First year students are required to choose at least one of the following courses
20 h
Program
After introducing two special,classical cases, The course will start with the axiomatic potential theory as it is presented in the monograph od Adams
and Hedberg. Then we will develop in some depth discerete potential theory and some of its applications. Several open problems will be discussed.
Period
November 4 - December 3, 2021
20 h
Program
Part 1: 10 hours (Landi)
The mathematical properties of the solutions of optimization problems constitute the basis of most optimization algorithms. The goal of this part of the course is to study such properties from a theoretical perspective and provide an overview of corresponding algorithmic schemes.
Part 2: 10 hours (Porcelli)
The second part of the course is devoted to an introduction of the definitions and tools of differential geometry in a need-based order for optimization. In particular, theoretical properties of some numerical methods analyzed in the first part of the course will be extended to the general manifold case.
Period
February 2022
30 h
University of Pennsylvania
Program
Classification theory of smooth manifolds, for example manifolds homotopy equivalent to projective spaces or tori, with applications to various geometric problems. Invariants of singular spaces: intersection homology and characteristic classes, extending those that appear in the manifold case. Embedding and immersions for singular spaces, especially in codimension two. Applications to varieties, for example L-classes and Todd classes of toric varieties, counting lattice points, Euler-MacLaurin formulae. Discussion of open problems as time and interest permit.
Period
February 23 - April 13, 2022
20 h
Program
Part 1 (10 hours, Fioresi):
Preliminaries of linear algebra: SVD, PCA, circulant matrices. Deep Learning and Geometric Deep Learning. Geometry differential on graphs.
Part 2 (10 hours, Frosini)
Elements of persistent homology. Group Equivariants Non Expansive operators and their application in machine learning.
Period
March 31 - May 6, 2022.
30 h
CNRS Aix-Marseille Université, Institut de Mathématiques
Program
Una misura aleatoria è per definizione una distribuzione di probabilità sullo spazio delle misure . Un processo di punto ne è un esempio chiave. Il caos gaussiano molteplicativo è un altro esempio importante di una misura aleatoria che ha un ruolo prominente nello studio dei modelli gerarchici. Il corso, accessibile a chiunque abbia seguito un corso di algebra lineare e di analisi reale, approfondirà questi argomenti agli sviluppi recenti significativi.
Period
November 16, 2021 - January 18, 2022
15 h
Program
Non-smooth large scale optimization problems arise naturally in many imaging applications. In particular, when variational approaches are used to solve imaging inverse problems (such as, for instance, image restoration and inpainting or computed tomography reconstruction), the solution is obtained as the (unconstrained or constrained) global minimizer of a cost function which is typically non-differentiable and dependent on hundreds of thousands of variables. The goal of this course is to describe theoretically some state-of-the-art iterative numerical methods with guaranteed convergence for the efficient solution of this kind of challenging optimization problems. The methods will then be experimentally tested and analyzed (using existing Matlab codes) on some realistic imaging problems.
Period
February - March 2022
20 h
King's College
Program
Period
March 2022
24 h
UNIBO, UNIPD
Program
The course will focus on some rigidity results for closed hypersurfaces with prescribed curvatures, starting from the connections with symmetry properties for solutions of partial differential equations of elliptic type. In particular two classical approaches will be discussed: the moving planes technique and an integral method based on sharp inequalities.
- Classical inequalities: convex functions, matrices, and perimeters.
- Interior and boundary maximum principles for elliptic operators.
- Serrin’s overdetermined problem (part I): moving planes.
- Serrin’s overdetermined problem (part II): Weinberger’s P-function.
- Hypersurfaces: the Second Fundamental Form and the Mean Curvature.
- Minkowski formulas and Jellett’s theorem.
- Aleksandrov’s theorem (part I): moving planes.
- Aleksandrov’s theorem (part II): Reilly’s integral approach.
Period
March 2022
8 h
Freie Universitat Berlin
Program
1) Some examples of singular and non-singular SPDEs. Distributions and function spaces
2) Paraproducts and Schauder estimates
3) Regularity of the stochastic heat equation and its monomials, solution of the Phi-4-2 equation
4) Beyond d=2: Paracontrolled distributions and the Phi 4-3 equation
Period
March 28, April 1 2022
6h
Cornell University
Program
This mini-course will tie together several topics, in approximately the following order. Very little background will be assumed, essentially basic commutative algebra, though exposure to Gröbner bases would probably be a bonus.
