Past

We prove that every rational homology cobordism class in the subgroup generated
by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in
any other element in the same class. As a consequence we show that several natural maps to
the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility
condition between the determinants of certain 2-bridge knots and other knots in the same
concordance class. This is joint work with Daniele Celoria and JungHwan Park.

In this talk I will discuss some recent results that allow to approximate a real singularity given by polynomial equations of degree d (e.g. the zero set of a polynomial, or the number of its critical points of a given Morse index) with a singularity which is diffeomorphic to the original one, but it is given by polynomials of degree O(d^(1/2)log d).
The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes toinfinity).

The talk is based on joint works with P. Breiding, D. N. Diatta and H. Keneshlou      

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators

The stack of Higgs bundles of a given rank and degree over a non-singular projective curve can be stratified in two ways: according to its Higgs Harder-Narasimhan type (its instability type) and according to the Harder-Narasimhan type of the underlying vector bundle (instability type of the underlying bundle). The semistable stratum is an open stratum of the former and admits a coarse moduli space, namely the moduli space of semistable Higgs bundles. It can be constructed using Geometric Invariant Theory (GIT) and is a widely studied moduli space due to its rich geometric structure.

In this talk I will explain how recent advances in Non-Reductive GIT can be used to refine the Higgs Harder-Narasimhan and Harder-Narasimhan stratifications in such a way that each refined stratum admits a coarse moduli space. I will explicitly describe these refined stratifications and their intersection in the case of rank 2 Higgs bundles, and discuss the topology and geometry of the corresponding moduli spaces

I will describe recent advances in the study of quantum gravity where one can explicitly show in examples that superpositions of states with fixed topology can change the topology of spacetime. These effects lead to paradoxes that are resolved in effective field theory by the introduction of code subspaces. I will also talk about more typical states and issues related on how to decide if a black hole horizon is smooth or not.

Valérie Voorsluijs
Deterministic limit of intracellular calcium spikes
Abstract: In non-excitable cells, global calcium spikes emerge from the collective dynamics of clusters of calcium channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Matthias Nagel
Knots in dimensions three and four
Abstract: Knot theory studies the various embeddings of a circle into three-dimensional space. I will describe an equivalence relation on knots, called "concordance", which takes the fourth dimension into account. The study of concordance is intimately related with many problems at the heart of the topology of four-manifolds, such as the difference between the smooth and the topological category, and I will discuss results that illuminate this relation.

This week’s Fridays@2 session, led by Dr Richard Earl and Dr Vicky Neale, is intended to provide advice on exam preparation and how to approach the Part A and Part B exams.

This session is aimed at second years and third years who will be sitting exams this term. Next week’s Fridays@2 will be for first years and will look at preparing for Prelims papers.
 

Competition between peptides for binding and presentation by MHC class I molecules decides the immune response to foreign or tumor antigens. Many previous studies have attempted to classify the immunogenicity of a peptide using machine learning algorithms to predict the affinity, or half-life, of the peptide binding to MHC. However immunopeptidome analyses have shown a poor correlation between sequence based predictions and the abundance on the cell surface of the experimentally identified peptides. Such metrics are, for instance, only comparable when the abundance of competing peptides can be accurately quantified. We have developed a model for predicting the relative presentation of competing peptides that takes into account off-rate, source protein abundance and turnover and cofactor-assisted MHC assembly with peptides. This model is mechanism based so that it can accommodate complex biology phenomena such as inflammation, up or downregulation of peptide loading complex chaperones, appearance of a mutanome. We have used aspects of the model to drive an investigation of the precise molecular mechanism of peptide selection by MHC I and its associated intracellular cofactors.

We discuss a nonlinear variant of Roth’s theorem on the existence of three-term progressions in dense sets of integers, focusing on an effective version of such a result. This is joint work with Sarah Peluse.
 

My goal for the talk is to give a "from the ground-up" introduction to symplectic topology. We will cover the Darboux lemma, pseudo-holomorphic curves, Gromov-Witten invariants, quantum cohomology and Floer cohomology.

We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation.

This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.

The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method.

This is joint work with Elle McDonald (CSIRO, Australia), Jennifer Pestana (Strathclyde University, UK) and Anthony Goddard (Durham University, UK)

According to the Harish-Chandra philosophy, cuspidal representations are the basic building blocks in the representation theory of finite reductive groups.  Similarly for supercuspidal representations of p-adic groups.  Self-dual representations play a special role in the study of parabolic induction.  Thus, it is of interest to know whether self-dual (super)cuspidal representations exist.  With a few exceptions involving some small fields, I will show precisely when a finite reductive group has irreducible cuspidal representations that are self-dual, of Deligne-Lusztig type, or both.  Then I will look at implications for the existence of irreducible, self-dual supercuspidal representations of p-adic groups.  This is joint work with Manish Mishra.

