Seminars from until

In this lecture: examples of Machine Learning applications with coding

In this talk we will review a model that was first introduced by Jonasson, Mossel and Peres. Starting with the usual square lattice on $Z^2$, entire rows (respectively columns) of edges extending along the horizontal (respectively vertical) direction are removed independently at random. On the remaining thinned lattice, Bernoulli bond percolation is performed, giving rise to a percolation model with infinite range dependencies under the annealed law. In 2005, Hoffman solved the main conjecture around this model: proving that this percolation process indeed undergoes a nontrivial phase transition. In this talk, besides reviewing this surprisingly challenging problem, we will present a novel proof, which replaces the dynamic renormalization presented previously by a static version. This makes the proof easier to follow and to extend to other models. We finally present some remarks on the sharpness of Hoffman’s result as well as a list of interesting open problems that we believe can provide a renewed interest in this family of questions. This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

We discuss the very basics of fully nonlinear elliptic equations, in close connection with regularity results. The latter include the celebrated Krylov-Safonov and Evans-Krylov theorems and the fundamental developments in Caffarelli’s theory. We also put forward more recent advances, such as the smoothness of flat solutions and the partial regularity result. In face of this panorama, we present some of our recent contributions to the theory. It covers fractional regularity estimates, the use of the Harnack approach and the connection with free boundary problems. We conclude with a discussion of a few open questions and their main challenges.

I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.

The search for the Theory of Everything has led to superstring theory, which then led physics, first to algebraic/differential geometry/topology, and then to computational geometry, and now to data science. With a concrete playground of the geometric landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of interest to theoretical physics and to pure mathematics. At the core of our programme is the question: how can AI help us with mathematics?

We present a general solution for correlators of external boundary operators in black hole states of Jackiw-Teitelboim gravity. We use the Hilbert space constructed using the particle-with-spin interpretation of the Jackiw-Teitelboim action, which consists of wavefunctions defined on Lorentzian $AdS_2$. The density of states of the gravitational system appears in the amplitude for a boundary particle to emit and reabsorb matter. Up to self-interactions of matter, a general correlator can be reduced in an energy basis to a product of amplitudes for interactions and Wilson polynomials mapping between boundary and bulk interactions.

After some basic recalls on the notion of Gromov width of a symplectic manifold, I will focus on the case of toric manifolds. I shall explain how this symplectic capacity can be estimated and even computed. This is a joint work with C. Bonala.

Extreme value statistics is essentially concerned with the modelling of rare events which are hard to predict and occur with only little warning. In this talk, I will address a number of challenges highlighted in the literature and how these align with the domain of attraction characterisation for extremes. Such a characterisation stems from a suite of mildly restrictive conditions, qualitative in nature, which not only provide computational convenience but also furnish sharp approximations to asymptotically justified models for extreme values, a key aspect to statistical testing procedures as well as interval estimation methodology in a nonparametric setting.

Extreme value statistics is essentially concerned with the modelling of rare events which are hard to predict and occur with only little warning. In this talk, I will address a number of challenges highlighted in the literature and how these align with the domain of attraction characterisation for extremes. Such a characterisation stems from a suite of mildly restrictive conditions, qualitative in nature, which not only provide computational convenience but also furnish sharp approximations to asymptotically justified models for extreme values, a key aspect to statistical testing procedures as well as interval estimation methodology in a nonparametric setting.

Finitary 2-representation theory, pioneered by Mazorchuk and Miemietz in 2010, is a categorification of finite dimensional representations of finite dimensional algebras. It primarily studies the 2-representation theory of finitary 2-categories, which are additive, linear, Krull-Schmidt 2-categories with various finiteness conditions. Much progress has been made in the area since, including various results that fall under the conceptual banner of 'internal vs. external' - that is, finding equivalences between arbitrary 'external' 2-representations and 'internal' 2-representations whose data is fully encoded with the finitary 2-category itself.

