Past

The Keating-Snaith conjecture for quadratic twists of elliptic curves predicts the central values should have a log-normal distribution. I present recent progress towards establishing this in the range of large deviations of order of the variance. This extends Selberg’s Central Limit Theorem from ranges of order of the standard deviation to ranges of order of the variance in a variety of contexts, inspired by random walk theory. It is inspired by recent work on large deviations of the zeta function and central values of L-functions.
 

Kasparov's bivariant K-theory (or KK-theory) is an extremely powerful invariant for both C*-algebras and C*-dynamical systems, which was originally motivated for a tool to solve classical problems coming from topology and geometry. Its paramount importance for classification theory was discovered soon after, impressively demonstrated within the Kirchberg-Phillips theorem to classify simple nuclear and purely infinite C*-algebras. Since then, it can be said that every methodological novelty about extracting information from KK-theory brought along some new breakthrough in classification theory. Perhaps the most important example of this is the Lin-Dadarlat-Eilers stable uniqueness theorem, which forms the technical basis behind many of the most important articles written over the past decade. In the recent landmark paper of Carrion et al, it was demonstrated how the stable uniqueness theorem can be upgraded to a uniqueness theorem of sorts under extra assumptions. It was then posed as an open problem whether the statement of a desired "KK-uniqueness theorem" always holds.

In this talk I want to present the affirmative answer to this question: If A and B are separable C*-algebras and (f,g) is a Cuntz pair of absorbing representations whose induced class in KK(A,B) vanishes, then f and g are strongly asymptotically unitarily equivalent. The talk shall focus on the main conceptual ideas towards this theorem, and I plan to discuss variants of the theorem if time permits. It turns out that the analogous KK-uniqueness theorem is true in a much more general context, which covers equivariant and/or ideal-related and/or nuclear KK-theory.

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

ALF gravitational instantons, of which the Taub-NUT and Atiyah-Hitchin metrics are prototypes, are the complete non-compact hyperkähler 4-manifolds with cubic volume growth. Examples have been known since the 1970's, but a complete classification was only given around 10 years ago. In this talk, I will present joint work with Haskins and Nordström where we extend some of these results to complete non-compact 7-manifolds with holonomy G2 and an asymptotic geometry, called ALC (asymptotically locally conical), that generalises to higher dimension the asymptotic geometry of ALF spaces.

Inverse problems involve reconstructing unknown physical quantities from indirect measurements. They appear in various fields, including medical imaging (e.g., MRI, Ultrasound, CT), material sciences and molecular biology (e.g., electron microscopy), as well as remote sensing just to name a few examples. While deep neural networks are currently able to achieve state-of-the-art performance in many imaging tasks, in this talk we argue that  many inverse imaging problems cannot be solved convincingly using a black-box solution. Instead, they require a well-crafted combination of computational tools taking the underlying signal, the physical constraints and acquisition characteristics into account.

In the first part of the talk, we introduce INDigo+, a novel INN-guided probabilistic diffusion algorithm for arbitrary image restoration tasks. INDigo+ combines the perfect reconstruction property of invertible neural networks (INNs) with the strong generative capabilities of pre-trained diffusion models. Specifically, we leverage the invertibility of the network to condition the diffusion process and in this way we generate high quality restored images consistent with the measurements.

In the second part of the talk, we discuss the unfolding techniques which is an approach that allows embedding priors and models in the neural network architecture. In this context we discuss the problem of monitoring the dynamics of large populations of neurons over a large area of the brain. Light-field microscopy (LFM), a type of scanless microscopy, is a particularly attractive candidate for high-speed three-dimensional (3D) imaging which is needed for monitoring neural activity. We review fundamental aspects of LFM and then present computational methods based on deep learning for neuron localization and activity estimation from light-field data.
Finally, we look at the multi-modal case and present an application in art investigation. Often X-ray images of Old Master paintings contain information of the visible painting and of concealed sub-surface design, we therefore introduce a model-based neural network capable of separating from the “mixed X-ray”  the X-ray image of the visible painting and the X-ray of the concealed design.

This is joint work with  A. Foust, P. Song, C. Howe, H. Verinaz, J. Huang, Di You and Y. Su from Imperial College London, M. Rodrigues and W. Pu from University College London, I. Daubechies from Duke University, Barak Sober from the Hebrew University of Jerusalem and C. Higgitt and N. Daly from The National Gallery in London.

