Scuola tematica "Modern Trends in Pure and Applied Mathematics"

26 May 2025, 09:00 → 30 May 2025, 12:55 Europe/Rome

Aula Dini (Scuola Normale Superiore, piazza del Castelletto - Pisa)

Nell’ambito del DM 231/2023 che vede come primo obiettivo la “promozione della collaborazione tra le Scuole Universitarie Superiori e altre Istituzioni di Istruzione Superiore, allo scopo di stabilire partenariati e convenzioni per l’organizzazione di attività formative comuni” e del DM 291/2024, che prevede, tra l’altro, lo "sviluppo di percorsi di formazione undergraduate innovativi e interdisciplinari (“seasonal school”) della durata di una settimana in modalità residenziale presso uno dei poli della rete, in linea con i temi di ricerca di MERITA e connotati da forte integrazione e sinergie per le competenze trasversali" la Scuola Galileiana di Studi Superiori (SGSS) organizza in collaborazione con la Scuola Normale Superiore (SNS) una Scuola Tematica denominata "Modern Trends in Pure and Applied Mathematics". I principali obiettivi formativi della Scuola sono fornire un’introduzione ad argomenti avanzati di Matematica, sia Pura che Applicata, che normalmente non sono oggetto degli insegnamenti curriculari della laurea magistrale. Al termine del corso gli studenti avranno familiarità con alcuni temi centrali nella ricerca in analisi geometrica (fenomeni di rigidità in Geometria Riemanniana ed in Relatività Generale), fisica matematica (limiti su larga scala di interazioni in gas di Bose), sistemi dinamici (biliardi matematici), geometria algebrica (dimensione essenziale).

La scuola si terrà in presenza dal 26 al 30 maggio 2025 presso la sede della SNS a Pisa, per un totale di 28 ore, corrispondenti a 4 Crediti Formativi Universitari (CFU). La scuola tematica prevede la partecipazione di un numero massimo di 30 iscritti; la lingua ufficiale della scuola è l'inglese.

Il corso si compone di 5 giornate ordinarie; oltre alle ore di lezione sono previsti alcuni seminari, più avanzati, ma di ampia visione, sui temi del corso.  

Relatori:

• Virginia Agostiniani, Università di Trento
• Serena Cenatiempo, Gran Sasso Science Institute, L’Aquila
• Alfonso Sorrentino, Università di Roma Tor Vergata
• Angelo Vistoli, Scuola Normale Superiore, Pisa

Monday, 26 May

1 V. Agostiniani (University of Trento) - Minicourse "Rigidity phenomena arising in Riemannian Geometry and in General Relativity"

In geometric analysis, rigidity phenomena occur when specific conditions, typically expressed in terms of partial differential equations, are imposed on a family of mathematical objects, such as functions or tensor fields, leading to the conclusion that solutions can exist only on a distinguished subclass of manifolds, typically characterised by a high number of symmetries.

In this mini-course, we will review some of the most significant examples of rigidity phenomena that have emerged in recent decades in the fields of Riemannian geometry and mathematical relativity, illustrating the main ideas and the techniques employed to establish them. Starting with the classical Liouville Theorem on complete manifolds with Ricci tensor bounded from below, we will move toward the classification of static metrics in general relativity. We will then relax the curvature assumptions of the ambient space to derive sharp and rigid geometric inequalities.

2 V. Agostiniani (University of Trento) - Minicourse "Rigidity phenomena arising in Riemannian Geometry and in General Relativity"

In geometric analysis, rigidity phenomena occur when specific conditions, typically expressed in terms of partial differential equations, are imposed on a family of mathematical objects, such as functions or tensor fields, leading to the conclusion that solutions can exist only on a distinguished subclass of manifolds, typically characterised by a high number of symmetries.

In this mini-course, we will review some of the most significant examples of rigidity phenomena that have emerged in recent decades in the fields of Riemannian geometry and mathematical relativity, illustrating the main ideas and the techniques employed to establish them. Starting with the classical Liouville Theorem on complete manifolds with Ricci tensor bounded from below, we will move toward the classification of static metrics in general relativity. We will then relax the curvature assumptions of the ambient space to derive sharp and rigid geometric inequalities.

