COURSES

Syllabus: This is 4/5 weeks 18 hours PhD course from a visiting professor. This course offers a foundational introduction to differential geometry, focusing on manifolds, submanifolds, and key examples like Lie groups, projective spaces, and Grassmannians. It covers tensors, exterior calculus, differential forms, and their roles in vector fields, symplectic manifolds, and Hamiltonian mechanics. Stokes’ Theorem, orientation, and manifolds with boundary are explored in depth. Students learn about vector bundles, connections, curvature, and Riemannian geometry, including Gaussian curvature and the Levi-Civita connection. The course introduces distributions, foliations, and Frobenius’ theorem, then moves into the structure of Lie groups and algebras, with applications to classical matrix groups. It delves into the geometry of surfaces in homogeneous spaces using Cartan’s method and the Darboux derivative. The concept of G-structures is introduced, along with examples like almost complex and symplectic structures. If time allows, the Gauss-Bonnet theorem and its extension via Chern-Weil theory are also covered.

References: Main Texts:

Liviu Nicolaescu, Lectures on Geometry of Manifolds, 2nd Edition, World ScientificPublishing, 2007

Supplementary Texts:

1. Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, Graduate Texts in Mathematics, v.94

2. Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and updated,Second Edition (Dover Books on Mathematics).

3. R.W. Sharpe, Differential Geometry: Cartan’s Generalization of Klein’s Erlangen

4. M. Do Carmo, Differential Forms and Applications (Universitext)

5. John Lee, Introduction to Smooth Manifolds , Graduate Texts in Mathematics, Vol.218, 2nd Edition

6. Michael Spivak , A Comprehensive Introduction to Differential Geometry, vol.1

7. Michael Spivak , A Comprehensive Introduction to Differential Geometry, vol.2

8. Walter Rudin, Principles of Mathematical Analysis.

Dates/Schedule: TBA

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Syllabus: The main goal of this course is to provide a useful toolbox for the study of the weak regularity theory á la De Giorgi-Nash-Moser (DGNM) for local and nonlocal kinetic operators. Firstly, we will analyze the local case of the Fokker-Planck operator in order to introduce the non-Euclidean geometry required for the study of these operators. Then, we will provide all the advanced necessary tools to deal with the nonlocal case, and we will conclude the course with some interesting applications to more complex physical models, as e.g. those involving the Boltzmann operator, where the main diffusion non-symmetric kernel satisfies very weak integrability and non-degeneracy conditions.

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Syllabus: The first part of the course will provide an introduction to complex and almost complex geometry. There will be a concise overview of the theory of holomorphic functions of several complex variables, omitting analytical details, leading to basic examples of complex and almost complex manifolds. We will also discuss the Dolbeault, Bott-Chern and Aeppli cohomologies. The second part of the course will focus on the Hermitian geometry, starting with the Kaehler manifold and describing their cohomological properties. Then, the existence of special structures on compact complex manifolds and their deformations will be investigated. Explicit examples and computational techniques of cohomological invariants will be carefully described.

Notes for the course will be provided.

See more information HERE (PDF file)

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Syllabus: LorElementary results on prime numbers. The Riemann zeta-function and its basic properties: analytic continuation, functional equation, Euler product and connection with prime numbers, the Riemann-von Mangoldt formula, the explicit formula, the Prime Number Theorem. Prime numbers in all and “almost all” short intervals.

Dates/Schedule: TBA

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Syllabus: The space BV: definition and examples.

– BV functions in one variable.

– Sets of finite perimeter.

– Embedding theorems and isoperimetric inequalities.

– Fine properties of BV functions.

– SBV functions: definition and examples.

Dates/Schedules: TBA

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Syllabus: The aim of the course is to introduce the main techniques used in complex and algebraic geometry to study the geometry of compact complex surfaces. We will review the main results of the theory: the Riemann—Roch theorem with its implications, the effects of a blow up on the topology and the cohomology of a surface and Castelnuovo’s contraction theorem, the concept of minimal surface and the problem of classification (Castelnuovo’s rationality criterion and Enriques—Kodaira classification). Finally, we will focus on automorphisms of surfaces, focusing on what is known about their automorphism groups in particular for surfaces of Kodaira dimension 0.

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Syllabus: The course introduces students to the topics of Decision Theory that are relevant for Artificial Intelligence. In particular, the course discusses decision-theoretic planning and learning through the following agenda: brief review of random variables and stochastic processes (if needed), discrete-time Markov chains, Markov decision processes, base algorithms for automated planning using Markov decision processes (e.g., value iteration and policy iteration), base algorithms for machine learning using Markov decision processes (e.g., Q-learning and SARSA), brief overview of additional topics (e.g., partially-observable Markov decision processes, game-theoretic planning).

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Syllabus: The course will present the duality approach to the study of Markov processes. This will combine, in a joint effort, probabilistic and algebraic tools. In particular we will consider several interacting particle systems that are used in (non-equilibrium) statistical mechanics, we will discuss “integrable probability”, we will show how (stochastic) PDE arise by taking scaling limits.