Black holes play a central role in our understanding of quantum gravity, but identifying their precise counterparts in a dual QFT remains a tricky business. These states are heavy, chaotic, and encode various universal aspects — but are also notoriously hard to characterise. In this talk, we’ll explore how supersymmetric field theories provide a controlled setting to study black hole states. In particular, we’ll introduce the idea of fortuitous states as a useful criterion for identifying BPS black hole states. We’ll then illustrate this concept with concrete examples, including the (supersymmetric) SYK model and the D1-D5 CFT.

The discussion will be based on the following recent papers:
arXiv:2402.10129, arXiv:2412.06902, and arXiv:2501.05448.

Exploring fascinating mathematics more independently by doing a Part B project or dissertation can be one of the most exciting and rewarding parts of undergraduate study. Supervision meetings are one of the main tools for making the most of this experience.

In this Fridays@2 session, a panel of staff and students with experience in Part B projects and dissertations will share practical tips on how to prepare, communicate effectively, and tackle common challenges. Whether you’re currently working on a project, planning one, or just curious, join us for insights and an interactive Q&A.

In recent years, a novel approach to studying integrable models has emerged which leverages a higher-dimensional gauge theory, specifically a holomorphic-topological theory. This new framework provides alternative methods for investigating quantum aspects of integrability and for constructing integrable models in more than two dimensions. This talk will review the foundations of this approach, its applications, and the exciting possibilities it opens up for future research in the field of integrable systems. 

We begin with the Fontaine--Wintenberger isomorphism, which gives an example of an extension of Q_p and of F_p((t)) with isomorphic absolute Galois groups. We explain how by trying to lift maps on mod p reductions one encounters Witt vectors. Next, by trying to apply the theory of Witt vectors to the two extensions, we encounter the idea of tilting. Perfectoid fields are then defined more-or-less so that tilting may be reversed. We indicate the proof of the tilting correspondence for perfectoid fields following the Witt vectors approach, classifying the untilts of a given characteristic p perfectoid field along the way. To end, we touch upon the Fargues--Fontaine curve and the geometrization of l-adic local Langlands as motivation for globalizing the tilting correspondence to perfectoid spaces.

In this talk I will give a number of short vignettes of work that has been undertaken in my group over the last 15 years. Mathematically, the theme that underlies our work is the importance of randomness to biological systems. I will explore a number of systems for which randomness plays a critical role. Models of these systems which ignore this important feature do a poor job of replicating the known biology, which in turn limits their predictive power. The underlying biological theme of the majority our work is development, but the tools and techniques we have built can be applied to multiple biological systems and indeed further afield. Topics will be drawn from, locust migration, zebrafish pigment pattern formation, mammalian cell migratory defects, appropriate cell cycle modelling and more. I won't delve to deeply into anyone area, but am happy to take question or to expand upon of the areas I touch on.

We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads -- these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals.

We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur.  These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group.  Joint work in progress with Tom Fisher.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint for learning time-dependent PDEs. 

In this talk, we will first explore connections with low-rank matrix recovery problems in numerical linear algebra. Then, we will show that one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis without querying the adjoint.

I will introduce basic concepts from category theory that are relevant to the study of generalized symmetries. Then, I will focus on constructions known as equivariantization and de-equivariantization, which allow one to move between categories with a group G-action and those with a Rep(G)-action. I will also discuss their relation to the concept of gauging, if time permits.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the p-enstrophies, p > 2. The talk is based on a recent publication with Christian Seis and Emil Wiedemann.

An adaptive regularization algorithm for unconstrained nonconvex optimization is presented in
which the objective function is never evaluated, but only derivatives are used and without prior knowledge of Lipschitz constant.  This algorithm belongs to the class of adaptive regularization methods, for which optimal worst-case complexity results are known for the standard framework where the objective function is evaluated. It is shown in this paper that these excellent complexity bounds are also valid for the new algorithm. Theoretical analysis of both exact and stochastic cases are discussed and  new probabilistic conditions on tensor derivatives are proposed.  Initial experiments on large binary classification highlight the merits of our method.

