Category: Seminar UniMi

June 5, 2023 - 11:15
Aula dottorato, first floor
Università degli Studi di Milano
Via Saldini 50

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SPEAKER: Andreas Deuchert (Universität Zürich)

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Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in General External Fields

We consider the Bardeen-Cooper-Schrieffer (BCS) free energy functional with weak and macroscopic external electric and magnetic fields and derive the Ginzburg-Landau functional. We also provide an asymptotic formula for the BCS critical temperature as a function of the external fields. This extends our previous results in arXiv:2105.05623 for the constant magnetic field to general magnetic fields with a nonzero magnetic flux through the unit cell. This is joint work with C. Hainzl and M. Maier.

13/02  14:00-16:00   Sala di rappresentanza, ground floor, Mathematics Department, Via C. Saldini 50
14/02  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
15/02  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
1/03   14:00-16:00   Aula Mp, Via Mangiagalli 32
2/03   10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
3/03   10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
13/03  14:00-16:00   Aula C10, Via Mangiagalli 25
14/03  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
15/03  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50

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SPEAKER: Matteo Gallone (SISSA Trieste)

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Introduction to Renormalisation Group for Fermionic Models

This series of seminars presents techniques used to rigorously approach the analysis of statistical mechanical systems of fermions. These include:
i) Gaussian integration using Feynman graphs
ii) Grassmann variables and Grassmann Gaussian integration
iii) ​Grassmann representation of the 2D Ising Model with quasi-periodic disorder
iv) Decay of the 2-point correlation function.

Students may get credit (in the category of seminar type F) for this course. Please contact Niels Benedikter if you intend to receive credit: niels.benedikter__A_T__unimi.it

November 24, 2022, 15:45
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Luca Fresta (Universität Bonn)

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Stochastic Analysis of Subcritical Euclidean Fermionic Field Theories

In my talk, I will introduce a forward-backward stochastic differential equation
which provides a stochastic quantisation of subcritical Grassmann measures. The method is inspired by the so-called continuous renormalisation group, but
avoids the technical difficulties encountered in the direct study of the Polchinski’s flow equation for the effective potentials. If time permits, I will also show how to prove the exponential decay of correlations by a coupling method.
Work in collaboration with De Vecchi and Gubinelli.

November 21, 2022, 11:15
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Nikolai Leopold (Universität Basel)

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Norm approximations for the Fröhlich dynamics

In this talk I will discuss recent results about the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. For large particle number and initial data in which the particles are in a Bose-Einstein condensate and the excitations of the phonon field are in a coherent state I will show that the time evolved many-body state can be approximated in norm by an effective dynamics. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics.
If time permits I will, in addition, present a joint work with D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer about the Fröhlich model in the strong coupling limit and compare the Bogoliubov dynamics in the strong coupling and mean-field regime.

December 12, 2022, 11:15
- Aula Dottorato, 1st floor -
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Vojkan Jakšić (McGill University Montreal)

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Some remarks on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

This talks concerns slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice Z^d . This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states, we will discuss a number of general structural results regarding such realizations.

This talk is based on a joint work with C-A Pillet and C. Tauber.

May 30, 2022, 16:30 (non-standard time)
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Laurent Lafleche (U Texas at Austin)

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Semiclassical regularity and mean-field limit
with singular potentials

In this talk I will present several techniques and concepts used in the context of the mean-field and the classical limit allowing to go from the N -body Schrödinger equation with singular potential to the Hartree–Fock and Vlasov equations, linked to works in collaboration with Chiara Saffirio and Jacky Chong. At the level of the Vlasov–Poisson equation, typical mean-field techniques from quantum mechanics for pure states can be translated to a weak-strong stability estimate in L^1 for the Vlasov equation. Another weak-strong stability can be obtained for the difference of the square roots of the solutions in L^2. They allow to better understand the mean-field and semiclassical estimates. These estimates are weak-strong in the sense that they require only the regularity of one of the solutions. This requires the propagation of a semi-classical notion of regularity uniformly in N and h. A typical obstacle is the lack of positivity of the Wigner transform and its few conserved quantities. A solution to this problem is to consider operators as the right generalization of the phase space distribution, and a quantum analogue of Sobolev spaces defined using Schatten norms. The advantage of these techniques is that they allow to obtain regularity estimates without higher order error terms.

May 16, 2022, 16:30 (non-standard time)
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Robin Reuvers (U Roma Tre)

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Ground state energy of dilute Bose gases in 1D

In 1963, Lieb and Liniger formulated an exactly solvable model for interacting bosons in 1D. Thanks to its exact, Bethe ansatz solution, the model and its generalizations soon became popular objects of study in mathematical physics. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, the Lieb-Liniger model found experimental use and became even better known.

In the meantime, Lieb and collaborators had moved on, and were rigorously studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to acquire, and a popular goal was the rigorous derivation of the ground state energy of gases of bosons in various settings in 2 and 3D. Many of the results focused on the dilute limit, in which the density of the boson gas is very low.

Somehow, Bose gases in 1D were excluded from this development. Of course, the original Lieb-Liniger model provided a solvable example, but we can nevertheless use insights from the 2 and 3D approaches to prove new results about the ground state energy of dilute Bose gases in 1D.

In the talk, I will review the developments above, and explain the new results.

February 21, 2022, 16:00 (UNUSUAL TIME!)
Zoom only (online talk)

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SPEAKER: Ian Jauslin (Rutgers University)

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An effective equation to study Bose gases at all densities

I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a one-particle problem, which incorporates the many-body
correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.

February 7, 2022, 14:00
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Ngoc Nhi Nguyen (U Paris Saclay)

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Fermionic semiclassical L^p estimates

Spectral properties of Schrödinger operators are studied a lot in mathematical physics. They can give the description of trapped fermionic particles. Researches on the spatial concentration of semiclassical Schrödinger operators’ eigenfunctions are still carried out, whether in physics or in mathematics. There are very precise results in special cases like the harmonic oscillator. However, it is not always possible to obtain explicitly point wise information for more general potentials. We can measure the concentration by estimating these functions with L^p bounds.