The Einstein equations in wave coordinates are an example of a system 
which does not obey the "null condition". This leads to many 
difficulties, most famously when attempting to prove global existence, 
otherwise known as the "nonlinear stability of Minkowski space". 
Previous approaches to overcoming these problems suffer from a lack of 
generalisability - among other things, they make the a priori assumption 
that the space is approximately scale-invariant. Given the current 
interest in studying the stability of black holes and other related 
problems, removing this assumption is of great importance.

The p-weighted energy method of Dafermos and Rodnianski promises to 
overcome this difficulty by providing a flexible and robust tool to 
prove decay. However, so far it has mainly been used to treat linear 
equations. In this talk I will explain how to modify this method so that 
it can be applied to nonlinear systems which only obey the "weak null 
condition" - a large class of systems that includes, as a special case, 
the Einstein equations. This involves combining the p-weighted energy 
method with many of the geometric methods originally used by 
Christodoulou and Klainerman. Among other things, this allows us to 
enlarge the class of wave equations which are known to admit small-data 
global solutions, it gives a new proof of the stability of Minkowski 
space, and it also yields detailed asymptotics. In particular, in some 
situations we can understand the geometric origin of the slow decay 
towards null infinity exhibited by some of these systems: it is due to 
the formation of "shocks at infinity".

In the spirit of the Ax-Kochen-Ershov principle, one could conjecture that the imaginaries in equicharacteristic zero Henselian fields can be entirely classified in terms of the Haskell-Hrushovski-Macpherson geometric imaginaries, residue field imaginaries and value group imaginaries. However, the situation is more complicated than that. My goal in this talk will be to present what we believe to be an optimal conjecture and give elements of a proof.

It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras (QHA). We can prove that cluster monomials of the quantum unipotent coordinate ring correspondi to real simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.

Numerical simulations indicate that deep artificial neural networks (DNNs) seem to be able to overcome the curse of dimensionality in many computational  problems in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate. However, there are only a few special situations where results in the literature can rigorously explain the success of DNNs when approximating high-dimensional functions.

In this talk it is revealed that DNNs do indeed overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. A crucial ingredient in our proof of this result is the fact that the artificial neural network used to approximate the PDE solution really is a deep artificial neural network with a large number of hidden layers.

For $G$ connected, reductive algebraic group defined over $\mathbb{C}$ the Springer Correspondence gives a bijection between the irreducible representations of the Weyl group $W$ of $G$ and certain pairs comprising a $G$-orbit on the nilpotent cone of the Lie algebra of $G$ and an irreducible local system attached to that $G$-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the fibres of the Springer resolution. These Springer fibres are geometrically very rich and provide interesting Weyl group combinatorics: for instance, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of $W$. In this talk, I'll give a general survey of the Springer Correspondence and then discuss recent joint projects with Daniele Rosso, Vinoth Nandakumar and Arik Wilbert on Kato's Exotic Springer correspondence.

Graph sampling with noise is a fundamental problem in graph signal processing (GSP). A popular biased scheme using graph Laplacian regularization (GLR) solves a system of linear equations for its reconstruction. Assuming this GLR-based reconstruction scheme, we propose a fast sampling strategy to maximize the numerical stability of the linear system--i.e., minimize the condition number of the coefficient matrix. Specifically, we maximize the eigenvalue lower bounds of the matrix that are left-ends of Gershgorin discs of the coefficient matrix, without eigen-decomposition. We propose an iterative algorithm to traverse the graph nodes via Breadth First Search (BFS) and align the left-ends of all corresponding Gershgorin discs at lower-bound threshold T using two basic operations: disc shifting and scaling. We then perform binary search to maximize T given a sample budget K. Experiments on real graph data show that the proposed algorithm can effectively promote large eigenvalue lower bounds, and the reconstruction MSE is the same or smaller than existing sampling methods for different budget K at much lower complexity.

Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form.

We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of von Neumann algebras. This is shown to work at least in the Ising model.

George Soules [1] introduced a set of vectors $r_1,...,r_N$ with the remarkable property that for any set of ordered numbers $\lambda_1\geq\dots\geq\lambda_N$, the matrix $\sum_n \lambda_nr_nr_n^T$ has nonnegative off-diagonal entries. Later, it was found [2] that there exists a whole class of such vectors - Soules vectors - which are intimately connected to binary rooted trees. In this talk I will describe the construction of Soules vectors starting from a binary rooted tree, and introduce some basic properties. I will also cover a number of applications: the inverse eigenvalue problem, equitable partitions in Laplacian matrices and the eigendecomposition of the Clauset-Moore-Newman hierarchical random graph model.