3 A. Sorrentino (University of Roma Tor Vergata) - Minicourse "When Mathematicians play... billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

4 A. Sorrentino (University of Roma Tor Vergata) - Minicourse "When Mathematicians play... billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

5 A. Vistoli (Scuola Normale Superiore) - Minicourse "Di quanti parametri abbiamo bisogno per definire un oggetto algebrico? Problemi vecchi e risultati nuovi"

Abstract
La dimensione essenziale, un invariante che si può definire in grande generalità; risponde alla domanda: quanti parametri indipendenti sono necessari per definite un oggetto algebrico in una certa classe? Fu introdotta, nel caso delle estensioni di Galois, nel 1997 da Z. Reichstein e J. Buhler nell'articolo "On the essential dimension of a finite group". Le sue radici sono in problemi molto classici sulle equazioni algebriche, ma per il suo studio occorrono strumenti avanzati di algebra e geometria algebrica. Molte questioni fondamentali sull'argomento rimangono aperte.
Nel minicorso comincerò con alcuni risultati classici su equazioni polinomiali e forme quadratiche, e spiegherò la formulazione della dimensione essenziale di un gruppo finito in termini di geometria algebrica. Illustrerò poi a grandi linee la dimostrazione di uno dei risultati fondamentali sull’argomento, il teorema di Karpenko-Merkurjev sulla dimensione essenziale dei p-gruppi, introducendo alcuni importanti concetti algebrici (algebre semplici centrali, gruppi di Brauer). Spiegherò anche il collegamento della dimensione essenziale di un gruppo finito con un altro invariante che viene dalla geometria birazionale, la dimensione di Cremona, che sarà l'argomento del seminario conclusivo.

6 A. Vistoli (Scuola Normale Superiore) - Minicourse "Di quanti parametri abbiamo bisogno per definire un oggetto algebrico? Problemi vecchi e risultati nuovi"

Abstract
La dimensione essenziale, un invariante che si può definire in grande generalità; risponde alla domanda: quanti parametri indipendenti sono necessari per definite un oggetto algebrico in una certa classe? Fu introdotta, nel caso delle estensioni di Galois, nel 1997 da Z. Reichstein e J. Buhler nell'articolo "On the essential dimension of a finite group". Le sue radici sono in problemi molto classici sulle equazioni algebriche, ma per il suo studio occorrono strumenti avanzati di algebra e geometria algebrica. Molte questioni fondamentali sull'argomento rimangono aperte.
Nel minicorso comincerò con alcuni risultati classici su equazioni polinomiali e forme quadratiche, e spiegherò la formulazione della dimensione essenziale di un gruppo finito in termini di geometria algebrica. Illustrerò poi a grandi linee la dimostrazione di uno dei risultati fondamentali sull’argomento, il teorema di Karpenko-Merkurjev sulla dimensione essenziale dei p-gruppi, introducendo alcuni importanti concetti algebrici (algebre semplici centrali, gruppi di Brauer). Spiegherò anche il collegamento della dimensione essenziale di un gruppo finito con un altro invariante che viene dalla geometria birazionale, la dimensione di Cremona, che sarà l'argomento del seminario conclusivo.

Tuesday, 27 May

7 S. Cenatiempo (Gran Sasso Science Institute, L'Aquila) - Minicourse "Large scale limits of the interacting Bose gas"

Abstract
While the theory of quantum mechanics describes interactions between the fundamental constituents of matter at microscopic scales, these interactions can lead to fascinating effects at the macroscopic level. Understanding the emergence of these phases from the microscopic description of quantum systems is a fundamental yet highly challenging mathematical problem. In this course, we will explore this challenge in the case of the interacting Bose gas, a system whose low-temperature phases exhibit the so-called Bose-Einstein condensation phenomenon (Nobel Prize, 2001), a phase of matter in which particles remain confined purely due to quantum effects. No prior knowledge of quantum mechanics will be assumed.

