In distributed quantum computing, the final solution of a problem is usually achieved by catenating these partial solutions resulted from different computing nodes, but intolerable errors likely yield in this catenation process. In the first part of this talk, I would like to introduce a universal error correction scheme to reduce errors and obtain effective solutions. Then, we apply this error correction scheme to designing a distributed phase estimation algorithm that presents a basic tool for studying distributed Shor’s algorithm and distributed discrete logarithm algorithm as well as other distributed quantum algorithms (for example, distributed quantum counting algorithm and distributed HHL algorithm). In the second part, I would like to introduce a quantum computing model--new quantum pushdown automata. For defining this quantum computing model, I would present a new definition of classical pushdown automata.
In this talk, we discuss a stochastic velocity tracking problem for the 2D-Navier-Stokes equations perturbed by a multiplicative Gaussian noise. From physical point of view, the control acts through a boundary injection/suction device with uncertainty, modelled by non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of solution to the state equation and prove the existence of an optimal solution to the control problem. In addition, the first-order necessary optimization conditions are analysed.
N.V. Chemetov acknowledges support from FAPESP, Grant 2024/16483-5: Theoretical study of mathematical models in fluid dynamics.
Joint work with Fernanda Cipriano (New University of Lisbon, Portugal).
References
N.V. Chemetov, F. Cipriano, Optimal control of Newtonian fluids in a stochastic environment. SIAM Journal on Mathematical Analysis, 57 (1), 364-403, 2025.
N.V. Chemetov, F. Cipriano, A boundary control problem for stochastic 2D-Navier-Stokes equations. Journal of Optimization Theory and Applications 203 (2), 1847-1879, 2024.
Inspired by Jaeger’s composition formula for the HOMFLY polynomial, Turaev defined a coproduct on the HOMFLY skein algebra of a framed surface S, turning it into a bialgebra. Jaeger’s formula can be viewed as a universal version of the restriction of the defining representation from $\operatorname{GL}_{m+n}$ to $\operatorname{GL}_m × \operatorname{GL}_n$. The restriction functor, however, is not braided, and therefore there is a priori no reason for the induced linear map between the corresponding skein algebras to be multiplicative. In this talk, I will address this problem using defect skein theory and the formalism of parabolic restriction.
In the first part of the talk, I will introduce skein theory for 3-manifolds with both surface and line defects. Local relations near the defects are produced from the algebraic data of a central algebra (codimension 1) and a centred bimodule (codimension 2). Examples of such structures are provided by the formalism of parabolic restriction. In the second part of the talk, I will explain how to construct a universal version of this formalism. Finally, we will see how Turaev’s coproduct extends to the entire skein category using the previous constructions.
The celebrated “hardness vs. randomness” paradigm lets us derandomize algorithms under the assumption that certain computational problems are hard to solve. Classical applications of this paradigm led to many exciting results in complexity theory, and in recent years we have been able to overcome several barriers of the original approach, using a host of new techniques.
In this talk I will survey a small selection of recent results in hardness vs. randomness. The common theme will be a close look at reconstructive pseudorandom generators, studying the complexity of reconstruction and the possibility of deterministic reconstructions.
The talk will be high-level and will aim to assume no prior knowledge.
In this talk, we consider the facilitated exclusion process on the one-dimensional discrete $N$-torus. Because of the facilitating mechanism, the process freezes in finite time if the particle density of the initial configuration is subcritical, i.e., if it is smaller than (or equal to) 1/2. We prove that, starting from any subcritical Bernoulli product measure, the correct scale of the transience/freezing time is of order $\log^3(N)$. Based on a joint work with Oriane Blondel, Clément Erignoux and Sanha Lee.
In this talk, I explore how internal deformations in spinless extended bodies, such as periodic pulsations or oscillations of the body’s shape, can trigger chaotic motion even when the background spacetime is fixed and the underlying geodesic dynamics is integrable. Using Dixon’s multipolar framework, I show how time-periodic finite-size effects can split separatrices and generate homoclinic chaos, diagnosed via the Melnikov method. I discuss Schwarzschild black holes for nearly spherical bodies with oscillating oblateness, as well as spherically symmetric pulsations in non-vacuum spacetimes, including electrovac/charged black holes and black holes embedded in dark-matter halos. Finally, I highlight a genuinely relativistic boundary of the point-particle idealization: in Ricci-flat vacuum, spinless spherical bodies move geodesically through quadrupole order, yet finite-size deviations can arise at hexadecapole order, where even vacuum Schwarzschild admits nontrivial corrections and chaos under pulsations.
