Past

Persistence theory provides useful tools to extract information from real-world data sets, and profits of techniques from different mathematical disciplines, such as Morse theory and quiver representation. In this seminar, I am going to present a new approach for studying persistence theory using model categories. I will briefly introduce model categories and then describe how to define a model structure on the category of the tame parametrised chain complexes, which are chain complexes that evolve in time. Using this model structure, we can define new invariants for tame parametrised chain complexes, which are in perfect accordance with the standard barcode when restricting to persistence modules. I will illustrate with some examples why such an approach can be useful in topological data analysis and what new insight on standard persistence can give us. 

Cuntz introduced pure infiniteness for simple C*-algebras as a C*-algebraic analogue of type III von Neumann factors. Notable examples include the Calkin algebra B(H)/K(H), the Cuntz algebras O_n, simple Cuntz-Krieger algebras, and other C*-algebras you would encounter in the wild. The separable, nuclear ones were classified in celebrated work by Kirchberg and Phillips in the mid 90s. I will talk about these topics including the non-simple case if time permits.

Unitary solutions of the Yang-Baxter equation ("R-matrices") play a prominent role in several fields, such as quantum field theory and topological quantum computing, but are difficult to find directly and remain somewhat mysterious. In this talk I want to explain how one can use subfactor techniques to learn something about unitary R-matrices, and a research programme aiming at the classification of unitary R-matrices up to a natural equivalence relation. This talk is based on joint work with Roberto Conti, Ulrich Pennig, and Simon Wood.

Liouville quantum gravity (LQG) is a theory of random fractal surfaces with origin in the physics literature in the 1980s. Most literature is about LQG with matter central charge $c\in (-\infty,1]$. We study a discretization of LQG which makes sense for all $c\in (-\infty,25)$. Based on a joint work with Gwynne, Pfeffer, and Remy.

Systems with lattice geometry can be renormalized exploiting their embedding in metric space, which naturally defines the coarse-grained nodes. By contrast, complex networks defy the usual techniques because of their small-world character and lack of explicit metric embedding. Current network renormalization approaches require strong assumptions (e.g. community structure, hyperbolicity, scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Here we introduce a graph renormalization scheme valid for any hierarchy of coarse-grainings, thereby allowing for the definition of block-nodes across multiple scales. This approach reveals a necessary and specific dependence of network topology on an additive hidden variable attached to nodes, plus optional dyadic factors. Renormalizable networks turn out to be consistent with a unique specification of the fitness model, while they are incompatible with preferential attachment, the configuration model or the stochastic blockmodel. These results highlight a deep conceptual distinction between scale-free and scale-invariant networks, and provide a geometry-free route to renormalization. If the hidden variables are annealed, the model spontaneously leads to realistic scale-free networks with cut-off. If they are quenched, the model can be used to renormalize real-world networks with node attributes and distance-dependence or communities. As an example we derive an accurate multiscale model of the International Trade Network applicable across arbitrary geographic resolutions.

https://arxiv.org/abs/2009.11024 (23 sept.)

There is a growing body of results in extremal combinatorics and Ramsey theory which give better bounds or stronger conclusions under the additional assumption of bounded VC-dimension. Schur and Erdős conjectured that there exists a suitable constant $c$ with the property that every graph with at least $2^{cm}$ vertices, whose edges are colored by $m$ colors, contains a monochromatic triangle. We prove this conjecture for edge-colored graphs such that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension. This result is best possible up to the value of $c$.
    Joint work with Jacob Fox and Andrew Suk.

This talk is a personal how-to (and how-not-to) manual for doing Maths with industry, or indeed with government. The Maths element is essential but lots of other skills and activities are equally necessary. Examples: problem elicitation; understanding the environmental constraints; power analysis; understanding world-views and aligning personal motivations; and finally, understanding the wider systems in which the Maths element will sit. These issues have been discussed for some time in the management science community, where their generic umbrella name is Problem Structuring Methods (PSMs).

Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects.

This is joint work with Kurusch Ebrahimi-Fard (NTNU Trondheim) and Nikolas Tapia (WIAS Berlin).