8 S. Cenatiempo (Gran Sasso Science Institute, L'Aquila) - Minicourse "Large scale limits of the interacting Bose gas"

Abstract
While the theory of quantum mechanics describes interactions between the fundamental constituents of matter at microscopic scales, these interactions can lead to fascinating effects at the macroscopic level. Understanding the emergence of these phases from the microscopic description of quantum systems is a fundamental yet highly challenging mathematical problem. In this course, we will explore this challenge in the case of the interacting Bose gas, a system whose low-temperature phases exhibit the so-called Bose-Einstein condensation phenomenon (Nobel Prize, 2001), a phase of matter in which particles remain confined purely due to quantum effects. No prior knowledge of quantum mechanics will be assumed.

9 A. Sorrentino (University of Roma Tor Vergata) - Minicourse "When Mathematicians play... billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

10 A. Sorrentino (University of Roma Tor Vergata) - Minicourse "When Mathematicians play... billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

11 A. Vistoli (Scuola Normale Superiore) - Minicourse "Di quanti parametri abbiamo bisogno per definire un oggetto algebrico? Problemi vecchi e risultati nuovi"

Abstract
La dimensione essenziale, un invariante che si può definire in grande generalità; risponde alla domanda: quanti parametri indipendenti sono necessari per definite un oggetto algebrico in una certa classe? Fu introdotta, nel caso delle estensioni di Galois, nel 1997 da Z. Reichstein e J. Buhler nell'articolo "On the essential dimension of a finite group". Le sue radici sono in problemi molto classici sulle equazioni algebriche, ma per il suo studio occorrono strumenti avanzati di algebra e geometria algebrica. Molte questioni fondamentali sull'argomento rimangono aperte.
Nel minicorso comincerò con alcuni risultati classici su equazioni polinomiali e forme quadratiche, e spiegherò la formulazione della dimensione essenziale di un gruppo finito in termini di geometria algebrica. Illustrerò poi a grandi linee la dimostrazione di uno dei risultati fondamentali sull’argomento, il teorema di Karpenko-Merkurjev sulla dimensione essenziale dei p-gruppi, introducendo alcuni importanti concetti algebrici (algebre semplici centrali, gruppi di Brauer). Spiegherò anche il collegamento della dimensione essenziale di un gruppo finito con un altro invariante che viene dalla geometria birazionale, la dimensione di Cremona, che sarà l'argomento del seminario conclusivo.

12 A. Vistoli (Scuola Normale Superiore) - Minicourse "Di quanti parametri abbiamo bisogno per definire un oggetto algebrico? Problemi vecchi e risultati nuovi"

Abstract
La dimensione essenziale, un invariante che si può definire in grande generalità; risponde alla domanda: quanti parametri indipendenti sono necessari per definite un oggetto algebrico in una certa classe? Fu introdotta, nel caso delle estensioni di Galois, nel 1997 da Z. Reichstein e J. Buhler nell'articolo "On the essential dimension of a finite group". Le sue radici sono in problemi molto classici sulle equazioni algebriche, ma per il suo studio occorrono strumenti avanzati di algebra e geometria algebrica. Molte questioni fondamentali sull'argomento rimangono aperte.
Nel minicorso comincerò con alcuni risultati classici su equazioni polinomiali e forme quadratiche, e spiegherò la formulazione della dimensione essenziale di un gruppo finito in termini di geometria algebrica. Illustrerò poi a grandi linee la dimostrazione di uno dei risultati fondamentali sull’argomento, il teorema di Karpenko-Merkurjev sulla dimensione essenziale dei p-gruppi, introducendo alcuni importanti concetti algebrici (algebre semplici centrali, gruppi di Brauer). Spiegherò anche il collegamento della dimensione essenziale di un gruppo finito con un altro invariante che viene dalla geometria birazionale, la dimensione di Cremona, che sarà l'argomento del seminario conclusivo.

