04&07/03/2024 : Stefan Teufel @ PoliMi

Monday, March 4, 2024 – 14:15 and Thursday, March 7, 2024 – 10:15
Sala Consiglio
Dipartimento di Matematica
Politecnico di Milano
Ed. 14 “Nave”, Campus Leonardo


SPEAKER: Stefan Teufel (Eberhard Karls Universität Tübingen)


Quantisation of Hall conductivity in infinite interacting fermion systems

Understanding the exact quantisation of the experimentally observed Hall
conductivity has led to many interesting developments also in mathematical
physics and to two Nobel prizes in physics. Although the problem is now more
than 40 years old, significant progress has been made on a rigorous level for
interacting systems in the last ten years. In my talk I will briefly review
some of the mathematical highlights of this development and then present a new result: We consider infinite systems of interacting electrons on a lattice
governed by a translation-invariant Hamiltonian with a unique gapped ground
state. Using the NEASS approach, we show that the Hall conductivity of such a
system at zero temperature is given by a many-body version of the famous
double-commutator formula without power-law corrections. We also show that this formula takes only quantised values. The main novelty compared to
existing mathematical results is that we consider the conductivity instead of
the conductance, and that working directly in infinite volume simplifies and
clarifies some arguments, provided one is willing to work in the C^*-algebraic
framework used in the mathematical description of such systems.

In the first lecture I will mainly talk about the history of the problem and set up the mathematical framework. In the second lecture I will present the new results. The latter are based on joint work with Giovanna Marcelli, Tadahiro Miyao, Domenico Monaco and Marius Wesle.

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

28/2/2024: Filippo Boni @ UnIns

Wednesday, February 28, 2024 - 9:45
Aula V2.10
Dipartimento di Scienza e Alta Tecnologia
Università degli Studi dell'Insubria
Via Valleggio 11, Como

SPEAKER: Filippo Boni (Scuola Superiore Meridionale)


Normalized ground states for Nonlinear Schrödinger equations on metric graphs

In this talk we present some results about the existence of normalized ground states on noncompact metric graphs for nonlinear Schrödinger equations involving possibly both a standard power nonlinearity and delta nonlinearities located at the vertices of the graph. In the first part, we review some results when the sole standard nonlinearity is present.
In the second part, we present more recent results, both when only delta nonlinearities at the vertices are considered and when standard and delta nonlinearities coexist. In the first case, we show that the ground state problem strongly depends on the degree of periodicity of the graph, the total number of delta nonlinearities and their dislocation in the graph. In the second case, we highlight that the existence of ground states is strongly affected by the value of the mass, the relation between the powers of the two nonlinear terms and by topological and metric properties of the graph itself. These results have been obtained in collaboration with R. Adami, S. Dovetta and E. Serra (Politecnico di Torino).

04/03/2024: Joachim Kerner @ UniMi

Monday, March 4, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Joachim Kerner (FernUniversität in Hagen)


On Bose-Einstein Condensation in the Random Kac-Luttinger Model

This talk is concerned with a random many-particle model originally considered by Kac and Luttinger in 1973 in order to study a well-known quantum phase transition known as Bose–Einstein condensation (BEC). Generally speaking, to understand this phase transition in interacting many-particle systems is a current hot topic in mathematical physics. However, due to the complexity of the underlying random one-particle model, the nature of the BEC in the non-interacting Kac-Luttinger model was understood only recently based on results obtained by Alain-Sol Sznitman (ETH). In this talk, our goal will be to understand the impact of repulsive two-particle interactions on this condensate. We will see that, due to the spatial localization of the condensate, strong enough interactions will immediately destroy it. On the other hand, for two-particle interactions of a mean-field type, we prove BEC in the interacting Kac–Luttinger model into a minimizer of a Hartree-type functional. This talk is based on joint work with C. Boccato (Milan), M. Pechmann (Tennessee), and W. Spitzer (Hagen).

The seminar is part of the activities of the project PRIN 2022AKRC5P “Interacting Quantum Systems: Topological Phenomena and Effective Theories” financed by the European Union – Next Generation EU.

06/02/2024: Marco Olivieri @ PoliMi

Tuesday, February 6, 2024 – 14:15
Sala Consiglio
Dipartimento di Matematica
Politecnico di Milano
Ed. 14 “Nave”, Campus Leonardo


SPEAKER: Marco Olivieri (University of Copenhagen)


A novel method for the derivation of the free energy expansion of the Bose gases

We present an innovative method to derive the expansion of the free energy
density of a dilute Bose gas in thermodynamic regime in dimension three. In the first part of the talk, we give a gentle introduction on the thermodynamic limit as the optimal tool for the macroscopic derivation of the thermodynamics from the quantum statistical mechanics. In the second part of the talk, we will focus our attention on the macroscopic behavior of Bose gases, which have a phase transition to Bose Einstein condensates for low temperature. We then study a gas of many bosons interacting through a spherical, pairwise, positive potential. Using a combination of the renormalization of the potential, the Neumann localization and Bogoliubov diagonalization, we derive a lower bound for the second order expansion of the free energy.

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on thetopics presented at the talk.