Biological matter has the fascinating ability to autonomously generate material deformations via intrinsic active forces, where the latter are often present within effectively two-dimensional structures. The dynamics of such “active surfaces” inevitably entails a complex, self-organized interplay between geometry of a surface and its mechanical interactions with the surrounding. The impact of these factors on the self-organization capacity of surfaces made of an active material, and how related effects are exploited in biological systems, is largely unknown.

In this talk, I will first discuss general numerical challenges in analysing self-organising active surfaces and the bifurcation structure of emergent shape spaces. I will then focus on active surfaces with broken up-down symmetry, of which the eukaryotic cell cortex and epithelial tissues are highly abundant biological examples. In such surfaces, a natural interplay arises between active stresses and surface curvature. We demonstrate that this interplay leads to a comprehensive library of spontaneous shape transformations that resemble stereotypical morphogenetic processes. These include cell-division-like invaginations and the autonomous formation of tubular surfaces of arbitrary length, both of which robustly overcome well-known shape instabilities that would arise in analogue passive systems.

Congruence Subgroup Property is a characterisation of finite-index subgroups of automorphism groups. It first arose from the study of subgroups of linear groups. In this talk, I will show a few examples where it holds and where it fails, and give an overview of what is known about the family $SL_n\mathbb{Z}$, $Out(F_n)$, $MCG(\Sigma)$. Then I will describe some related results in the case of Mapping Class Groups, and explain their relation to profinite rigidity of 3-manifolds.

For over 150 years, the study of phase transitions—such as water freezing into ice or magnets losing their magnetism—has been a cornerstone of statistical physics. In this talk, we explore the critical behavior of two-dimensional percolation models, which use random graphs to model the behavior of porous media. At the critical point, remarkable symmetries and emergent properties arise, providing precise insights into the nature of these systems and enriching our understanding of phase transitions. The presentation is designed to be accessible and does not assume any prior background in percolation theory.

About the Speaker: 

Hugo Duminil-Copin is is a French mathematician recognised for his groundbreaking work in probability theory and mathematical physics. He was appointed full professor at the University of Geneva in 2014 and since 2016 has also been a permanent professor at the Institut des Hautes Études Scientifiques (IHES) in France. In 2022 he was awarded the Fields Medal, the highest distinction in mathematics.

In this talk, we discuss the condition for the marginal distribution of the solution to singular SPDEs on the d-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearized equation. As applications of our result, we see the singularity of the Phi^4_3-measure with respect to the Gaussian free field measure and the border of parameters for the fractional Phi^4-measure to be singular with respect to the base Gaussian measure. This talk is based on a joint work with Martin Hairer and Seiichiro Kusuoka.

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. 
In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. 
 

Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net. 

In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). If time permits, we will briefly outline ongoing work on bicommutant category modules (which are the structures assigned by the Segal Chiral CFT at the level of 1-manifolds), hinting towards a categorified analogue of Connes fusion of von Neumann algebra modules.

(Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces)

A trace on a group is a positive-definite conjugation-invariant function on it. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.

Valulations (of polytopes or matroids) are very useful and very mysterious. After taking some time to explain this concept, I will categorify it, with the aim of making it both more useful and less mysterious.

"The question of whether an impurity can be screened by bulk degrees of freedom is central to the study of defects and to (variations of) the Kondo problem. In this talk I discuss how symmetry, generalized or not, can give serious constraints on the possible scenarios at long distances. These can be quantified in the UV where the defect is weakly coupled. I will give some examples of interesting symmetric defect RG flows in (1+1) and (2+1)d.

Based on https://arxiv.org/pdf/2412.18652 and work in progress."

Minimal discrete energy problems arise in a variety of scientific contexts – such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few.     Our goal is to analyze the structure of configurations generated by optimal (and near optimal)-point configurations that minimize the Riesz s-energy over a sphere in Euclidean space R^d and, more generally, over a bounded manifold. The Riesz s-energy potential, which is a generalization of the Coulomb potential, is simply given by 1/r^s, where r denotes the distance between pairs of points. We show how such potentials for s>d and their minimizing point configurations are ideal for use in sampling surfaces.

