Title: Introduction to derived categories and applications
Timetable: every Wednesday 11.00-13.00
Program: The course will be divided in two parts. The first part will be devoted to an introduction to derived categories while the second one will focus on different applications. There is no definitive schedule for the second part yet: a first selection of topics was made by the participants, however we plan to add others topics, always based on the interests of the participants.
Part I : Introduction to derived categories
- Recall of abelian categories, the homotopic category, triangulated categories;
- Localization of categories: definition and construction;
- Derived categories: definition, construction, derived functors, existence criteria;
- Spectral sequences: know-how and applications.
Part II : Applications
- Formalism of the six functors in algebraic geometry;
- Grothendieck-Serre duality;
- Infinite categorical approach to derived categories.
A shared folder with course material is available. It is possible to attend the lectures online. Whoever is interested in following the course can write to luca.casarin@uniroma1.it or davide.gori@uniroma1.it for more details.
Seminar cycle
Didattica della Matematica
Timetable: 9/02 -25/05 2023
The detailed program of the course is attached to the "Insights" section
Course: Local fundamental group of klt singularities, after L. Braun
Prof. Benoit Claudon ( IRMAR Université de Rennes 1 )
Timetable: 6 hours. 15- 22- 29 march 2023, 09:00-11:00, room B.
Abstract: Topics discussed:
- local fundamental groups of (isolated or not) singularities
- ideas of a simple proof by Kollár of the fact that any finitely presented group can be realized as the local fundamental group of an isolated three-dimensional singularity (using geometric realization of a simplicial complex, blow-up, resolution of explicit singularities + Grauert/Artin contraction theorem)
- notion of klt singularities and example of orbifolds (quotient singularities)
- discussion concerning Fano manifold and idea of the proof (Braun) that if X is klt and Fano then the fundamental group of its regular locus is finite
- link between Fano varieties and klt singularities and an explicit example in the surface case (A_n singularities) showing that even if we are only interested in studying varieties we have to considered pairs (a
variety + an effective Q-divisor)
Course: Front propagation in integro-differential models
Prof. Jean-Michel Roquejoffre, Université Paul Sabatier, Toulouse
Timetable: Mercoledì 26 aprile, Venerdì 28 aprile, Mercoledì 3 maggio dalle ore 14 alle 16 in aula B.
Giovedì 4 maggio dalle ore 11 alle 13 in aula B.
Abstract: the invasion of a piece of land by a pest, the propagation of a fire or
the spread of an epidemics, and many other real life situations, have
the common point that they can be described by rection-diffusion
equations where the diffusion mechanisms are given by integral
operators. We will explain some of the models, and analyse their large
time behaviour. In the situations under study, a front will form, and we
will try to analyse its development as precisely as possible.
Course: "Discrete maximum principle of finite difference schemes
on staggered Cartesian grids for heterogeneous and anisotropic
diffusion equations"
Prof. Chiara Simeoni
Timetable: March 2023
Abstract: The aim is to give an introduction to finite difference and finite
volume methods for approaching anisotropic and heterogeneous diffusion
equations on staggered Cartesian grids, with the main issue to satisfy
the discrete maximum/comparison principle and to guarantee
non-negativity properties for the numerical solutions.
The lectures are complemented with experimental sessions performed
through SciLab -- www.scilab.org/ -- or another programming language.
LECTURE NOTES : R. Dani, C. Simeoni, Discrete maximum principle and
the Ultraviolet Catastrophe of finite difference schemes on staggered
Cartesian grids for heterogeneous and anisotropic diffusion equations,
Université Nice-Sophia Antipolis (2014)
https://hal.archives-ouvertes.fr/hal-00950849
CONTENTS (tentative) :
1) Linear diffusion equations and qualitative properties : physical
background, parabolic operators, properties of the diffusion tensor,
relationship between parabolic operators and positive definite
matrices, maximum principle for linear operators, applications of the
maximum principle, uniqueness, monotonicity and comparison principle,
non-negativity property.
2) Numerical schemes for parabolic conservation laws : discretization
of the spatial domain, the finite difference method, approximation
of parabolic equations, mixed derivatives and the θ-scheme for time
evolution, the finite volume method, finite difference schemes for
two-dimensional heterogeneous equations, the chain rule method, the
standard discretization method, the non-negative method, discrete
maximum principle, theoretical results for semi-discrete problems,
application to the finite difference schemes.
3) Stability analysis of one-dimensional and two-dimensional methods :
the simplest homogeneous case, numerical schemes for one-dimensional
heat equation, L2- stability of the θ-methods, the question of the
Ultraviolet Catastrophe, modified equation and consistency, discrete
maximum principle, the heterogeneous linear case, the two-dimensional
anisotropic and heterogeneous case, diagonal diffusion tensors,
discrete maximum principle for two-dimensional problems, diagonal
anisotropic homogeneous diffusion, diagonal anisotropic heterogeneous
diffusion, fully anisotropic homogeneous diffusion, L2-stability
analysis of numerical schemes.