In this talk, I will start by outlining the basic theory of finitary 2-categories and their finitary 2-representations, and I will discuss two examples of 'internal' 2-representations, namely cell 2-representations and 2-representations formed of comodule 1-morphisms over a coalgebra 1-morphism. I will then discuss relaxing the finiteness assumptions of finitary 2-categories, resulting in a type of 2-category called 'locally wide finitary 2-categories'. After discussing some of the difficulties this introduces, I will focus on a specific type of locally wide finitary 2-category, namely locally wide quasi-fiat 2-categories, and discuss what we know about coalgebra 1-morphisms and their associated 2-representations in this case.

How do deep neural networks learn to construct useful features? Why do self-attention-based networks such as transformers perform so well on combinatorial tasks such as language learning? Why do some capabilities of networks emerge "discontinuously" as the computational resources used for training are scaled up? We will present perspectives on these questions through the lens of a particular class of simple synthetic tasks: learning sparse boolean functions. In part one, we will show that the hypothesis class of one-layer transformers can learn these functions in a statistically efficient manner. This leads to a view of each layer of a transformer as creating new "variables" out of sparse combinations of the previous layer's outputs. In part two, we will focus on the classic task of learning sparse parities, which is statistically easy but computationally difficult. We will demonstrate that SGD on various neural networks (transformers, MLPs, etc.) successfully learns sparse parities, with computational efficiency that is close to known lower bounds. Moreover, the training curves display no apparent progress for a long time, and then quickly drop late in training. We show that despite this apparent delayed breakthrough in performance, hidden progress is actually being made throughout the course of training.

Based on joint work with Surbhi Goel, Sham Kakade, Cyril Zhang, Boaz Barak, and Eran Malach:
https://arxiv.org/abs/2110.10090
https://arxiv.org/abs/2207.08799

Motivated by some conjectures originating in the Physics literature, I have recently been looking for closed geodesics in the K3 surfaces constructed by Lorenzo Foscolo. It turns out to be possible to locate several such with high precision and compute their index (their length is also approximately known). Interestingly, in my view, the construction of these geodesics is related to an open problem in electrostatics posed by Maxwell in 1873.

The ability to simultaneously record the activity from tens to hundreds to thousands of neurons has allowed us to analyze the computational role of population activity as opposed to single neuron activity. Recent work on a variety of cortical areas suggests that neural function may be built on the activation of population-wide activity patterns, the neural modes, rather than on the independent modulation of individual neural activity. These neural modes, the dominant covariation patterns within the neural population, define a low dimensional neural manifold that captures most of the variance in the recorded neural activity. We refer to the time-dependent activation of the neural modes as their latent dynamics and argue that latent cortical dynamics within the manifold are the fundamental and stable building blocks of neural population activity.

Quantum Hall effect is well known physics experiment, featuring precise quantization of the Hall conductance in materials with imprecisely known characteristics. It is said to be one of the most striking examples of macroscopic manifestation of quantum phenomena.

One of the approaches to explain QHE is by constructing explicit N-particle wave functions, called Quantum Hall states. It was laid out and explored in the famous works of Laughlin, Haldane, Haldane-Rezayi, Wen-Niu, Avron-Seiler-Zograf and others. It became customary to study QH states analytically on geometric backgrounds, e.g. compact Riemann surfaces. This approach turned out to be unexpectedly fruitful, leading in particular, to the discovery of topological phases of matter, anyons, non-abelian statistics, topological quantum computing and much more.

I will review this approach and talk about the program to set this approach on a rigorous mathematical footing, and to prove various conjectures in the field. I will also talk about what in my opinion are the most exciting things to do going forward. The keywords for the lectures include holomorphic line bundles, Riemann surfaces, moduli spaces, Bergman kernel, determinant point processes, Coulomb gas, etc.

