The Speakers
Professor Ari Laptev
Title: Magnetic rings
Abstract: We study functional and spectral properties of perturbations of a magnetic second order differential operator on a circle. This operator appears when considering the restriction to the unit circle of a two dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove some Hardy-type inequalities and sharp Keller-Lieb-Thirring inequalities.
Affliation: Imperial College London
Professor Bruno Franchi
Title: Poincaré and Sobolev inequalities for differential forms in Euclidean spaces and Heisenberg groups
Abstract: PDF
Affliation: University of Bologna
Professor Raffaella Carbone
Title: On log-Sobolev and other coercive inequalities for quantum Markov semigroups
Abstract: We revisit some results about variational inequalities for quantum Markov semigroups defined on von Neumann algebras.
In particular, we concentrate on the generalization to the non-commutative context of some well-established classical results about the relations linking spectral gap and different forms of log-Sobolev inequalities to (hyper)contractivity and entropy decay properties.
As illustrative examples, we shall briefly discuss a two-level atom model and a quantum Ornstein-Uhlenbeck semigroup.
Affliation: University of Pavia
Professor Adam Sikora
Title: Probabilistic approach to quantum separation effect for Feynman-Kac semigroup (joint work with Jacek Zienkiewicz)
Abstract: Quantum tunnelling phenomenon allows a particle in Schrödinger mechanics tunnels through a barrier that it classically could not overcome. Even the infinite potentials do not always form impenetrable barriers. We discuss an answer to the following question: What is a critical magnitude of potential, which creates impenetrable barrier and for which the corresponding Schrödinger evolution system separates? In addition we describe some quantitative estimates for the separating effect in terms of cut-off potentials.
Affliation: Macquarie University
Professor Fabio Eugenio Giovanni Cipriani
Title: Markovian convergence to KMS equilibria
Abstract: The aim of the talk is to show a method of construction of noncommutative Dirichlet forms with respect to KMS equilibrium states and to provide coercive inequalities implying a control on the convergence to equilibrium for the associated Markovian semigroup.
Affliation: The Polytechnic University of Milan
Professor Carme Cascante
Affliation: University of Barcelona
Professor Boguslaw Zegarlinski
Title: TBA
Abstract: TBA
Affliation: Imperial College London
Dr. Serena Frederico
Title: Smoothing and Strichartz estimates for a class of time-degenerate Schrödinger operators
Abstract: In this talk we will investigate the homogeneous and the inhomogeneous smoothing effect of some time-degenerate Schrödinger operators. These estimates will allow us to prove local well-posedness results for the associated semilinear and nonlinear Cauchy problem. The results above were obtained in collaboration with Gigliola Staffilani. Additionally, we shall see that Strichartz estimates can be derived for a related class of time-degenerate Schrödinger operators. These inequalities will give a local well-posedness result for a different associated semilinear Cauchy problem. These last results were obtained in collaboration with Michael Ruzhansky.
Affliation: Ghent University
Marianna Chatzakou
Title: q-Poincaré inequalities on Carnot groups
Abstract: We prove q-Poincaré inequalities for probability measures on nilpotent Lie groups whose filiform Lie algebra is of any length, and has a density (with respect to the Haar measure) given as a function of a suitable homogeneous norm.
Joint work with Serena Federico and Boguslaw Zegarlinski.
Affliation: Imperial College London
Esther Bou Dagher (PhD Student)
Title: Coercive Inequalities and U-Bounds
Abstract: Coercive inequalities such as the Poincaré and Logarithmic Sobolev inequalities have been the subject of study because of their importance in applications. W. Hebisch and B. Zegarliński introduced the method of U-bounds to prove such inequalities in metric measure spaces. I adapt and apply their methods to enlarge the class of measures for which such inequalities hold true and prove that those imply the Talagrand inequality and hypercontractivity. In the case where the metric space is the H-type group, I extend, by giving a simpler proof, a result of J.Inglis which involves a measure depending on the pth-power of the Kaplan norm by one which depends on a more general increasing function of the Kaplan norm.
Affliation: Imperial College London
Yifu Wang (PhD student)
Title: An improved Adam's Inequality and Its Application
Abstract: In this talk we consider a probability measure with a potential imposed with Adam's regularity conditions. We generalize the Adam's inequality and apply it to prove the equivalence between two norms: the traditional Sobolev norm and the norm concerning a Friedrichs extension of a Dirichlet operator. The talk also involves some discussion of the best constant of higher order Poincare's inequality.
Affliation: Imperial College London
Mengchun Zhang (PhD student )
Title: Logarithmic Schrodinger Equation on Infinite Lattice
Abstract: Logarithmic Schrödinger equation (LSE) has been widely used in mathematical physics to model nonlinear systems owing to its unique properties. The configuration space of LSE is usually set as R^n, but this fails to model systems with infinitely many particles. To solve this problem, we extend LSE from R^n to an infinite-dimensional lattice Ω=R^(Z^d). The topology on Ω is constructed by introducing a proper potential field and the associated global Gibbs measure μ. Then, with φ being the solution of LSE, we show that μ|φ| is invariant over time and μ|∇φ|≤ Ae^(ct) for some positive constants A and c. In addition, we prove that μ satisfies a logarithmic Sobolev inequality, which can be used to study the behaviour of the spectrum of the logarithmic Schrödinger operator.
Affliation: Imperial College London
Larry Read (PhD student)
Title: A Lieb-Thirring inequality for Schrödinger operators with Soliton term
Abstract: In this talk we will consider Schrödinger operators in 1D with a Soliton term and use a factorisation method to prove a bound on the normalised sum of 3/2 moments of its negative eigenvalues. We will also discuss what similar inequality we may attain for general $-\Delta-W-V$.
Affliation: Imperial College London
Anderson Melchor Hernancez (PhD student)
Title: A Berry-Esseen bound: A rate of convergence in Kolmogorov distance
Abstract: In this talk, I will discuss a Berry-Esseen upper bound for the variable speed random walk (VSRW). In the present analysis, I assume a moment condition and ergodicity in the form of a logarithmic Sobolev inequality (LSI). I make use of a LSI that only requires finite oscillations on random variables and allows us to show hypercontractivity features on harmonic functions. As a consequence, if the dimension of the state space is large enough, we prove convergence in Kolmogorov distance of order t -1/10.
Affliation: University of Pavia
Federico Girotti (PhD student)
Title: Absorption in invariant domains for semigroups of quantum channels
Abstract: Quantum channels are the mathematical objects used to describe a wide class of changes of state that a quantum system (even an open one) can undergo. The concept of absorption probability exists and is well-known in probability, for classical Markov chains, widely used to study asymptotics of the evolution. We introduce a notion of absorption operators in the context of quantum channels in order to takle the absorption problem in invariant domains (enclosures) for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous time. We define a well-behaving set of positive operators which can correspond to classical absorption probabilities and we study their basic properties, in general, and with respect to transience and recurrence. This is a joint work with Raffaella Carbone.
Affliation: University of Pavia
Xianfeng Ren (PhD student)
Title: Trilinear Estimate for Derivative Schrödinger Equation in All subcritical space
Abstract: We are going to discuss the trilinear estimate to Nonlinear Schreodinger Equations with derivative term after using gauge transformation. This inequality is key to establishing the local well-posedness for this evolutionary equation. The main ingredients in proof are modulation dispersive decay and frequency analysis under uniform decomposition where we aim to solve the loss of derivatives.
Affliation: Imperial College London