Distinguishing classes of surfaces in $\mathbb{R}^n$ is a task which arises in many situations. There are many characteristics we can use to solve this classification problem. The Persistent Homology Transform allows us to look at shapes in $\mathbb{R}^n$ from $S^{n-1}$ directions simultaneously, and is a useful tool for surface classification. Using the Julia package DiscretePersistentHomologyTransform, we will look at some example curves in $\mathbb{R}^2$ and examine distinguishing features. 

The Gelfand correspondence between compact Hausdorff spaces and unital C*-algebras justifies the slogan that C*-algebras are to be thought of as "non-commutative topological spaces", and Rieffel's theory of compact quantum metric spaces provides, in the same vein, a non-commutative counterpart to the theory of compact metric spaces. The aim of my talk is to introduce the basics of the theory and explain how the classical Gromov-Hausdorff distance between compact metric spaces can be generalized to the quantum setting. If time permits, I will touch upon some recent results obtained in joint work with Jens Kaad and Thomas Gotfredsen.

C*-algebras associated to etale groupoids appear as a versatile construction in many contexts. For instance, groupoid C*-algebras allow for implementation of natural one-parameter groups of automorphisms obtained from continuous cocycles. This provides a path to quantum statistical mechanical systems, where one studies equilibrium states and ground states. The early characterisations of ground states and equilibrium states for groupoid C*-algebras due to Renault have seen remarkable refinements. It is possible to characterise in great generality all ground states of etale groupoid C*-algebras in terms of a boundary groupoid of the cocycle (joint work with Laca and Neshveyev). The steps in the proof employ important constructions for groupoid C*-algebras due to Renault.

We study the minimum Wasserstein distance from the empirical measure to a space of probability measures satisfying linear constraints. This statistic can naturally be used in a wide range of applications, for example, optimally choosing uncertainty sizes in distributionally robust optimization, optimal regularization, testing fairness, martingality, among many other statistical properties. We will discuss duality results which recover the celebrated Kantorovich-Rubinstein duality when the manifold is sufficiently rich and associated test statistics as the sample size increases. We illustrate how this relaxation can beat the statistical curse of dimensionality often associated to empirical Wasserstein distances.

The talk builds on joint work with S. Ghosh, Y. Kang, K. Murthy, M. Squillante, and N. Si.

High dimensionality of biological data is a crucial element that is in need of different methods to unravel their complexity. The current and rich biomedical material that hospitals generate every other day related to cancer detection can benefit from these new techniques. This is the case of diseases such as Acute Lymphoblastic Leukaemia (ALL), one of the most common cancers in childhood. Its diagnosis is based on high-dimensional flow cytometry tumour data that includes immunophenotypic expressions. Not only the intensity of these markers is meaningful for clinicians, but also the shape of the points clouds generated, being then fundamental to find leukaemic clones. Thus, the mathematics of shape recognition in high dimensions can turn itself as a critical tool for this kind of data. This is why we resort to the use of tools from Topological Data Analysis such as Persistence Homology.

Given that ALL relapse incidence is of almost 20% of its patients, we provide a methodology to shed some light on the shape of flow cytometry data, for both relapsed and non-relapsed patients. This is done so by combining the strength of topological data analysis with the versatility of machine learning techniques. The results obtained show us topological differences between both patient sets, such as the amount of connected components and 1-dimensional loops. By means of the so-called persistence images, and for specially selected immunophenotypic markers, a classification of both cohorts is obtained, highlighting the need of new methods to provide better prognosis. 

In the structure theory of operator algebras, it has been a reliable theme that a classification of interesting classes of objects is usually followed by a classification of group actions on said objects. A good example for this is the complete classification of amenable group actions on injective factors, which complemented the famous work of Connes-Haagerup. On the C*-algebra side, progress in the Elliott classification program has likewise given impulse to the classification of C*-dynamics. Although C*-dynamical systems are not yet understood at a comparable level, there are some sophisticated tools in the literature that yield satisfactory partial results. In this introductory talk I will outline the (known) classification of finite group actions with the Rokhlin property, and in the process highlight some themes that are still relevant in today's state-of-the-art.