In these lectures we review recent results on the fluctuations of a reaction-diffusion model. We consider a one-dimensional dynamics obtained as a superposition of a Glauber and a Kawasaki dynamics. The reaction term is tuned so that a dynamical phase transition occurs in the model as a suitable parameter is varied. We study dynamical fluctuations of the density field at the critical point.
We characterise the slowdown of the dynamics at criticality, and prove that this slowdown is induced by a single observable, the global density (or magnetisation). We show that magnetisation fluctuations are non-Gaussian and characterise their limit as the solution of a non-linear SDE. We prove, furthermore, that other observables remain fast: the density field acting on the fast modes (i.e. on mean-0 test functions) and with Gaussian scaling converges, in the sense of finite dimensional distributions, to a Gaussian field with space-time covariance that we compute explicitly.
I will first introduce the bigraded cohomology for real algebraic varieties developed by Johannes Huisman and Dewi Gleuher. This is a cohomology theory that refines the equivariant cohomology "à la Kahn-Krasnov" of the complex points of a real variety, the latter often being preferred (by the algebraic geometers) in the cohomological study of real algebraic varieties. Since the construction of this bigraded cohomology and its associated characteristic classes relies on the sheaf exponential morphism, I will explain how to produce an arithmetic (or algebraic) variant of these cohomology groups, whose main advantage is toeliminate topological or transcendental conditions. I will conclude by comparing these two versions of bigraded cohomology.
In these lectures we review recent results on the fluctuations of a reaction-diffusion model. We consider a one-dimensional dynamics obtained as a superposition of a Glauber and a Kawasaki dynamics. The reaction term is tuned so that a dynamical phase transition occurs in the model as a suitable parameter is varied. We study dynamical fluctuations of the density field at the critical point.
We characterise the slowdown of the dynamics at criticality, and prove that this slowdown is induced by a single observable, the global density (or magnetisation). We show that magnetisation fluctuations are non-Gaussian and characterise their limit as the solution of a non-linear SDE. We prove, furthermore, that other observables remain fast: the density field acting on the fast modes (i.e. on mean-0 test functions) and with Gaussian scaling converges, in the sense of finite dimensional distributions, to a Gaussian field with space-time covariance that we compute explicitly.
In these lectures we review recent results on the fluctuations of a reaction-diffusion model. We consider a one-dimensional dynamics obtained as a superposition of a Glauber and a Kawasaki dynamics. The reaction term is tuned so that a dynamical phase transition occurs in the model as a suitable parameter is varied. We study dynamical fluctuations of the density field at the critical point.
We characterise the slowdown of the dynamics at criticality, and prove that this slowdown is induced by a single observable, the global density (or magnetisation). We show that magnetisation fluctuations are non-Gaussian and characterise their limit as the solution of a non-linear SDE. We prove, furthermore, that other observables remain fast: the density field acting on the fast modes (i.e. on mean-0 test functions) and with Gaussian scaling converges, in the sense of finite dimensional distributions, to a Gaussian field with space-time covariance that we compute explicitly.
Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$ ) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$ ). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.
But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$. The hierarchical structure of the $p$-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.
Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$ ) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$ ). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.
But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$. The hierarchical structure of the $p$-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.
The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(T)$ generated by an arbitrary sequence of points in the unit disk that do not approach the boundary very fast. The nth point of the sequence is responsible for multiplying the nth and subsequent terms of the system by a Möbius transform taking the point to 0. One can recover the classical trigonometric system, its perturbations or conformal transformations, as particular examples of the MT system. However, for many interesting choices of the generating sequence, the MT system is less understood. We prove almost everywhere convergence of the MT series for three different classes of generating sequences.
The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize.
From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems.
Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems.
In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.
The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize.
From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems.
Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems.
In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.
