Seminars from until

We consider stochastic processes arising from chaotic systems by evaluating an heavy tailed observable function along the orbits of the system. We prove the convergence of a normalised sum process to a Lévy process with excursions, designed to describe the oscillations observed during the clusters of extremal observations. The applications to specific systems include both hyperbolic and non-uniformly expanding systems.

The increasing dimensionality of data in the modern machine learning age presents new challenges and opportunities. The high-dimensional settings allow one to use powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations and develop new algorithmic approaches. There is indeed a decades-long tradition in statistical physics with building and solving such simplified models of neural networks.

I will give examples of recent works that build on powerful methods of physics of disordered systems to analyze different problems in machine learning and neural networks, including overparameterization, kernel methods, and the gradient descent algorithm in a high dimensional non-convex setting.

We will review the status of the uniqueness theory of static vacuum black holes, with or without a cosmological constant $\Lambda$, and we will outline the proof of a uniqueness theorem with $\Lambda \gt 0$, proved jointly in collaboration with Stefano Borghini.

I will review the construction of invariants of knots, loop braids and knotted surfaces derived from finite crossed modules. I will also show a method to calculate the algebraic homotopy 2-type of the complement of a knotted surface $\Sigma$ embedded in the 4-sphere from a movie presentation of $\Sigma$. This will entail a categorified form of the Wirtinger relations for a knot group. Along the way I will also show applications to welded knots in terms of a biquandle related to the homotopy 2-type of the complement of the tube of a welded knots.

The last stages of this talk are part of the framework of the Leverhulme Trust research project grant: RPG-2018-029: Emergent Physics From Lattice Models of Higher Gauge Theory.

Emergence of anomalous transport laws in deterministic interacting many-body systems has become a subject of intense study in the past few years. One of the most prominent examples is the unexpected discovery of superdiffusive spin dynamics in the isotropic Heisenberg quantum spin chain with at half filling, which falls into the universality class of the celebrated Kardar-Parisi-Zhang equation. In this talk, we will theoretically justify why the observed superdiffusion of the Noether charges with anomalous dynamical exponent $z=3/2$ is indeed superuniversal, namely it is a feature of all integrable interacting lattice models or quantum field theories which exhibit globally symmetry of simple Lie group $G$, in thermal ensembles that do not break $G$-invariance. The phenomenon can be attributed to thermally dressed giant quasiparticles, whose properties can be traced back to fusion relations amongst characters of quantum groups called Yangians. Giant quasiparticles can be identified with classical solitons, i.e. stable nonlinear solutions to certain integrable PDE representing classical ferromagnet field theories on certain types of coset manifolds. We shall explain why these inherently semi-classical objects are in one-to-one correspondence with the spectrum of Goldstone modes. If time permits, we shall introduce another type of anomalous transport law called undular diffusion that generally occurs amongst the symmetry-broken Noether fields in $G$-invariant dynamical systems at finite charge densities.

For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one.

Joint work with Josh Greene.

High-dimensional learning remains an outstanding phenomena where experimental evidence outpaces our current mathematical understanding, mostly due to the recent empirical successes of Deep Learning algorithms. Neural Networks provide a rich yet intricate class of functions with statistical abilities to break the curse of dimensionality, and where physical priors can be tightly integrated into the architecture to improve sample efficiency. Despite these advantages, an outstanding theoretical challenge in these models is computational, ie providing an analysis that explains successful optimization and generalization in the face of existing worst-case computational hardness results.

In this talk, I will focus on the framework that lifts parameter optimization to an appropriate measure space. I will cover existing results that guarantee global convergence of the resulting Wasserstein gradient flows, as well as recent results that study typical fluctuations of the dynamics around their mean field evolution. We will also discuss extensions of this framework beyond vanilla supervised learning, to account for symmetries in the function, as well as for competitive optimization.

In this talk, we study the homogenization of the Schrödinger equation beyond the periodic setting. More precisely, rigorous derivation of the effective mass theorems in solid state physics for non-crystalline materials are obtained.

This is a joint work with Vernny Ccajma and Jean Silva.

Finitary birepresentation theory of finitary bicategories is a categorical analog of finite-dimensional representation theory of finite-dimensional algebras. The role of the simples is played by the so-called simple transitive birepresentations and the classification of the latter, for any given finitary bicategory, is a fundamental problem in finitary birepresentation theory (the classification problem).

After briefly reviewing the basics of birepresentation theory, I will explain an analog of the double centralizer theorem for finitary bicategories, which was inspired by Etingof and Ostrik's double centralizer theorem for tensor categories. As an application, I will show how it can be used to (almost completely) solve the classification problem for Soergel bimodules in any finite Coxeter type.

Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the AdS/CFT correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code. Relating its logical code basis to Majorana dimers, we efficiently compute boundary state properties even for the non-Gaussian case of generic logical input. The dimers characterizing these boundary states coincide with discrete bulk geodesics, leading to a geometric picture from which properties of entanglement, quantum error correction, and bulk/boundary operator mapping immediately follow. We also elaborate upon the emergence of the Ryu-Takayanagi formula from our model, which realizes many of the properties of the recent bit thread proposal. Our work thus elucidates the connection between bulk geometry, entanglement, and quantum error correction in AdS/CFT, and lays the foundation for new models of holography.

In the presence of a strong magnetic field, and for an integer filling of the Landau levels, Coulomb interactions favor a ferromagnetic ground-state. It has been shown already twenty years ago, both theoretically and experimentally, that when extra charges are added or removed to such systems, the ferromagnetic state becomes unstable and is replaced by spin textures called Skyrmions. We have generalized this notion to an arbitrary number $N$ of internal states for the electrons, which may correspond to the combination of spin, valley, or layer indices. The first step is to associate a many electron wave-function, projected on the lowest Landau level, to any classical spin texture described by a smooth map from the plane to the projective space $\mathbb{CP}^{N-1}$. In the large magnetic field limit, we assume that the spin texture is slowly varying on the scale of the magnetic length, which allows us to evaluate the expectation value of the interaction Hamiltonian on these many electron quantum states. The first non trivial term in this semi-classical expansion is the usual $\mathbb{CP}^{N-1}$ non-linear sigma model, which is known to exhibit a remarkable degeneracy of the many electron states obtained from holomorphic textures. Surprisingly, this degeneracy is not lifted by reintroducing quantum fluctuations. It is eventually lifted by the sub-leading term in the effective Hamiltonian, which selects a hexagonal Skyrmion lattice and therefore breaks both translational and internal $SU(N)$ symmetries. I will show that these optimal classical textures can be interpreted in an appealing way using geometric quantization.

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some special cases even cutoff.

No prior knowledge is assumed.

Based on joint work with Evita Nestoridi (Princeton) and Dominik Schmid (Munich).

Deep learning continues to dominate machine learning and has been successful in computer vision, natural language processing, etc. Its impact has now expanded to many research areas in science and engineering. In this talk, I will mainly focus on some recent impact of deep learning on computational mathematics. I will present our recent work on bridging deep neural networks with numerical differential equations. On the one hand, I will show how to design transparent deep convolutional networks to uncover hidden PDE models from observed dynamical data. On the other hand, I will present our preliminary attempt to establish a deep reinforcement learning based framework to solve 1D scalar conservation laws, and a meta-learning approach for solving linear parameterized PDEs based on the multigrid method.

Random missing data can constitute a problem when modelling rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models which cover these situations. In this talk, we consider a family of models $\{Y_n\}, n\geq 1$, that can be associated to automatic systems which have a periodic control, in the sense that it is guaranteed that at instants multiple of $T$, $T\geq 2$, no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered.

We characterize the extremal behaviour of $\{Y_n\}, n\geq 1$, and obtain its extremal index expression. A consistent estimator for the model parameter is also proposed and its finite sample behaviour analysed.

Joint work with Helena Ferreira (UBI) and Maria da Graça Temido (UC).

Random missing data can constitute a problem when modelling rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models which cover these situations. In this talk, we consider a family of models $\{𝑌_𝑛\}$, $𝑛≥1$, that can be associated to automatic systems which have a periodic control, in the sense that it is guaranteed that at instants multiple of $T$, $T ≥ 2$, no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered.

We characterize the extremal behaviour of $\{𝑌_𝑛\}$, $𝑛\geq1$, and obtain its extremal index expression. A consistent estimator for the model parameter is also proposed and its finite sample behaviour analysed.

Joint work with Helena Ferreira (UBI) and Maria da Graça Temido (UC).

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents, though we do not know how accurate it is. I will show that the low-energy properties of the Kitaev honeycomb lattice model, such as the shape of Majorana zero modes or the deformations of the correlation length, are faithfully described in terms of Riemann-Cartan geometry. Moreover, I will present how effective axial gauge fields can couple to Majorana fermions, thus giving a unified picture between vortices and lattice dislocations that support Majorana zero modes.