Quantum groups, which has been a major overarching theme across various branches of mathematics since late 20th century, appear in many ways. Deformation of compact Lie groups is a particularly fruitful paradigm that sits in the intersection between operator algebraic approach to quantized spaces on the one hand, and more algebraic one arising from study of quantum integrable systems on the other.
On the side of operator algebra, Woronowicz defined the C*-bialgebra representing quantized SU(2) based on his theory of pseudospaces. This gives a (noncommutative) C*-algebra of "continuous functions" on the quantized group SUq(2).
Its algebraic counterpart is the quantized universal enveloping algebra Uq(??2), due to Kulish–Reshetikhin and Sklyanin, coming from a search of algebraic structures on solutions of the Yang-Baxter equation. This is (an essentially unique) deformation of the universal enveloping algebra U(??2) as a Hopf algebra.
These structures are in certain duality, and have far-reaching generalization to compact simple Lie groups like SU(n). The interaction of ideas from both fields led to interesting results beyond original settings of these theories.
In this introductory talk, I will explain the basic quantization scheme underlying this "q-deformation", and basic properties of the associated C*-algebras. As part of more recent and advanced topics, I also want to explain an interesting relation to complex simple Lie groups through the idea of quantum double.

The Riemann hypothesis (RH) is one of the great open problems in mathematics. It arose from the study of prime numbers in an analytic context, and—as often occurs in mathematics—developed analogies in an algebraic setting, leading to the influential Weil conjectures. RH for curves over finite fields was proven in the 1940’s by Weil using algebraic-geometric methods. In this talk, we discuss an alternate proof of this result by Stepanov (and Bombieri), using only elementary properties of polynomials. Over the decades, the proof has been whittled down to a 5 page gem! Time permitting, we also indicate connections to exponential sums and the original RH.
 

This talk is motivated by the following question: "how can one reconstruct the geometry of a state space given a collection of observed time series?" A well-studied technique for metric fusion is Similarity Network Fusion (SNF), which works by mixing random walks. However, SNF behaves poorly in the presence of correlated noise, and always reconstructs an intrinsic metric. We propose a new methodology based on delay embeddings, together with a simple orthogonalization scheme that uses the tangency data contained in delay vectors. This method shows promising results for some synthetic and real-world data. The authors suspect that there is a theorem or two hiding in the background -- wild speculation by audience members is encouraged. 

Topological features as computed via persistent homology offer a non-parametric approach to robustly capture multi-scale connectivity information of complex datasets. This has started to gain attention in various machine learning applications. Conventionally, in topological data analysis, this method has been employed as an immutable feature descriptor in order to characterize topological properties of datasets. In this talk, however, I will explore how topological features can be directly integrated into deep learning architectures. This allows us to impose differentiable topological constraints for preserving the global structure of the data space when learning low-dimensional representations.

The TMD algorithm (Kanari et al. 2018) computes the barcode of a neuron (tree) with respect to the radial or path distance from the soma (root). We are interested in the inverse problem: how to understand the space of trees that are represented by the same barcode. Our tool to study this spaces is the stochastic TNS algorithm (Kanari et al. 2020) which generates trees from a given barcode in a biologically meaningful way. 

I will present some theoretical results on the space of trees that have the same barcode, as well as the effect of adding noise to the barcode. In addition, I will provide a more combinatorial perspective on the space of barcodes, expressed in terms of the symmetric group. I will illustrate these results with experiments based on the TNS.

This is joint work with L. Kanari and K. Hess. 

A bundle of C*-algebras is a collection of algebras continuously parametrised by a topological space. There are (at least) two different definitions in operator algebras that make this intuition precise: Continuous C(X)-algebras provide a flexible analytic point of view, while locally trivial C*-algebra bundles allow a classification via homotopy theory. The section algebra of a bundle in the topological sense is a C(X)-algebra, but the converse is not true. In this talk I will compare these two notions using the classical work of Dixmier and Douady on bundles with fibres isomorphic to the compacts  as a guideline. I will then explain joint work with Marius Dadarlat, in which we showed that the theorems of Dixmier and Douady can be generalized to bundles with fibers isomorphic to stabilized strongly self-absorbing C*-algebras. An important feature of the theory is the appearance of higher analogues of the Dixmier-Douady class.