13 A. Sorrentino (University of Roma Tor Vergata) - Minicourse "When Mathematicians play... billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

Wednesday, 28 May

14 A. Sorrentino (University of Roma Tor Vergata) - Minicourse "When Mathematicians play... billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

15 A. Sorrentino (University of Roma Tor Vergata) - Seminar "Dynamical and spectral rigidity of mathematical billiards"

Abstract
Mathematical billiards offer a rich and fascinating playground where geometry, dynamics, and analysis theory meet. In its simplest form, a billiard system consists of a point particle moving freely within a domain, undergoing elastic reflections off the boundary according to the classical law of reflection. Despite the apparent simplicity of the model, the resulting dynamics are deeply intricate and sensitive to the geometry of the domain.
This mini-course will provide an introduction to the mathematical study of billiard systems, with a focus on smooth strictly convex planar domains (also known as Birkhoff Billiards). Topics will include: periodic orbits, invariant curves, and caustics, as well as classical results concerning integrable billiards. Time permitting, we will also briefly discuss related billiard-like models, such as outer billiards, symplectic billiards, or billiards on surfaces of constant curvature, highlighting their similarities and key differences from the classical ones.

16 V. Agostiniani (University of Trento) - Minicourse "Rigidity phenomena arising in Riemannian Geometry and in General Relativity"

In geometric analysis, rigidity phenomena occur when specific conditions, typically expressed in terms of partial differential equations, are imposed on a family of mathematical objects, such as functions or tensor fields, leading to the conclusion that solutions can exist only on a distinguished subclass of manifolds, typically characterised by a high number of symmetries.

In this mini-course, we will review some of the most significant examples of rigidity phenomena that have emerged in recent decades in the fields of Riemannian geometry and mathematical relativity, illustrating the main ideas and the techniques employed to establish them. Starting with the classical Liouville Theorem on complete manifolds with Ricci tensor bounded from below, we will move toward the classification of static metrics in general relativity. We will then relax the curvature assumptions of the ambient space to derive sharp and rigid geometric inequalities.

17 V. Agostiniani (University of Trento) - Minicourse "Rigidity phenomena arising in Riemannian Geometry and in General Relativity"

In geometric analysis, rigidity phenomena occur when specific conditions, typically expressed in terms of partial differential equations, are imposed on a family of mathematical objects, such as functions or tensor fields, leading to the conclusion that solutions can exist only on a distinguished subclass of manifolds, typically characterised by a high number of symmetries.

In this mini-course, we will review some of the most significant examples of rigidity phenomena that have emerged in recent decades in the fields of Riemannian geometry and mathematical relativity, illustrating the main ideas and the techniques employed to establish them. Starting with the classical Liouville Theorem on complete manifolds with Ricci tensor bounded from below, we will move toward the classification of static metrics in general relativity. We will then relax the curvature assumptions of the ambient space to derive sharp and rigid geometric inequalities.

Thursday, 29 May

18 S. Cenatiempo (Gran Sasso Science Institute, L'Aquila) - Minicourse "Large scale limits of the interacting Bose gas"

Abstract
While the theory of quantum mechanics describes interactions between the fundamental constituents of matter at microscopic scales, these interactions can lead to fascinating effects at the macroscopic level. Understanding the emergence of these phases from the microscopic description of quantum systems is a fundamental yet highly challenging mathematical problem. In this course, we will explore this challenge in the case of the interacting Bose gas, a system whose low-temperature phases exhibit the so-called Bose-Einstein condensation phenomenon (Nobel Prize, 2001), a phase of matter in which particles remain confined purely due to quantum effects. No prior knowledge of quantum mechanics will be assumed.

19 S. Cenatiempo (Gran Sasso Science Institute, L'Aquila) - Minicourse "Large scale limits of the interacting Bose gas"

Abstract
While the theory of quantum mechanics describes interactions between the fundamental constituents of matter at microscopic scales, these interactions can lead to fascinating effects at the macroscopic level. Understanding the emergence of these phases from the microscopic description of quantum systems is a fundamental yet highly challenging mathematical problem. In this course, we will explore this challenge in the case of the interacting Bose gas, a system whose low-temperature phases exhibit the so-called Bose-Einstein condensation phenomenon (Nobel Prize, 2001), a phase of matter in which particles remain confined purely due to quantum effects. No prior knowledge of quantum mechanics will be assumed.