4) Experimental validation and numerical results : definition of
initial data, boundary conditions and numerical parameters, numerical
tests for the one-dimensional heat equation, the time-implicit method,
the time-explicit method, the semi-implicit Crank-Nicolson method,
numerical tests for one-dimensional heterogeneous diffusion, numerical
tests for two-dimensional anisotropic diffusion, the purely diagonal
case, taking into account the mixed derivatives.
Corso: "Concentration phenomenons in Geometry"
Prof. Gerard Besson (Institut Fourier of Grenoble)
Timetable: 2 - 18 may 2023, tuesday 16/18; thursday 14/17, room B.
Abstract: Il programma dettagliato delle lezioni è allegato alla sezione "approfondimenti"
Due corsi di dottorato (6 ore ciascuno) su due temi di teoria dei numeri tra di loro in parte legati perché connessi dalla classica analogia tra campi di numeri e campi globali in caratteristica non nulla.
Primo corso - An introduction to classical and finite multiple zeta values
Prof. Masanobu Kaneko, Kyushu University ( 6 ore).
Timetable: march 31, 10-12, april 4, 9-11, april 5, 15:15 -17:15, room B
Abstract: This course gives an introduction to the theory of the classical multiple zeta values and their finite analogues. Starting with the basics, we will review the theory of regularization and prove the fundamental theorem of regularization in an almost purely algebraic way. A generalization to the Hurwitz type multiple zeta values and its relation to Kawashima’s relation will also be mentioned. The second part discusses two very different finite versions of the multiple zeta values and presents a conjecture connecting those two analogues. If time permits, some topics around the level two or four analogues of multiple zeta values will be discussed.
Secondo corso - Recent results and conjectures in Function Field Arithmetic.
Prof. Dinesh Thakur, University of Rochester (6 ore).
Timetable: Maggio 2023
Abstract: Starting with the basics, this course gives an introduction to recent developments in Function Field Arithmetic, focusing on the following three broad areas: The first topic deal with the Zeta and multiple zeta values and related structures, discussing analogies and contrasts with the number fields situation first discussed by Professor Kaneko. The second topic deals with Diophantine approximation of algebraic elements by rational or algebraic elements, again discussing various analogies and contrasts with the number fields case. The third topics deals with the results and conjectures about the distribution of general or special Primes, Congruences and Super-congruences, arithmetic derivatives, zeta values.
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.
Info https://people.dm.unipi.it/trevisan/teaching.html
Corso - Data Geometry and Deep Learning
Prof. Mircea Petrache
Dates: November 14th 2022 - December 20th 2022,
Timetable: Tuesdays 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)
Place: Math 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:
-
Introduction and a brief history of Neural Networks. Overview of the course.
-
Stochastic Gradient Descent, Backpropagation, Convergence improvement methods
-
Some very common Neural Network architectures and their motivations
-
Neuromorphic Neural Networks
Part II: Staples of classical Deep Learning theory
-
Regularization, Generalization: some mathematical interpretations
-
Expressivity, PAC learning, VC dimension and comparison to real life learning for DNNs
-
Introduction to Information Theory, and the Information Bottleneck Principle
Part III: Selected topics of research
-
Network pruning: the "Lottery ticket hypothesis", and directions for theoretical justifications
-
Equivariant Neural Networks
-
Curvature and its role in Generative Adversarial Networks
-
Hyperbolic Neural Networks
-
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: March 2023: Monday, Tuesday Thursday 16-18, room B
The detailed program of the course is attached to the "Insights" section
Corso - Branching Random Walks in d>2
Prof. Amine Asselah
Timetable: May 29th, June 1, June 8th, June 9th, June 12th, June 15 th, June 19th, June 22nd, Room B, from 11:00 to 13:00.
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.
Course - Symplectic Geometry
Prof. Siye Wu (National Tsing Hua University)
Timetable: 8 novembre 2022 10:30 - 12:30 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.
Course - History of mathematics course for the doctorate
Module I:
Prof. Alberto Cogliati (Pisa).
Titolo: Origini 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).
Module II:
Prof. Paolo Freguglia (L'Aquila).
Titolo: Il primo periodo del calcolo delle variazioni.
Timetable: 15, 16, 22, 23, 29, Venerdì 30 novembre 2022 dalle 14:00 alle 15:00 in aula B seminario conclusivo rivolto a tutto il Dipartimento:
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.
Module 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)