The signature of a path in Euclidean space resides in the tensor algebra of that space; it is obtained by systematic iterated integration of the components of the given path against one another. This straightforward definition conceals a host of deep theoretical properties and impressive practical consequences. In this talk I will describe the homotopical origins of path signatures, their subsequent application to stochastic analysis, and how they facilitate efficient machine learning in topological data analysis. This last bit is joint work with Ilya Chevyrev and Harald Oberhauser.

Quantum limits are objects describing the limit of quadratic quantities (Af_n,f_n) where (f_n) is a sequence of unit vectors in a Hilbert space and A ranges over an algebra of bounded operators. We will discuss the motivation underlying this notion with some important examples from quantum mechanics and from analysis.

The random initialisation of Artificial Neural Networks (ANN) allows one to describe, in the functional space, the limit of the evolution of ANN when their width tends towards infinity. Within this limit, an ANN is initially a Gaussian process and follows, during learning, a gradient descent convoluted by a kernel called the Neural Tangent Kernel.

Connecting neural networks to the well-established theory of kernel methods allows us to understand the dynamics of neural networks, their generalization capability. In practice, it helps to select appropriate architectural features of the network to be trained. In addition, it provides new tools to address the finite size setting.

Since the seminal work of Penrose, it has been understood that conformal compactifications (or "asymptotic simplicity") is the geometrical framework underlying Bondi-Sachs' description of asymptotically flat space-times as an asymptotic expansion. From this point of view the asymptotic boundary, a.k.a "null-infinity", naturally is a conformal null (i.e degenerate) manifold. In particular, "Weyl rescaling" of null-infinity should be understood as gauge transformations. As far as gravitational waves are concerned, it has been well advertised by Ashtekar that if one works with a fixed representative for the conformal metric, gravitational radiations can be neatly parametrized as a choice of "equivalence class of metric-compatible connections". This nice intrinsic description however amounts to working in a fixed gauge and, what is more, the presence of equivalence class tend to make this point of view tedious to work with.

I will review these well-known facts and show how modern methods in conformal geometry (namely tractor calculus) can be adapted to the degenerate conformal geometry of null-infinity to encode the presence of gravitational waves in a completely geometrical (gauge invariant) way: Ashtekar's (equivalence class of) connections are proved to be in 1-1 correspondence with choices of (genuine) tractor connection, gravitational radiation is invariantly described by the tractor curvature and the degeneracy of gravity vacua correspond to the degeneracy of flat tractor connections. The whole construction is fully geometrical and manifestly conformally invariant.

The concept of path signatures has been widely used in several areas of pure mathematics including in applications to data science. However, we remain unable to answer even the most basic questions about it. For instance, how to fully characterise the set of (untruncated) signatures of bounded variation paths? Can certain norms on signatures be related to the length of a path, like in Fourier isometry? In this talk, we will review some known results, explain the open problems and discuss their difficulties.

Abstract: John Roe was a much admired figure in topology and noncommutative geometry, and the creator of the C*-algebraic approach to coarse geometry. John died in 2018 at the age of 58. My aim in the first part of the lecture will be to explain in very general terms the major themes in John’s work, and illustrate them by presenting one of his best-known results, the partitioned manifold index theorem. After the break I shall describe a later result, about relative eta invariants, that has inspired an area of research that is still very active.

Assumed Knowledge: First part: basic familiarity with C*-algebras, plus a little topology. Second part: basic familiarity with K-theory for C*-algebras.

