation imbalance and improve binding predictions for novel proteins and ligands. We illustrate the value of AI-Bind by predicting drugs and natural compounds with binding affinity to SARS-CoV-2 viral proteins and the associated human proteins. We also validate these predictions via auto-docking simulations and comparison with recent experimental evidence. Overall, AI-Bind offers a powerful high-throughput approach to identify drug-target combinations, with the potential of becoming a powerful tool in drug discovery.

arXiv link: https://arxiv.org/abs/2112.13168

There's an idea going back at least to Kirchberger in 1902 that since the only operations we can ultimately compute are +, -, *, and /, all of numerical computation must reduce to rational functions.  I've been looking into this idea and it has led in some interesting directions.

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.

Symmetry protected topological (SPT) phases have recently attracted a lot of
attention from physicists and mathematicians as a topological classification
scheme for gapped ground states. In this talk I will briefly introduce the
operator algebraic approach to SPT phases in the infinite-volume limit. In
particular, I will focus on the quasifree (free-fermionic) setting, where we

can adapt tools from algebraic quantum field theory to describe phases of
gapped ground states using K-homology and the coarse index.

TBC

We'll sketch how the $K$-rational solutions of a system $X$ of multivariate polynomials can be viewed as the solutions of a "classical mechanics" problem on an associated affine space.

When $X$ has a suitable topology, e.g. if its $\mathbb{C}$-solutions form a Riemann surface of genus $>1$, we'll observe some of the advantages of this new point of view such as a relatively computable algorithm for effective finiteness (with some stipulations). This is joint work with Minhyong Kim.
 

Lattice homology is a combinatorial invariant of plumbed 3-manifolds due to Nemethi. The definition is a formalization of Ozsvath and Szabo's computation of the Heegaard Floer homology of plumbed 3-manifolds. Nemethi conjectured that lattice homology is isomorphic to Heegaard Floer homology. For a restricted class of plumbings, this isomorphism is known to hold, due to work of Ozsvath-Szabo, Nemethi, and Ozsvath-Stipsicz-Szabo. By using the Manolescu-Ozsvath link surgery formula for Heegaard Floer homology, we prove the conjectured isomorphism in general. In this talk, we will talk about aspects of the proof, and some related topics and extensions of the result.

The homogeneous coordinate ring of a projective variety may be constructed by geometrically quantizing the multiples of a symplectic form, using the complex structure as a polarization. In this talk, I will explain how a holomorphic Poisson structure allows us to deform the complex polarization into a generalized complex structure, leading to a non-commutative deformation of the homogeneous coordinate ring. The main tool is a conjectural construction of a category of generalized complex branes, which makes use of the A-model of an associated symplectic groupoid. I will explain this in the example of toric Poisson varieties. This is joint work with Marco Gualtieri (arXiv:2108.01658).

The global symmetries of a d-dimensional quantum field theory (QFT), and their ’t Hooft anomalies, are conveniently captured by a topological field theory (TFT) in (d+1) dimensions, which we may refer to as the Symmetry TFT of the given d-dimensional QFT. This point of view has a vast range of applicability: it encompasses both ordinary symmetries, as well as generalized symmetries. In this talk, I will discuss systematic methods to compute the Symmetry TFT for QFTs realized by M-theory on a singular, non-compact space X. The desired Symmetry TFT is extracted from the topological couplings of 11d supergravity, via reduction on the space L, the boundary of X. The formalism of differential cohomology allows us to include discrete symmetries originating from torsion in the cohomology of L. I will illustrate this framework in two classes of examples: M-theory on an ALE space (engineering 7d SYM theory); M-theory on Calabi-Yau cones (engineering 5d superconformal field theories).

In many machine learning tasks, it is crucial to extract low-dimensional and descriptive features from a data set. In this talk, I present a method to extract features from multi-dimensional space-time signals which is motivated, on the one hand, by the success of path signatures in machine learning, and on the other hand, by the success of models from the theory of regularity structures in the analysis of PDEs. I will present a flexible definition of a model feature vector along with numerical experiments in which we combine these features with basic supervised linear regression to predict solutions to parabolic and dispersive PDEs with a given forcing and boundary conditions. Interestingly, in the dispersive case, the prediction power relies heavily on whether the boundary conditions are appropriately included in the model. The talk is based on the following joint work with Andris Gerasimovics and Hendrik Weber: https://arxiv.org/abs/2108.05879

Deep learning continues to dominate machine learning and has been successful in computer vision, natural language processing, etc. Its impact has now expanded to many research areas in science and engineering. In this talk, I will mainly focus on some recent impacts of deep learning on computational mathematics. I will present our recent work on bridging deep neural networks with numerical differential equations, and how it may guide us in designing new models and algorithms for some scientific computing tasks. On the one hand, I will present some of our works on the design of interpretable data-driven models for system identification and model reduction. On the other hand, I will present our recent attempts at combining wisdom from numerical PDEs and machine learning to design data-driven solvers for PDEs and their applications in electromagnetic simulation.