Some references

1. R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50, 1395 (1983).
2. F. D. M. Haldane and E. H. Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Phys. Rev. B 31, 2529 (1985).
3. X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B41, 9377 (1990).
4. J. E. Avron, R. Seiler, and P. G. Zograf, Adiabatic Quantum Transport: Quantization and Fluctuations, Phys. Rev. Lett. 73, 3255 (1994).
5. S. Klevtsov, X. Ma, G. Marinescu, and P. Wiegmann, Quantum Hall effect and Quillen metric, Commun. Math. Phys. 349, 819 (2017).
6. S. Klevtsov, Laughlin states on higher genus Riemann surfaces, Commun. Math. Phys. 367, 837 (2019).
7. S. Klevtsov and D. Zvonkine, Geometric Test for Topological States of Matter, Phys. Rev. Lett. 128, 036602 (2021).

Latent Gaussian models (LGMs) are perhaps the most commonly used class of models in statistical applications. Nevertheless, in areas ranging from longitudinal studies in biostatistics to geostatistics, it is easy to find datasets that contain inherently non-Gaussian features, such as sudden jumps or spikes, that adversely affect the inferences and predictions made from an LGM. These datasets require more general latent non-Gaussian models (LnGMs) that can handle these non-Gaussian features automatically. However, fast implementation and easy-to-use software are lacking, which prevent LnGMs from becoming widely applicable. In this seminar, I will present the generic class of LnGMs and variational Bayes algorithms for fast and scalable inference of LnGMs. The methods can be applied to a wide range of models, such as autoregressive processes for time series, simultaneous autoregressive models for areal data, and spatial Matérn models. To facilitate Bayesian inference, we have built the ngvb package, where LGMs implemented in R-INLA can be easily extended to LnGMs by adding a single line of code.

Paper: https://arxiv.org/abs/2211.11050

Package: https://github.com/rafaelcabral96/ngvb

Quantum Hall effect is well known physics experiment, featuring precise quantization of the Hall conductance in materials with imprecisely known characteristics. It is said to be one of the most striking examples of macroscopic manifestation of quantum phenomena.

One of the approaches to explain QHE is by constructing explicit N-particle wave functions, called Quantum Hall states. It was laid out and explored in the famous works of Laughlin, Haldane, Haldane-Rezayi, Wen-Niu, Avron-Seiler-Zograf and others. It became customary to study QH states analytically on geometric backgrounds, e.g. compact Riemann surfaces. This approach turned out to be unexpectedly fruitful, leading in particular, to the discovery of topological phases of matter, anyons, non-abelian statistics, topological quantum computing and much more.

I will review this approach and talk about the program to set this approach on a rigorous mathematical footing, and to prove various conjectures in the field. I will also talk about what in my opinion are the most exciting things to do going forward. The keywords for the lectures include holomorphic line bundles, Riemann surfaces, moduli spaces, Bergman kernel, determinant point processes, Coulomb gas, etc.

Some references

1. R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50, 1395 (1983).
2. F. D. M. Haldane and E. H. Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Phys. Rev. B 31, 2529 (1985).
3. X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B41, 9377 (1990).
4. J. E. Avron, R. Seiler, and P. G. Zograf, Adiabatic Quantum Transport: Quantization and Fluctuations, Phys. Rev. Lett. 73, 3255 (1994).
5. S. Klevtsov, X. Ma, G. Marinescu, and P. Wiegmann, Quantum Hall effect and Quillen metric, Commun. Math. Phys. 349, 819 (2017).
6. S. Klevtsov, Laughlin states on higher genus Riemann surfaces, Commun. Math. Phys. 367, 837 (2019).
7. S. Klevtsov and D. Zvonkine, Geometric Test for Topological States of Matter, Phys. Rev. Lett. 128, 036602 (2021).

It is known that the 't Hooft anomalies of invertible global symmetries can be characterized by an invertible TQFT in one higher dimension. The analogous statement remains to be understood for non-invertible symmetries. In this note we discuss how the linking invariants in a non-invertible TQFT known as the Symmetry TFT (SymTFT) can be used as a diagnostic for 't Hooft anomalies of non-invertible symmetries. When the non-invertible symmetry is non-intrinsically non-invertible, and hence the SymTFT is a Dijkgraaf-Witten model, the linking invariants can be computed explicitly.