1. Schubert varieties, matrix Schubert varieties, and pipe dreams
2. Frobenius splitting, Kazhdan-Lusztig varieties, and quiver cycles
3. Positroid varieties, Bruhat atlases, and wonderful compactifications of groups.
Period
April 6 - 8 2022
20 h
Queen's University, Canada
Program
The course will focus on recent advances concerning the study of random behaviour of number-theoretical sequences. We will mainly focus on the distribution of square-free integers and their generalisations (e.g. k-free, B-free). We will discuss Sarnak's conjecture on the disjointness of the Mobius function mu(n) from sequences generated by zero-entropy dynamical systems. We will prove that mu^2(n) (the indicator of square-free integers) is completely deterministic and study the statistics of its patterns in long intervals. We will also discuss some very recent progress on the distribution of square-free integers in small intervals. Here are some references:
Time permitting, we may discuss the limiting distribution of quadratic Weyl sums and their generalisations (e.g. classical Jacobi theta functions). Quadratic Weyl sums are a special kind of exponential sums that appear naturally in number theory, mathematical physics, and representation theory. They can be interpreted as deterministic walks (with a random `seed') in the complex plane. Generalising Sarnak's equidistribution of horocycles under the action of the geodesic ow, we can study the limiting distribution of such Weyl sums. A stochastic process of number-theoretical origin can be defined using such sums. Understanding the behaviour of trajectories of the geodesic ow in a homogeneous space, we can study this process, that shares only some of its properties with those of the Brownian motion. Here are some references:
The level of the course can be adjusted based on the background of the audience. Basic knowledge of probability and analysis will be assumed, as well as minimal knowledge of number theory. The course's goal is to foster an interactive learning environment around some exciting recent interdisciplinary advances in pure mathematics.
Period
April 11 - May 12, 2022
8 h
Michigan State University
Program
Banach space valued Poincaré inequality, extension of Pisier’s inequality, and dimension free estimates of singular integrals on discrete cube are the topics of these lectures.
For Banach space valued functions Poincaré inequality is usually replaced by Pisier’s inequality. It is interesting to understand precisely for which Banach spaces X Pisier inequality on Hamming cube is dimension free. This has been done by Ivanisvili-Van Handel-Volberg (IVHV). This, in particular, gave a solution to Enflo’s conjecture. There is a whole scale of related inequalities filling the gap between Pisier’s inequality and singular integral inequalities on Hamming cube. For those inequalities the description of class of Banach spaces X that allows the dimension free estimates is not known, the reason is related to the following fact: we are used to the “fact” that singular integrals on X-valued functions have to be bounded in Lp(X) if X is UMD. But on Hamming cube this is not true anymore. However, we will show a wide class of spaces for which those inequalities hold. The proofs are the mixture of the formula of IVHV and quantum random variables technique à la Francoise Lust-Piquard.
Lecture 1. What is Enflo's problem? Ribe program. Bourgain's discretization theorem.
Lecture 2. The solution of Enflo's problem. Semi-group approach and special formula.
Lecture 3. Poincaré estimates and beyond Enflo's problem on Hamming cube.
Lecture 4. Francoise Lust-Piquard non-commutative approach to singular integrals estimate on Hamming cube.
Period
April 23, May 21 2022
Columbia University
Program
Many geometric structures on algebraic varieties (such as vector bundles or coherent sheaves) are best studied by considering the collection of all such structures, modulo some natural equivalence, and giving it a geometric structure itself. Depending on the moduli problem considered, this may lead to a moduli scheme, a moduli space, or a moduli stack. Focusing on examples related to vector bundles on smooth curves, we will discuss the geometry of the corresponding moduli spaces and stacks, explaining how the notion of stability throws a bridge from stacks to spaces. This will be preceded by some relevant background (though very informal and example-driven) on stacks and on geometric invariant theory.
Period
May 19, h. 11-12, Seminario II (Topics in Mathematics)
May 20, h. 14-16, Seminario II
May 23, h. 11-13, Aula VIII piano
May 24, h. 11-12, Seminario II
8 h
Institute of Mathematics, Bulgarian Academy of Sciences
Program
The study of water waves involves various disciplines such as mathematics, physics and engineering and within this there are many specific areas of direct or associated interest such as pure mathematics, applied mathematics, modelling, numerical simulation, laboratory experiments, data collection in the field, the design and construction of ships, harbours, the prediction of natural disasters, climate studies and so on. In this course we shall study travelling wave solutions of shallow water waves. Camassa-Holm considered a third order nonlinear PDE of two
variables modelling the propagation of unidirectional irrotational shallow water waves over a flat bed, as well as water waves moving over an underlying shear flow. In the special case of the motion of a shallow water over a flat bottom the corresponding system was simplified by Green and Naghdi and related to an appropriate two component first order Camassa-Holm system. Another interesting system of nonlinear PDE is the viscoelastic generalization of Burger's equation. In the above mentioned systems we are looking for travelling wave solutions and we are studying their profiles. To do this we use several results from the classical Analysis of ODE that enable us to give the geometrical picture and in several cases to express the solutions by the inverse of Legendre's elliptic functions. Moreover, we shall apply microlocal approach in studying the propagation of nonlinear waves. As an application we shall present propagation of tsunami waves from their small disturbance at the sea level to the size they reach approaching the coast. Even with the aid of the most advanced computers it is not possible to find the exact solutions to the nonlinear governing equations for water waves. For this purpose we introduce Cellular Nonlinear Network (CNN) approach.
Main topics:
1. Existence and profiles of traveling waves
2. Traveling wave solutions of special type to third order nonlinear PDE
3. Method of characteristics applied to the Hunter-Saxton equation
4. Integrable multicomponent Generalizations of Camassa-Holm equation
5. Microlocal approach in studying the propagation of non-linear waves
6. Application to tsunami waves
Period
min 6 - max 24 h
Program
Calabi Yau: mini-corso di dottorato sulla geometria della teoria delle stringe, fruibile anche a distanza, con tempi, dettagli e modalità da definire.
Period
TBD
List of publications. It gives an idea of what might be subsequently pursued in various directions by anyone interested.
Join the course team on MS Teams