[1] Soules (1983), Constructing Symmetric Nonnegative Matrices
[2] Elsner, Nabben and Neumann (1998), Orthogonal bases that lead to symmetric nonnegative matrices

In joint work with Peter Topping we introduce pyramid Ricci flows, defined throughout uniform regions of spacetime that are not simply parabolic cylinders, and enjoying curvature estimates that are not required to remain spatially constant throughout the domain of definition. This weakened notion of Ricci flow may be run in situations ill-suited to the classical theory. As an application of pyramid Ricci flows, we obtain global regularity results for three-dimensional Ricci limit spaces (extending results of Miles Simon and Peter Topping) and for higher dimensional PIC1 limit spaces (extending not only the results of Richard Bamler, Esther Cabezas-Rivas and Burkhard Wilking, but also the subsequent refinements by Yi Lai).
 

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li. 

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined
topological operads with homological stability (OHS) and proved that the
group completion of an algebra over an OHS is weakly equivalent to an
infinite loop space.

In this talk, I shall outline a construction which to an algebra A over
an OHS associates a new infinite loop space. Under mild conditions on
the operad, this space is equivalent as an infinite loop space to the
group completion of A. This generalises a result of Wahl on the
equivalence of the two infinite loop space structures constructed by
Tillmann on the classifying space of the stable mapping class group. I
shall also talk about an application of this construction to stable
moduli spaces of high-dimensional manifolds in thesense of Galatius and
Randal-Williams.

We will start by presenting two basic probabilistic effects for questions concerning the regularity of functions and nonlinear operations on functions. We will then overview well-posedenss results for the nonlinear wave equation, the nonlinear Schr\"odinger equation and the nonlinear heat equation, in the presence of singular randomness.

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties.
These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This is joint work with Corey Harris (University of Oslo).

A virtue of the special geometry underlying the string theory moduli space of  Calabi--Yau manifolds is the existence of a canonical choice of moduli space coordinates. In heterotic theories, as much as we would desire it, there is no obvious choice of coordinates and so we should be covariant. I will discuss some issues in doing this.

Aura Raulo (Ecological and Evolutionary Dynamics) and Marie-Claire Koschowitz (Vertebrate Palaeobiology) discuss their work and its mathematical challenges.

Aura Raulo

" Aura Raulo is a graduate student in Zoology Department working on transmission of symbiotic bacteria in the social networks of their animal hosts"
Title: Heaps in networks - How we share our microbiota through kisses
Abstract: Humans, like all vertebrates have a microbiome, a diverse community of symbiotic bacteria that live in and on us and are crucial for our functioning. These bacteria help us digest food, regulate our mood and function as a key part of our immune system. Intriguingly, while they are part of us, they are, unlike our other cells, in constant flux between us, challenging the traditional definition of a biological individual. Many of these bacteria need intimate social contact to be transmitted from human to human, making social network analysis tools handy in explaining their community dynamics.What then is a recipe for a ``good microbiome”? Theories and evidence implies that the most healthy and immunologically robust microbiome composition is both diverse, semi-stable and somewhat synchronized among closely interacting individuals, but little is known about what kind of transmission landscapes determine these bacterial cocktails. In my talk, I will present humanmicrobiome as a network trait: a metacommunity of cells shaped by an equilibrium of isolation and contact among their hosts. I propose that we do notnecessarily need to think of levels of life (e.g. cells, individuals, populations) as being neatly nested inside of each other. Rather, aggregations of cooperating cells (both bacteria and human cells) can be considered as mere tighter clusters in their interaction network, dynamically creating de novo defined units of life. I will present a few game theoretical evolutionary dilemmas following from this perspective and highlight outstanding questions in mapping how network position of the host translates into community composition of bacteria in flux.

Marie Koschowitz
“Marie Koschowitz is a PhD student in the Department of Zoology and the Department of Earth Sciences, working on comparative physiology and large scale evolutionary patterns in reptiles such as crocodiles, birds and dinosaurs."
Title: Putting the maths into dinosaurs – A zoologist's perspective
Abstract: Contemporary palaeontology is a subject area that often deals with sparse data.Therefore, palaeontologists became rather inventive in pursuit of getting the most out of what is available. If we find a dinosaur’s skull that shows prominent, but puzzling, bony ridges without any apparent function, how can we make meaningful interpretations of its purpose in the living animal that was? If we are confronted with a variety of partially preserved bones from animals looking anatomically similar, but not quite alike, how can we infer relationships in the absence of genetic data?Some methods that resolve these questions, such as finite element analysis, were borrowed from engineering. Others, like comparative phylogenetics or MCMC generalised mixed effects models, are even more directly based on mathematical computations. All of these approaches help us to calculate things like a raptors bite-force and understand the ins and outsof their skulls anatomy, or why pterosaurs and plesiosaurs aren’t exactly dinosaurs. This talk aims to presents a selection of current approaches to applied mathematics which have been inspired by interdisciplinary research – and to foster awareness of all the ways how mathematicians can get involved in “dinosaur research”, if they feel inclined to do so.