The path signature is a characterization of paths that originated in Chen's iterated integral cochain model for path spaces and loop spaces. More recently, it has been used to form the foundations of rough paths in stochastic analysis, and provides an effective feature map for sequential data in machine learning. In this talk, we return to the topological foundations in Chen's construction to develop generalizations of the signature.

Adaptive agents through active inference: The main fields of research that are used to model and realise adaptive agents are optimal control, reinforcement learning and active inference. Active inference is a probabilistic description of adaptive agents that is relatively less known to mathematicians, as it originated from neuroscience in the last decade. This talk presents the mathematical underpinnings of active inference, starting from fundamental considerations about agents that maintain their structural integrity in the face of environmental perturbations. Through this, we derive a probability distribution over actions, that describes decision-making under uncertainty in adaptive agents . Interestingly, this distribution has an interesting information geometric structure, combining, for instance, drives for exploration and exploitation, which may yield a principled answer to the exploration-exploitation trade-off. Preserving this geometric structure enables to realise adaptive agents in practice. We illustrate their behaviour with simulation examples and empirical comparisons with reinforcement learning.

Scaling Limits of Random Graphs: The scaling limit of directed random graphs remains relatively unexplored compared to their undirected counterparts. In contrast, many real-world networks, such as links on the world wide web, financial transactions and “follows” on Twitter, are inherently directed. Previous work by Goldschmidt and Stephenson established the scaling limit for the strongly connected components (SCCs) of the Erdős -- Rényi model in the critical window when appropriately rescaled. In this talk, we present a result showing the SCCs of another class of critical random directed graphs will converge when rescaled to the same limit. Central to the proof is an exploration of the directed graph and subsequent encodings of the exploration as real valued random processes. We aim to present this exploration algorithm and other key components of the proof.

From Mathematics to Data Science and Back: We give an overview of the interaction between rough path theory and data science at the current time.
 

We consider spectral methods that uncover hidden structures in directed networks. We establish and exploit connections between node reordering via (a) minimizing an objective function and (b) maximizing the likelihood of a random graph model. We focus on two existing spectral approaches that build and analyse Laplacian-style matrices via the minimization of frustration and trophic incoherence. These algorithms aim to reveal directed periodic and linear hierarchies, respectively. We show that reordering nodes using the two algorithms, or mapping them onto a specified lattice, is associated with new classes of directed random graph models. Using this random graph setting, we are able to compare the two algorithms on a given network and quantify which structure is more likely to be present. We illustrate the approach on synthetic and real networks, and discuss practical implementation issues. This talk is based on a joint work with Desmond Higham and Konstantinos Zygalakis. 

Article link: https://royalsocietypublishing.org/doi/10.1098/rsos.211144

The Hull--Strominger system is a system of non-linear PDEs on heterotic string theory involving a pair of Hermitian metrics $(g,h)$ on a six dimensional manifold $M$. One of these equations dictates the metric $g$ on $M$ to be conformally balanced. We will begin the talk by giving a description of the geometry of cohomogeneity one manifolds and SU(3)-structures. Then, we will look for solutions to the Hull--Strominger system in the cohomogeneity one setting. We show that a six-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group $G$ admits no $G$-invariant balanced non-Kähler SU(3)-structures. This is a joint work with F. Salvatore.

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.

This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.

Candida Bowtell

Title: Chess puzzles: from recreational maths to fundamental mathematical structures

Abstract:
Back in 1848, in a German chess magazine, Max Bezzel asked how many ways there are to place 8 queens on a chessboard so that no two queens can attack one another. This question caught the attention of many, including Gauss, and was subsequently generalised. What if we want to place n non-attacking queens on an n by n chessboard? What if we embed the chessboard on the surface of a torus? How many ways are there to do this? It turns out these questions are hard, but mathematically interesting, and many different strategies have been used to attack them. We'll survey some results, old and new, including progress from this year.

Joshua Bull

Title: From Cancer to Covid: topological and spatial descriptions of immune cells in disease

Abstract:
Advances in medical imaging techniques mean that we have increasingly detailed knowledge of the specific cells that are present in different diseases. The locations of certain cells, like immune cells, gives clinicians clues about which treatments might be effective against cancer, or about how the immune system reacts to a Covid infection - but the more detailed this spatial data becomes, the harder it is for medics to analyse or interpret. Instead, we can turn to tools from topological data analysis, mathematical modelling, and spatial statistics to describe and quantify the relationships between different cell types in a wide range of medical images. This talk will demonstrate how mathematics can be used as a tool to advance our understanding of medicine, with a focus on immune cells in both cancer and covid-19.

The study of gravity currents has long been of interest due to their prevalence in industry and in nature, one such example being the spreading of viscoplastic (yield-stress) fluid films. When a viscoplastic fluid is extruded onto a flat plate, the resulting gravity current expands axisymmetrically when the surface is dry and rough. In this talk, I will discuss two instabilities that arise when (1) the no-slip surface is replaced by a free-slip surface; and (2) the flat plate is wet by a thin coating of water.