Connections to the results by Field's medalist M. Viazovska and her collaborators on best-packing and universal optimality in 8 and 24 dimensions will be discussed. Finally we analyze the minimization of a "k-nearest neighbor" truncated version of Riesz energy that reduces the order N^2 computation for energy minimization to order N log N , while preserving global and local properties.

The security of many widely used communication systems hinges on the presumed difficulty of factoring integers or computing discrete logarithms. However, Shor's celebrated algorithm from 1994 demonstrated that quantum computers can perform these tasks in polynomial time. In 2023, Regev proposed an even faster quantum algorithm for factoring integers. Unfortunately, the correctness of his new method is conditional on an ad hoc number-theoretic conjecture. Using tools from analytic number theory, we establish a result in the direction of Regev's conjecture. This enables us to design a provably correct quantum algorithm for factoring and solving the discrete logarithm problem, whose efficiency is comparable to Regev's approach. In this talk, we will give an accessible account of these developments.

 While extremal black hole microstates are reproduced by index calculations, the study of near-BPS black holes requires special care to account for quantum fluctuations. A semiclassical analysis indicates that the spectrum of such black holes has a large extremal degeneracy followed by a mass gap up to a continuum of non-BPS states. The inclusion of a theta angle term alters the properties of the spectrum (Witten effect shifting the mass gap and mixed 't Hooft anomaly). This journal club will study two papers by Toldo and Heydeman, [2412.03695] and [2412.03697] where they study 4d near-BPS black holes. As we shall see, a key point of their derivation is the reduction to 2d JT gravity. The dual CFTs are ABJM and some class R (non lagrangian) theories. Since these theories are strongly coupled, the gravity analysis offers a powerful tool to describe their specturm at finite temperature.

Discrete and continuous energy problems that arise in a variety of scientific contexts are introduced, along with their fundamental existence and uniqueness results. Particular emphasis will be on Riesz and Gaussian pair potentials and their connections with best-packing and the discretization of manifolds. The latter application leads to the asymptotic theory (as N → ∞) for N-point configurations that minimize energy when the potential is hypersingular (short-range). For fixed N, the determination of such minimizing configurations on the d-dimensional unit sphere S d is especially significant in a range of contexts that include coding theory, discrete geometry, and physics. We will review linear programming methods for proving the optimality of configurations on S d , including Cohn and Kumar’s theory of universal optimality. The following reference will be made available during the short course: Discrete Energy on Rectifiable Sets, by S. Borodachov, D.P. Hardin and E.B. Saff, Springer Monographs in Mathematics, 2019.

Sessions:

Friday, 24 January 14:00-16:00

Friday, 31 January 14:00-16:00

The Junior Algebra and Representation Theory Seminar will kick-off the start of Hilary Term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Deep learning algorithms provide unprecedented opportunities to characterise complex structure in large data, but typically in a manner that cannot easily be interpreted beyond the 'black box'. We are developing methods to leverage the benefits of deep generative learning and computational modelling (e.g. fluid dynamics, solid mechanics, biochemistry), particularly in conjunction with biomedical imaging, to enable new insights into disease to be made. In this talk, I will describe our applications in several areas, including modelling drug delivery in cancer and retinal blood vessel loss in diabetes, and how this is leading us into the development of personalised digital twins.

I will describe the contents of a joint project with Pablo Destic and Nuno Hultberg. In the paper we confirm a conjecture of Roberto Gualdi regarding a formula for the average height of the intersection of twisted (by roots of unity) hyperplanes in a toric variety. I will introduce the 'GVF analytification' of a variety, which is defined similarly as the Berkovich analytification, but with norms replaced by heights. Moreover, I will discuss some motivations coming from (continuous) model theory and Arakelov geometry.

We present an exponential-integrator-based multi-index Monte Carlo (MIMC) method for the weak approximation of mild solutions to semilinear stochastic partial differential equations (SPDEs). Theoretical results on multi-index coupled solutions of the SPDE are provided, demonstrating their stability and the satisfaction of multiplicative error estimates. Leveraging this theory, we develop a tractable MIMC algorithm. Numerical experiments illustrate that MIMC outperforms alternative approaches, such as multilevel Monte Carlo, particularly in low-regularity settings.