Given a Gabor orthonormal basis of $L^2(\mathbb{R})$\[\mathcal{G}(g,T,S):=\big\{ g(x-t) e^{2\pi is x}: g\in L^2(\mathbb{R}), \,t\in T,\, s\in S\big\},\]we study periodicity properties of the translation and modulation sets $T$ and $S$. In particular, we show that if the window function $g$ is compactly supported, then $T$ and $S$ must be periodic sets, i.e., of the form\[T = a\mathbb{Z}+ \{t_1,\ldots,t_n\}, \qquad S = b\mathbb{Z} + \{s_1,\ldots,s_m\}.\]To achieve this, we first obtain a result of independent interest: if the system $\mathcal{G}(g,T,S)$ is an orthonormal basis of $L^2(\mathbb{R})$, then both $|g|^2$ and $|\widehat{g}|^2$ tile $\mathbb{R}$ by translations (when translated along the sets $T$ and $S$, respectively), and moreover,\[\sum_{t\in T} |g(x-t)|^2=D(T), \qquad \sum_{s\in S} |\widehat{g}(x-s)|^2=D(S), \qquad \text{a.e. }x\in \mathbb{R},\]where $D(\Lambda)$ denotes the uniform density of a set $\Lambda\subset \mathbb{R}$.

Partial results towards the Liu-Wang conjecture are also obtained.

This talked in based on a joint work with Nir Lev (Bar-Ilan University, Israel)

We study the twisted elliptic genera of 2d $(0,4)$ SCFTs associated with the BPS strings in the twisted circle compactification of 6d rank-one $(1,0)$ SCFTs. Such objects can arise when the 6d gauge algebra allows outer automorphism, thus are classified by twisted affine Lie algebras. We study several fascinating aspects of the twisted elliptic genera including 2d localization, twisted elliptic blowup equations, Higgsing and spectral flow symmetry. We derive a recursion formula with respect to the number of strings to exactly compute the twisted elliptic genera. We also investigate the modular bootstrap of twisted one-string elliptic genera. Geometrically, our study solves the refined BPS partition functions of the underlying genus-one fibered Calabi-Yau threefolds with $N$-section.

Generative modeling is the task of drawing new samples from an underlying distribution known only via an empirical measure. There exists a myriad of models to tackle this problem with applications in image and speech processing, medical imaging, forecasting and protein modeling to cite a few. Among these methods diffusion models are a new powerful class of generative models that exhibit remarkable empirical performance. They consist of a “noising” stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a “denoising” process defined by approximating the time-reversal of the diffusion. In this talk we discuss three aspects of diffusion models. First, we will dive into the methodology behind diffusion models. Second, we will present some of their theoretical guarantees with an emphasis on their behavior under the so-called manifold hypothesis. Such theoretical guarantees are non-vacuous and provide insight on the empirical behavior of these models. Finally, I will present an extension of diffusion models to the Optimal Transport setting and introduce Diffusion Schrodinger Bridges.

Donaldson (folklore) asked whether Lagrangian Dehn twists always generate the symplectic mapping class groups in real dimension four. So far, all known examples indicate this is true, even though the symplectic Torelli group is generally much larger than the algebraic one. Yet there are only very few cases people could prove this as a theorem.

We will define a notion of "positive rational surfaces", which is equivalent to the ambient symplectic manifolds of (symplectic) log Calabi-Yau pairs. We compute the symplectic Torelli group for the positive rational surfaces and confirm Donaldson's conjecture as a result. We also answer several other questions about the symplectic Torelli groups in dimension $4$.

The principle of the holography of information states that in a theory of quantum gravity a copy of all the information available on a Cauchy slice is also available near the boundary of the Cauchy slice. This redundancy in the theory is already present at low energy. In the context of the AdS/CFT correspondence, this principle can be translated into a statement about the dual conformal field theory. We carry out this translation and demonstrate that the principle of the holography of information holds in bilocal holography.