20 V. Agostiniani (University of Trento) - Minicourse "Rigidity phenomena arising in Riemannian Geometry and in General Relativity"

In geometric analysis, rigidity phenomena occur when specific conditions, typically expressed in terms of partial differential equations, are imposed on a family of mathematical objects, such as functions or tensor fields, leading to the conclusion that solutions can exist only on a distinguished subclass of manifolds, typically characterised by a high number of symmetries.

In this mini-course, we will review some of the most significant examples of rigidity phenomena that have emerged in recent decades in the fields of Riemannian geometry and mathematical relativity, illustrating the main ideas and the techniques employed to establish them. Starting with the classical Liouville Theorem on complete manifolds with Ricci tensor bounded from below, we will move toward the classification of static metrics in general relativity. We will then relax the curvature assumptions of the ambient space to derive sharp and rigid geometric inequalities.

21 V. Agostiniani (University of Trento) - Minicourse "Rigidity phenomena arising in Riemannian Geometry and in General Relativity"

In geometric analysis, rigidity phenomena occur when specific conditions, typically expressed in terms of partial differential equations, are imposed on a family of mathematical objects, such as functions or tensor fields, leading to the conclusion that solutions can exist only on a distinguished subclass of manifolds, typically characterised by a high number of symmetries.

In this mini-course, we will review some of the most significant examples of rigidity phenomena that have emerged in recent decades in the fields of Riemannian geometry and mathematical relativity, illustrating the main ideas and the techniques employed to establish them. Starting with the classical Liouville Theorem on complete manifolds with Ricci tensor bounded from below, we will move toward the classification of static metrics in general relativity. We will then relax the curvature assumptions of the ambient space to derive sharp and rigid geometric inequalities.

22 A. Vistoli (Scuola Normale Superiore) - Minicourse "Di quanti parametri abbiamo bisogno per definire un oggetto algebrico? Problemi vecchi e risultati nuovi".

Abstract
La dimensione essenziale, un invariante che si può definire in grande generalità; risponde alla domanda: quanti parametri indipendenti sono necessari per definite un oggetto algebrico in una certa classe? Fu introdotta, nel caso delle estensioni di Galois, nel 1997 da Z. Reichstein e J. Buhler nell'articolo "On the essential dimension of a finite group". Le sue radici sono in problemi molto classici sulle equazioni algebriche, ma per il suo studio occorrono strumenti avanzati di algebra e geometria algebrica. Molte questioni fondamentali sull'argomento rimangono aperte.
Nel minicorso comincerò con alcuni risultati classici su equazioni polinomiali e forme quadratiche, e spiegherò la formulazione della dimensione essenziale di un gruppo finito in termini di geometria algebrica. Illustrerò poi a grandi linee la dimostrazione di uno dei risultati fondamentali sull’argomento, il teorema di Karpenko-Merkurjev sulla dimensione essenziale dei p-gruppi, introducendo alcuni importanti concetti algebrici (algebre semplici centrali, gruppi di Brauer). Spiegherò anche il collegamento della dimensione essenziale di un gruppo finito con un altro invariante che viene dalla geometria birazionale, la dimensione di Cremona, che sarà l'argomento del seminario conclusivo.

23 A. Vistoli (Scuola Normale Superiore) - Minicourse "Di quanti parametri abbiamo bisogno per definire un oggetto algebrico? Problemi vecchi e risultati nuovi".

Abstract
La dimensione essenziale, un invariante che si può definire in grande generalità; risponde alla domanda: quanti parametri indipendenti sono necessari per definite un oggetto algebrico in una certa classe? Fu introdotta, nel caso delle estensioni di Galois, nel 1997 da Z. Reichstein e J. Buhler nell'articolo "On the essential dimension of a finite group". Le sue radici sono in problemi molto classici sulle equazioni algebriche, ma per il suo studio occorrono strumenti avanzati di algebra e geometria algebrica. Molte questioni fondamentali sull'argomento rimangono aperte.
Nel minicorso comincerò con alcuni risultati classici su equazioni polinomiali e forme quadratiche, e spiegherò la formulazione della dimensione essenziale di un gruppo finito in termini di geometria algebrica. Illustrerò poi a grandi linee la dimostrazione di uno dei risultati fondamentali sull’argomento, il teorema di Karpenko-Merkurjev sulla dimensione essenziale dei p-gruppi, introducendo alcuni importanti concetti algebrici (algebre semplici centrali, gruppi di Brauer). Spiegherò anche il collegamento della dimensione essenziale di un gruppo finito con un altro invariante che viene dalla geometria birazionale, la dimensione di Cremona, che sarà l'argomento del seminario conclusivo.