Anomalies in quantum systems are present when a classical symmetry is broken by quantum effects. They give rise to physical predictions and constraints. This talk will focus on the mathematical features of anomalies of continuous, ordinary, symmetries. In the first part, we will review the topological nature of anomalies, in particular the connection to the Atiyah-Singer index theorem and its non-perturbative path-integral computation by Fujikawa. We will then discuss how anomalies and their associated (topological) Chern-Simons polynomials are related to BRST cohomology via the Stora-Zumino chain of descent equations, explaining the connection to the two-step descent procedure reviewed in the talk by Alice Lüscher last term.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Active solids consume energy to allow for actuation and shape change not possible in equilibrium. I will first introduce active solids in comparison with their active fluid counterparts. I will then focus on active solids composed of non-reciprocal springs and show how so-called odd elastic moduli arise in these materials. Odd active solids have counter-intuitive elastic properties and require new design principles for optimal response. For example, in floppy lattices, zero modes couple to microscopic non-reciprocity, which destroys odd moduli entirely in a phenomenon reminiscent of rigidity percolation. Instead, an optimal odd lattice will be sufficiently soft to activate elastic deformations, but not too soft. These results provide a theoretical underpinning for recent experiments and point to the design of novel soft machines.

High-order tensor methods employing local Taylor approximations have attracted considerable attention for convex and nonconvex optimisation. The pth-order adaptive regularisation (ARp) approach builds a local model comprising a pth-order Taylor expansion and a (p+1)th-order regularisation term, delivering optimal worst-case global and local convergence rates. However, for p≥2, subproblem minimisation can yield multiple local minima, and while a global minimiser is recommended for p=2, effectively identifying a suitable local minimum for p≥3 remains elusive.
This work extends interpolation-based updating strategies, originally proposed for p=2, to cases where p≥3, allowing the regularisation parameter to adapt in response to interpolation models. Additionally, it introduces a new prerejection mechanism to discard unfavourable subproblem minimisers before function evaluations, thus reducing computational costs for p≥3.
Numerical experiments, particularly on Chebyshev-Rosenbrock problems with p=3, indicate that the proper use of different minimisers can significantly improve practical performance, offering a promising direction for designing more efficient high-order methods.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Donaldson proved that there are pairs of 4-manifolds that are homeomorphic but not diffeomorphic, a phenomenon that does not appear for any lower dimensional manifolds. Until recently, proving this for compact manifolds has required smooth 4-manifold invariants coming from gauge theory. In this talk, we will give an introduction to an exciting new smooth 4-manifold invariant of Morrison Walker and Wedich, called a skein lasagna module that does not rely on gauge theory. Further, this talk will not assume any knowledge of 4-manifold topology.

Paraphrasing the title of Riemann’s famous lecture of 1854 I ask: What is the most rudimentary notion of a geometry? A possible answer is a path system: Consider a finite set of “points” $x_1,…,x_n$ and provide a recipe how to walk between $x_i$ and $x_j$ for all $i\neq j$, namely decide on a path $P_{ij}$, i.e., a sequence of points that starts at $x_i$ and ends at $x_j$, where $P_{ji}$ is $P_{ij}$, in reverse order. The main property that we consider is consistency. A path system is called consistent if it is closed under taking subpaths. What do such systems look like? How to generate all of them? We still do not know. One way to generate a consistent path system is to associate a positive number $w_{ij}>0$ with every pair and let $P_{ij}$ be the corresponding $w$-shortest path between $x_i$ and $x_j$. Such a path system is called metrical. It turns out that the class of consistent path systems is way richer than the metrical ones.

My main emphasis in this lecture is on what we don’t know and wish to know, yet there is already a considerable body of work that we have done on the subject.

The new results that I will present are joint with my student Daniel Cizma as well as with him and with Maria Chudnovsky.

In subfactor theory, it has been observed that operator algebras often admit symmetries beyond mere groups, sometimes called quantum symmetries. Besides recent substantial progress on the classification programs of simple amenable C*-algebras and group actions on them, there has been increasing interest in their quantum symmetries. This talk is devoted to an attempt to ensure the existence of various quantum symmetries on simple amenable C*-algebras, at least in the purely infinite case, by providing a systematic way to produce them. As a technical ingredient, a simplicity criterion for certain Pimsner algebras is given.

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.