Past

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.