 

The class of sheaf Laplacians can be characterized as the convex closure of a certain set of sparse semidefinite matrices. From this viewpoint, the study of sheaf Laplacians becomes a question of linear algebra on sparse matrices. I will discuss the applications of this perspective to the problems of approximating, sparsifying, and learning sheaves.

This workshop will focus on the main causes of exam stress, anxiety and panic and look at practical strategies to manage and overcome these issues. We will also review strategies to best support exam preparation.

Dr Ruth Collins is a Chartered Psychologist who specialises in the management of anxiety and panic. She is also a trained mindfulness teacher and an associate of the Oxford Mindfulness Centre.

Inertia-gravity waves (IGWs) are ubiquitous in the ocean and the atmosphere. Once generated (by tides, topography, convection and other processes), they propagate and scatter in the large-scale, geostrophically-balanced background flow. I will discuss models of this scattering which represent the background flow as a random field with known statistics. Without assumption of spatial scale separation between waves and flow, the scattering is described by a kinetic equation involving a scattering cross section determined by the energy spectrum of the flow. In the limit of small-scale waves, this equation reduces to a diffusion equation in wavenumber space. This predicts, in particular, IGW energy spectra scaling as k^{-2}, consistent with observations in the atmosphere and ocean, lending some support to recent claims that (sub)mesoscale spectra can be attributed to almost linear IGWs.  The theoretical predictions are checked against numerical simulations of the three-dimensional Boussinesq equations.
(Joint work with Miles Savva and Hossein Kafiabad.)

The discrete fundamental groups of a metric space can be thought of as fundamental groups that `ignore' closed loops up to some specified size R. As the parameter R grows, these groups have been used to produce interesting invariants of coarse geometry. On the other hand, as R gets smaller one would expect to retrieve the usual fundamental group as a limit. In this talk I will try to briefly illustrate both these aspects.

The problem of decomposing a given dataset as a superposition of basic motifs arises in a wide range of application areas, including neural spike sorting and the analysis of astrophysical and microscopy data. Motivated by these problems, we study a "short-and-sparse" deconvolution problem, in which the goal is to recover a short motif a from its convolution with a random spike train $x$. We formulate this problem as optimization over the sphere. We analyze the geometry of this (nonconvex) optimization problem, and argue that when the target spike train is sufficiently sparse, on a region of the sphere, every local minimum is equivalent to the ground truth, up to symmetry (here a signed shift). This characterization obtains, e.g., for generic kernels of length $k$, when the sparsity rate of the spike train is proportional to $k^{-2/3}$ (i.e., roughly $k^{1/3}$ spikes in each length-$k$ window). This geometric characterization implies that efficient methods obtain the ground truth under the same conditions. 

Our analysis highlights the key roles of symmetry and negative curvature in the behavior of efficient methods -- in particular, the role of a "dispersive" structure in promoting efficient convergence to global optimizers without the need to explicitly leverage second-order information. We sketch connections to broader families of benign nonconvex problems in machine learning and signal processing, in which efficient methods obtain global optima independent of initialization. These problems include variants of sparse dictionary learning, tensor decomposition, and phase recovery.

Joint work with Yuqian Zhang, Yenson Lau, Han-Wen Kuo, Dar Gilboa, Sky Cheung, Abhay Pasupathy

Atom Probe Tomography is a powerful 3D mass spectrometry technique. By pulsing the sample apex with an electric field, surface atoms are ionised and collected by a detector. A 3D image of estimated initial ion positions is constructed via an image reconstruction protocol. Current protocols assume ion trajectories follow a stereographic projection. However, this method assumes a hemispherical sample apex that fails to account for varying material ionisation rates and introduces severe distortions into atomic distributions for complex material systems.

We aim to develop continuum models and use this to derive a time-dependent mapping describing how ion initial positions on the sample surface correspond to final impact positions on the detector. When correctly calibrated with experiment, such a mapping could be used for performing reconstruction.