It was observed by J. Kędra that there are many symplectic 4-manifolds $(M, \omega)$, where $M$ is neither rational nor ruled, that admit no circle action and $\pi_1 (\mathrm{Ham}( M))$ is nontrivial. In the case $M={\mathbb C\mathbb P}^2\#\,k\overline{\mathbb C\mathbb P}\,\!^2$, with $k \leq 4$, it follows from the work of several authors that the full rational homotopy of $\mathrm{Symp}(M,\omega)$, and in particular their fundamental group, is generated by circle actions on the manifold. In this talk we study loops in the fundamental group of $\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2) $ of symplectomorphisms that act trivially on homology, and show that, for some particular symplectic forms, there are loops which cannot be realized by circle actions. Our work depends on Delzant classification of toric symplectic manifolds and Karshon's classification of Hamiltonian circle actions

This talk is based in joint work with Miguel Barata, Martin Pinsonnault and Ana Alexandra Reis.

Inverse problems in imaging range from tomographic reconstruction (CT, MRI, etc) to image deconvolution, segmentation, and classification, just to name a few. In this talk I will discuss approaches to inverse imaging problems which have both a mathematical modelling (knowledge driven) and a machine learning (data-driven) component. Mathematical modelling is crucial in the presence of ill-posedness, making use of information about the imaging data, for narrowing down the search space. Such an approach results in highly generalizable reconstruction and analysis methods which come with desirable solutions guarantees. Machine learning on the other hand is a powerful tool for customising methods to individual data sets. Highly parametrised models such as deep neural networks in particular, are powerful tools for accurately modelling prior information about solutions. The combination of these two paradigms, getting the best from both of these worlds, is the topic of this talk, furnished with examples for image classification under minimal supervision and for tomographic image reconstruction.

False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the Circle Method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions.

Transport in disordered one-dimensional wires is described by diffusion at short distances/times and by Anderson localization at long distances/times. I will investigate how this picture is altered in a disordered multi-channel wire where some of the channels are topologically protected. This scenario can be realized at the interface between two quantum Hall systems, in a Weyl semimetal in a magnetic field or at the edge of a quantum spin Hall insulator. Technically, the problem is described by a $0+1$-dimensional field theory in the form of a supersymmetric non-linear sigma model with a topological term. I will show how to solve this field theory exactly to obtain DC (static) transport quantities such as DC conductance and shot noise as well as dynamical responses such as diffusion probability of return and correlations of local density of states. I will discuss several surprising findings of this exact solution. First, I find that disorder is much more effective in localizing the diffusive channels in the presence of topologically protected ones. This can be understood in terms of statistical level repulsion by mapping the problem to that of a random matrix ensemble with zero eigenvalues. Second, I find that localization corrections dramatically alter the long time behavior of the return probability in a system described by diffusion+drift equation at the classical level. Finally, I find that in a disordered wire with topologically protected channels, the wave functions display level attraction rather than level repulsion.

I will discuss the impact of nuisance parameters on the effectiveness of supervised classification in high energy physics problems, and techniques that may mitigate or remove their effect in the search for optimal selection criteria and variable transformations. The approaches discussed include nuisance parametrized models, modified or adversary losses, semi supervised learning approaches and inference-aware techniques.

We define and study asymptotic Killing and conformal Killing vectors in $d$-dimensional Minkowski, $(A)dS$, $\mathbb{R} \times S^{d−1}$ and $AdS_2 \times S^{d−2}$. We construct the associated quantum charges for an arbitrary CFT and show they satisfy a closed algebra that includes the BMS as a sub-algebra (i.e. supertranslations and superrotations) plus a novel transformation we call `superdilations'. We study representations of this algebra in the Hilbert space of the CFT, as well as the action of the finite transformations obtained by exponentiating the charges. In the context of the AdS/CFT correspondence, we propose a bulk holographic description in semi-classical gravity that reproduces the results obtained from CFT computations. We discuss the implications of our results regarding quantum hairs of asymptotically flat (near-) extremal black holes.

This talk is based on joint work with M.Khovanov and Y.Kononov. By evaluating a topological field theory in dimension $2$ on surfaces of genus $0,1,2$, etc., we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds $M_d$, including the contribution of the isometries of $M_d$. We then use the result to perform a counting of microstates for electrically charged and rotating supersymmetric black strings in $\operatorname{AdS}_5 \times S^5$ and $\operatorname{AdS}_7 \times S^4$.

The waves that describe systems in quantum physics can carry information about how their environment has been altered, for example by forces acting on them. This effect is the geometric phase. It also occurs in the optics of polarised light, where it goes back to the 1830s; and it gives insight into the spin-statistics relation for identical quantum particles. The underlying mathematics is geometric: the phenomenon of parallel transport, which also explains how falling cats land on their feet, and why parking a car in a narrow space is difficult. Incorporating the back-reaction of the geometric phase on the dynamics of the changing environment exposes the unsolved problem of how strictly a system can be separated from a slowly-varying environment, and involves different mathematics: divergent infinite series.