COURSES 2022-23


Corso - Mathematical aspects of quantum information theory
Prof. Dario Trevisan
Timetable: da definire

Topics:
1. Principles of QM: pure and mixed states, observables, tensor products
and entanglement.
2. Examples: qudits, spin chains, Gaussian systems.
3. Open quantum systems: CPTP maps, quantum Markov semigroups and their
generators.
4. Distances between quantum states: trace distance, fidelity, quantum
optimal transport.
5. Quantum entropy and its properties. A quantum channel coding theorem.

Bibliography:
- Nielsen, M.A. and Chuang, I.L., Quantum Computation and Quantum
Information: 10th Anniversary Edition, Cambridge University Press, 2010
- Naaijkens, Pieter. Quantum spin systems on infinite lattices.
eScholarship, University of California, 2013.


Corso -  Data Geometry and Deep Learning
 Prof Mircea Petrache
DatesNovember 14th 2022 - December 20th 2022,
TimetableTuesdays 3-5PM, Thursdays 3-5PM
(La lezione di giovedì 1 dicembre 2022 è SOSTITUITA dalla lezione di mercoledì 30 novembre ore 10-12 in Sala di Consiglio) 
PlaceMath Department of "La Sapienza" university (room: aula B).

Abstract: The idea is to give the basic ingredients to understand what Deep Learning theory is, and of what are the underlying mathematical/theoretical structures. Hopefully at the end of the course, the mathematics students that follow it will be able to read Deep Learning research papers without being lost, and will have the basic tools to start working on some important maths-rich topics. Lectures 7-12 below, each will cover an important direction of research for which some first mathematical steps have been done, but which has many subtopics and extensions left open. Particular emphasis is given to the view that the geometry of datapoints and of learning algorithms has important practical consequences, some of which have started to emerge in the last 3-4 years.

The plan is to spend one lecture on each topic. The below plan may change as the course progresses:

Part I: Introduction to Deep Learning:

  1. Introduction and a brief history of Neural Networks. Overview of the course.

  2. Stochastic Gradient Descent, Backpropagation, Convergence improvement methods

  3. Some very common Neural Network architectures and their motivations

  4. Neuromorphic Neural Networks

Part II: Staples of classical Deep Learning theory

  1. Regularization, Generalization: some mathematical interpretations

  2. Expressivity, PAC learning, VC dimension and comparison to real life learning for DNNs

  3. Introduction to Information Theory, and the Information Bottleneck Principle

Part III: Selected topics of research

  1. Network pruning: the "Lottery ticket hypothesis", and directions for theoretical justifications

  2. Equivariant Neural Networks

  3. Curvature and its role in Generative Adversarial Networks

  4. Hyperbolic Neural Networks

  5. Persistence diagrams and Topological Data Analysis (guest lecture by Sara Scaramuccia)

Useful bibliography for the first 6 lectures (precise chapter references and other references will appear in the lecture slides):

 





Corso K-theory in condensed matter physics
Prof. Domenico Monaco
Timetible: secondo semestre

il programma dettagliato del corso è allegato alla sezione "approfondimenti"


Corso - Branching Random Walks in d>2

Prof. Amine Asselah
Timetable: giugno 2023. Il corso consiste in 16 ore di lezione, suddivise in quattro settimane.
Abstract We discuss basic properties of branching random walk, made of a critical Galton Watson tree, associated with increments in Zd , in dimension three and more. We review what is known on the volume of the support of such trees, as well as a study of the local times in dimension 5 and more. Dimension 4 is critical, and we plan to explain the stretched scaling of the local times. Finally, we present some derivation of the branching capacity, and explain why such object is relevant. This discussion also includes a discussion of the analogous concepts for the simple random walk.

Il programma dettagliato delle lezioni è allegato alla sezione "approfondimenti"
 


Course - Symplectic Geometry

Prof. Siye Wu (National Tsing Hua University)
Timetable: 
Mercoledi' 16 Novembre, ore 15.30, Aula B
Giovedi' 17 Novembre, ore 11, Aula B
Mercoledi' 23 Novembre, ore 15.30, Aula B
Giovedi' 24 Novembre, ore 11, Aula B
Mercoledi' 30 Novembre, ore 15.30, Aula B
Giovedi' 1 Dicembre, ore 11, Aula B
Abstract:
Symplectic geometry is a branch of differential geometry that studies symplectic manifolds,
which are smooth manifolds equipped with closed non-degenerate two-forms. The course
begins with basic concepts such as Hamiltonian vector fields, Poisson brackets, Lagrangian
submanifolds and the Darboux theorem. Examples include cotangent bundles, K ̈ahler man-
ifolds, coadjoint orbits and fibrations of Lagrangian tori. Comparisons will be made with 
contact and Poisson manifolds. The second part is about symmetries of symplectic man-
ifolds. Important notions to be introduced are Hamiltonian group actions, moment maps
and their images, and symplectic quotients. Interesting examples are toric manifolds and
moduli space of flat connections on surfaces. The last part of the course is to be on the
applications of symplectic geometry to classical mechanics (Lagrangian and Hamiltonian
mechanics), solving problems on the motion of rigid bodies and integrable systems.
The course is suitable for students who have already taken an introductory course on man-
ifolds (with calculus of di↵erential forms) and who wish to engage their knowledge in a 
constructive and useful setting.
Il programma dettagliato del corso è allegato alla sezione "approfondimenti".


 

Corso di Storia della Matematica per il dottorato (composto di tre Moduli):

Modulo I:
Prof. Alberto Cogliati (Pisa).
TitoloOrigini e sviluppi del calcolo differenziale assoluto
Timetable: 13, 14, 15, 20, 21, 22 Settembre 2022
(l’ultima data è per un seminario conclusivo rivolto a tutto il Dipartimento).
 
Modulo II:
Prof. Paolo Freguglia (L'Aquila).
Titolo: Il primo periodo del calcolo delle variazioni.
Timetable: 15, 16, 22, 23, 29, Venerdì 30 novembre 2022, al termine della lezione del corso di dottorato, si terrà  il seminario conclusivo rivolto a tutto il Dipartimento, dalle 14:00 alle 15:00 in aula B:

Title: The birth of Hamiltonian Analytical Optics and its historical role

Abstract: William Rowan Hamilton’s works on Optics are dated about 1828 (On ordinary system of reflected rays, and On ordinary system of refracted rays). These two essays analyze in a mathematically new way the laws and properties of geometric optics. These Hamiltonian researches also influenced the birth of analytical mechanics. Although this is true, the analytical Hamiltonian studies on Optics have their autonomy and originality. Our aim is to examine and to reconstruct some essential aspects of these topics.

 

Modulo III:
Prof. Enrico Rogora (Roma).
Titolo: Intrecci tra la teoria delle equazioni e la teoria delle funzioni ellittiche .
Timetable: 10, 11, 12, 17, 18, 19 Gennaio 2023
(l’ultima data è per un seminario conclusivo rivolto a tutto il Dipartimento)
 

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