Currently we track the sample surface using a level set method, while the electric field is solved via BEM or a FEM-BEM coupling. These field calculations must remain accurate close to the boundary. Calibrating unknown evaporation parameters with experiment requires an ensemble of models per experiment. Therefore, we are also looking to maximise model efficiency via BEM compression methods i.e. fast multipole BEM. Efficiently constructing and reliably interpolating the non-bijective trajectory mapping, while accounting for ion trajectory overlap and instabilities (at sample surface corners), also presents intriguing problems.

This project is in collaboration with Cameca, the leading manufacturer of commercial atom probe instruments. If successful in minimising distortions such a technique could become valuable within the semiconductor industry.

PLEASE NOTE THAT THIS SEMINAR IS CANCELLED DUE TO UNFORSEEN CIRCUMSTANCES.

We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve".  In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

Gauge-theoretic invariants such as Donaldson or Seiberg–Witten invariants of 4-manifolds, Casson invariants of 3-manifolds, Donaldson–Thomas invariants of Calabi–Yau 3- and 4-folds, and putative Donaldson–Segal invariants of G_2 manifolds are defined by constructing a moduli space of solutions to an elliptic PDE as a (derived) manifold and integrating the (virtual) fundamental class against cohomology classes. For a moduli space to have a (virtual) fundamental class it must be compact, oriented, and (quasi-)smooth. We first describe a general framework for addressing orientability of gauge-theoretic moduli spaces due to Joyce–Tanaka–Upmeier. We then show that the moduli stack of perfect complexes of coherent sheaves on a Calabi–Yau 4-fold X is a homotopy-theoretic group completion of the topological realisation of the moduli stack of algebraic vector bundles on X. This allows one to extend orientations on the locus of algebraic vector bundles to the boundary of the (compact) moduli space of coherent sheaves using the universal property of homotopy-theoretic group completions. This is a necessary step in constructing Donaldson–Thomas invariants of Calabi–Yau 4-folds. This is joint work with Yalong Cao and Dominic Joyce.

We present a consistent neural network based calibration method for a number of volatility models-including the rough volatility family-that performs the calibration task within a few milliseconds for the full implied volatility surface.
The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several model families (such as rough volatility models) within the scope of applicability in industry practice. As customary for machine learning, the form in which information from available data is extracted and stored is crucial for network performance. With this in mind we discuss how our approach addresses the usual challenges of machine learning solutions in a financial context (availability of training data, interpretability of results for regulators, control over generalisation errors). We present specific architectures for price approximation and calibration and optimize these with respect different objectives regarding accuracy, speed and robustness. We also find that including the intermediate step of learning pricing functions of (classical or rough) models before calibration significantly improves network performance compared to direct calibration to data.

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Abstract

The dynamical fields that emanate from self-similarly expanding ellipsoidal regions undergoing phase change (change in density, i.e., volume collapse, and change in moduli) under pre-stress, constitute the dynamic generalization of the seminal Eshelby inhomogeneity problem (as an equivalent inclusion problem), and they consist of pressure, shear, and M waves emitted by the surface of the expanding ellipsoid and yielding Rayleigh waves in the crack limit. They may constitute the model of Deep Focus Earthquakes (DFEs) occurring under very high pressures and due to phase change. Two fundamental theorems of physics govern the phenomenon, the Cauchy-Kowalewskaya theorem, which based on dimensional analysis and analytic properties alone, dictates that there is zero particle velocity in the interior, and Noether’s theorem that extremizes (minimizes for stability) the energy spent to move the boundary so that it does not become a sink (or source) of energy, and determines the self-similar shape (axes expansion speeds). The expression from Noether’s theorem indicates that the expanding region can be planar, thus breaking the symmetry of the input and the phenomenon manifests itself as a newly discovered one of a “dynamic collapse/ cavitation instability”, where very large strain energy condensed in the very thin region can escape out. In the presence of shear, the flattened very thin ellipsoid (or band) will be oriented in space so that the energy due to phase change under pre-stress is able to escape out at minimum loss condensed in the core of dislocations gliding out on the planes where the maximum configurational force (Peach-Koehler) is applied on them. Phase change occurring planarly produces in a flattened expanding ellipdoid a new defect present in the DFEs. The radiation patterns are obtained in terms of the equivalent to the phase change six eigenstrain components, which also contain effects due to planarity through the Dynamic Eshelby Tensor for the flattened ellipsoid. Some models in the literature of DFEs are evaluated and excluded on the basis of not having the energy to move the boundary of phase discontinuity. Noether’s theorem is valid in anisotropy and nonlinear elasticity, and the phenomenon is independent of scales, valid from the nano to the very large ones, and applicable in general to other dynamic phenomena of stress induced martensitic transformations, shear banding, and amorphization.