24 S. Cenatiempo (Gran Sasso Science Institute, L’Aquila) - Minicourse "Large scale limits of the interacting Bose gas"

Abstract
While the theory of quantum mechanics describes interactions between the fundamental constituents of matter at microscopic scales, these interactions can lead to fascinating effects at the macroscopic level. Understanding the emergence of these phases from the microscopic description of quantum systems is a fundamental yet highly challenging mathematical problem. In this course, we will explore this challenge in the case of the interacting Bose gas, a system whose low-temperature phases exhibit the so-called Bose-Einstein condensation phenomenon (Nobel Prize, 2001), a phase of matter in which particles remain confined purely due to quantum effects. No prior knowledge of quantum mechanics will be assumed.

Friday, 30 May

25 V. Agostiniani (University of Trento) - Seminar "Green's functions and positive mass theorems"

Abstract
In this concluding seminar, we will illustrate some geometric and relativistic applications of the study of Green's functions on asymptotically flat 3-manifolds with nonnegative scalar curvature. Time permitting, we will also discuss some more recent results concerning the case of compact 3-manifolds with boundary and strictly positive scalar curvature.

26 A. Vistoli (Scuola Normale Superiore) - Seminar "Sulla dimensione di Cremona di un gruppo finito"

Abstract
La dimensione di Cremona è un invariante di un gruppo finito, che fornisce un limite inferiore per la sua dimensione essenziale. Illustrerò tre dei risultati noti, dovuti a Prokhorov e Shramov, a Haution, e a Bresciani, Reichstein e me stesso.

27 S. Cenatiempo (Gran Sasso Science Institute, L'Aquila) - Minicourse "Large scale limits of the interacting Bose gas"

Abstract
While the theory of quantum mechanics describes interactions between the fundamental constituents of matter at microscopic scales, these interactions can lead to fascinating effects at the macroscopic level. Understanding the emergence of these phases from the microscopic description of quantum systems is a fundamental yet highly challenging mathematical problem. In this course, we will explore this challenge in the case of the interacting Bose gas, a system whose low-temperature phases exhibit the so-called Bose-Einstein condensation phenomenon (Nobel Prize, 2001), a phase of matter in which particles remain confined purely due to quantum effects. No prior knowledge of quantum mechanics will be assumed.

28 S. Cenatiempo (Gran Sasso Science Institute, L'Aquila) - Seminar "Macroscopic behaviour of dilute Bose gases: the Gross-Pitaevskii equation"

Since the early experiments on Bose-Einstein condensation in cold atomic gases, the Gross-Pitaevskii equation has emerged as a unique tool for describing both the equilibrium and dynamical properties of dilute Bose gases at low temperature. From a mathematical perspective, pioneering works by Lieb, Seiringer and Yngvason (2000) and Erdös, Schlein and Yau (2010) have shown that the Gross-Pitaevskii equation can be rigorously derived from the many-body Schrödinger equation in a suitable scaling limit, known as the Gross-Pitaevskii regime. In this regime N interacting bosons are trapped in a region with volume of order one, and interact through a two body potential whose scattering length is of order 1/N, and N tends to infinity.

In this talk we discuss recent methods, developed since 2019, for characterizing fluctuations around the effective description provided by the Gross-Pitaevskii equation. These methods - valid for integrable interactions - offer a rigorous implementation of an heuristic theory due to Bogoliubov (1957), in a regime where Bogoliubov’s approximations do not hold. Based on joint works with C. Boccato, C. Brennecke, C. Caraci, B. Schlein.