In the first part of the talk, I will review the basic ideas behind Stein’s method for normal approximation and present a general result which we obtained in arXiv:1706.10251 (joint work with Michel Ledoux and Christian Webb). This result states that for a Gibbs measure, an eigenfunction of the corresponding infinitesimal generator is approximately Gaussian in a sense which will be made precise. In the second part, I will report on several applications in random matrix theory. This includes a proof of Johansson’s central limit theorem for linear statistics of beta-ensembles on \R, as well as an application to circular beta-ensembles in the high temperature regime (based on arXiv:1909.01142, joint work with Adrien Hardy).

It is a well-known fact that conformal structures on Riemann surfaces are in 1:1 correspondence with complex structures, but have you ever wondered whether this is just a fluke in 2 dimensions? In this talk, I will explain the concept of Penrose's "non-linear graviton", a fancy name for the twistor space of a hyperkahler manifold and one of the major historical achievements of Oxford maths. The twistor correspondence associates points of the hyperkahler manifold with certain holomorphic rational curves embedded in twistor space. We will see how information of the hyperkahler metric can be encoded purely in the complex structure on twistor space, giving a partial but welcome generalization of the 2-dimensional "fluke". Then I will outline a recently found Dolbeault-framework for the metric's reconstruction from local representatives of this complex structure. This provides an explicit integral formula for Kahler forms and consequently for the hyperkahler metric in terms of holomorphic data on twistor space. Finally, time permitting, I will discuss some interesting applications to (some or all of) PDEs, hyperkahler quotients, and the physics of "quantum gravity".
 

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimisation problem, we are able to calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where exact solutions are not known. In the latter case, we obtain favourable comparisons with standard execution strategies.

In this talk I will present some recent explorations of cluster-algebraic patterns in the building blocks of scattering amplitudes in N = 4 super Yang-Mills theory. In particular, I will first briefly introduce the main characters on stage, i.e. Leading Singularities, Landau singularities, the amplituhedron and cluster algebras. I will then present my main conjecture, "LL-adjacency", which makes all the above characters play together: given a maximal cut of a loop amplitude, Landau singularities and poles of each Yangian invariant appearing in any representation of the corresponding Leading Singularities can be found together in a cluster.  I will explain how the conjecture has been tested for all one-loop amplitudes up to 9 points using cluster algebraic and amplituhedron-based methods.  Finally, I will discuss implications for computing loop amplitudes and their singularity structure, and open research directions.

This is based on the joint work with Ömer Gürdoğan (arXiv: 2005.07154).

Most of the basic results on martingale problems extend to the setting in which the generator depends on a control.  The “control” could represent a random environment, or the generator could specify a classical stochastic control problem.  The equivalence between the martingale problem and forward equation (obtained by taking expectations of the martingales) provides the tools for extending linear programming methods introduced by Manne in the context of controlled finite Markov chains to general Markov stochastic control problems.  The controlled martingale problem can also be applied to the study of constrained Markov processes (e.g., reflecting diffusions), the boundary process being treated as a control.  The talk includes joint work with Richard Stockbridge and with Cristina Costantini. 

There is a well-known correspondence between Weil-Petersson geodesic loops in the moduli space of a surface S and hyperbolic 3-manifolds fibering over the circle with fibre S. Much is unknown, however, about the detailed relationship between geometric features of the loops and those of the 3-manifolds.

In work with Leininger-Souto-Taylor we study the relation between WP length and 3-manifold volume, when the length (suitably normalized) is bounded and the fiber topology is unbounded. We obtain a WP analogue of a theorem proved by Farb-Leininger-Margalit for the Teichmuller metric. In work with Modami, we fix the fiber topology and study connections between the thick-thin decomposition of a geodesic loop and that of the corresponding 3-manifold. While these decompositions are often in direct correspondence, we exhibit examples where the correspondence breaks down. This leaves the full conjectural picture somewhat mysterious, and raises many questions. 

In this talk, I will explain a new wall-crossing phenomenon on P^3 that induces non-Q-factorial singularities and thus cannot be understood as an operation in the MMP of the moduli space, unlike the case for many surfaces.  If time permits, I will explain how the wall-crossing could help to understand the geometry of the associated Hilbert scheme and PT moduli space.

One of the main concerns in social network science is the study of positions and roles. By "position" social scientists usually mean a collection of actors who have similar ties to other actors, while a "role" is a specific pattern of ties among actors or positions. Since the 1970s a lot of research has been done to develop these concepts in a rigorous way. An open question in the field is whether it is possible to perform role and positional analysis simultaneously. In joint work in progress with Mason Porter we explore this question by proposing a framework that relies on the principle of functoriality in category theory. In this talk I will introduce role and positional analysis, present some well-studied examples from social network science, and what new insights this framework might give us.

Femoral neck response to physiological loading during level walking can be better understood, if personalized muscle and bone anatomy is considered. Finite element (FE) models of in vivo cadaveric bones combined with gait data from body-matched volunteers were used in the earlier studies, which could introduce errors in the results. The aim of the current study is to report the first fully personalized multiscale model to investigate the strains predicted at the femoral neck during a full gait cycle. CT-based Finite element models (CT/FE) of the right femur were developed following a validated framework. Muscle forces estimated by the musculoskeletal model were applied to the CT/FE model. For most of the cases, two overall peaks were predicted around 15% and 50% of the gait. Maximum strains were predicted at the superior neck region in the model. Anatomical muscle variations seem to affect femur response leading to considerable variations among individuals, both in term of the strains level and the trend at the femoral neck.
 

Since the work of von Neumann, the theory of operator algebras has been closely linked to the theory of groups. On the one hand, operator algebras constructed from groups provide an important source of examples and insight. On the other hand, many problems about groups are most naturally studied within an operator-algebraic framework. In this talk I will give an overview of some problems relating the structure of a group to the structure of a corresponding C*-algebra. I will discuss recent results and some possible future directions.

Abstract: We use a deep neural network to generate controllers for optimal trading on high frequency data. For the first time, a neural network learns the mapping between the preferences of the trader, i.e. risk aversion parameters, and the optimal controls. An important challenge in learning this mapping is that in intraday trading, trader's actions influence price dynamics in closed loop via the market impact. The exploration--exploitation tradeoff generated by the efficient execution is addressed by tuning the trader's preferences to ensure long enough trajectories are produced during the learning phase. The issue of scarcity of financial data is solved by transfer learning: the neural network is first trained on trajectories generated thanks to a Monte-Carlo scheme, leading to a good initialization before training on historical trajectories. Moreover, to answer to genuine requests of financial regulators on the explainability of machine learning generated controls, we project the obtained ``blackbox controls'' on the space usually spanned by the closed-form solution of the stylized optimal trading problem, leading to a transparent structure. For more realistic loss functions that have no closed-form solution, we show that the average distance between the generated controls and their explainable version remains small. This opens the door to the acceptance of ML-generated controls by financial regulators.
 

The study of non-local games has involved fruitful interactions between operator algebra theory and quantum physics, with a starting point the link between the Connes Embedding Problem and the Tsirelson Problem, uncovered by Junge et al (2011) and Ozawa (2013). Particular instances of non-local games, such as binary constraint system games and synchronous games, have played an important role in the pursuit of the resolution of these problems. In this talk, I will summarise part of the operator algebraic toolkit that has proved useful in the study of non-local games and of their perfect strategies, highlighting the role C*-algebras and operator systems play in their mathematical understanding.

This week Professor Dominic Vella will talk about wrinkling  

In this talk I will provide an overview of recent work on the wrinkling of thin elastic objects. In particular, the focus of the talk will be on answering questions that arise in recent applications that seek not to avoid, but rather, exploit wrinkling. Such applications usually take place far beyond the threshold of instability and so key themes will be the limitations of “standard” instability analysis, as well as what analysis should be performed instead. I will discuss the essential ingredients of this ‘Far-from-Threshold’ analysis, as well as outlining some open questions.  

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I introduce a recently developed variational approach for hyperbolic PDE's. The method allows to show the existence of weak solutions to fluid-structure interactions where a visco-elastic bulk solid is interacting with an incompressible fluid governed by the unsteady Navier Stokes equations. This is a joint work with M. Kampschulte and B. Benesova.