Dates/Schedules: TBA

Venue: TBA

Syllabus: The first eigenvalue of the Laplacian on an open set, and more generally of a second order elliptic operator, is an important object both from an applied and theoretical point of view. In Mathematical Physics, it usually plays the role of the ground state energy of a physical system. Despite its importance, for general sets it is not easy to explicitly compute it: thus, we aim at finding estimates in terms of simple geometric quantities of the sets, which are the sharpest possible. The most celebrated instance of this kind of problems is the so-called Faber-Krahn inequality. This course offers an overview of the methods and results on sharp geometric estimates for the first eigenvalue of the Laplacian and more generally of sharp Poincaré-Sobolev embedding constants (sometimes called “generalized principal frequencies”). In particular, we will present: supersolutions methods, symmetrization techniques, convex duality methods, the method of interior parallels, conformal transplantation techniques.

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Syllabus: laboruHodge theory is a powerful tool to understand the topology of Riemannian Manifolds. The aim of this course is to give an introduction to the theory together with applications to complex algebraic geometry. In particular we will treat the following topics:

-Recap on differentiable manifolds, differential forms and vector bundles

-Compact Kähler manifolds

-Hodge theory for Kähler manifolds

-Hodge theory for algebraic projective varieties

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Venue: TBA

Syllabus:The subject of the course is linear second order Partial Differential Equations with non-negative characteristic form satisfying the Hormander’s hypoellipticity condition. Maximum principle, local regularity, boundary value problem will be discussed for several examples of equations.

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Syllabus: Over the past few years, machine learning and deep learning techniques have emerged as cutting-edge methodologies in several domains. Learning techniques usually require solving minimization problems which are characterized by both large scale datasets and many parameters to be optimized. The aim of this course is to introduce the optimization models which typically arise in machine learning and deep learning applications and several inexact and stochastic optimization methods suitable to deal with these models.

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Syllabus: This course aims to introduce some of the fundamental concepts in the area of design theory such as block designs, graph decompositions, difference families, and present some modern extensions of this subject.

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Syllabus: The course aims at introducing basic problems and classic existence and regularity techniques for uniformly elliptic PDEs with linear growth. In particular, the following topics will be analyzed, more or less in detail according to the students’ interest:

– Harmonic functions, weak formulation and Weyl’s Lemma.

– Second order regularity for the Poisson equation via Calderon-Zygmund decomposition and singular integrals.

– Second order Sobolev regularity for equations with constant coefficients.

– Linear equations with variable coefficients: W^{1,q} estimates for continuous coefficients.

– Campanato spaces and Schauder theory for linear equations with Holder coefficients and data.

– De Giorgi theory (Holder regularity of solutions for measurable coefficients).

– Harnack inequalities, expansion of positivity for equations with measurable coefficients.

– Gehring theory (higher integrability of the gradient).

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Syllabus: The course covers the following topics:

– review and complements of MeasureTheory;

– covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems;

– rectifiable sets and rectifiability criteria;

– the theory of sets of finite perimeter;

– applications to geometric variational problems;

– the isoperimetric problem;

– the partial regularity theory for quasi-minimiser of the perimeter.

Hand-written notes of the whole course are available in Italian on the Elly platform.

Dates/Schedule: TBA

Venue: TBA

Syllabus: Lie algebras are algebraic structures with wide-ranging applications in geometry and mathematical physics. The course will focus on semisimple Lie algebras and their real forms, addressing their structure and classification. We will introduce the fundamental concept of root systems and study finite reflection groups associated with them, i.e. the Weyl groups. These combinatorial tools will be used to classify semisimple Lie algebras and their real forms, classical and exceptional cases.

References: The primary reference for this course will be J.E. Humphreys’ book Introduction to Lie Algebras and Representation Theory. Additionally, Anthony W. Knapp’s book Lie Groups Beyond an Introduction will serve as a helpful supplementary resource.

Dates/Schedule: Precise dates will be determined in consultation with interested PhD students, who are encouraged to contact the teacher in advance.

Venue: TBA

Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.

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Venue: TBA

Syllabus: The course is principally focused on Boundary Element Methods (BEMs). Lectures involve: Boundary integral formulation of elliptic, parabolic and hyperbolic problems – Integral operators with weakly singular, strongly singular and hyper-singular kernels – Approximation techniques: collocation and Galerkin BEMs – Quadrature formulas for weakly singular integrals, Cauchy principal value integrals and Hadamard finite part integrals – Convergence results – Numerical schemes for the generation of the linear system coming from Galerkin BEM discretization.

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Syllabus: – Introduction to differential model problems for option pricing in the Black-Scholes framework.

– Analysis of peculiar troubles and advantages in application of standard numerical methods for partial differential problems: Finite Difference, Finite Element, Boundary Element, Binomial, Monte Carlo.

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Syllabus: Birational maps of the complex projective plane. Factorization of transformations and proofs of Noether-Castelnuovo theorem. Cremona equivalence of plane curves. Lengths in the plane Cremona group.

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Syllabus: Studio e approfondimento dei più consolidati quadri teorici della ricerca in educazione matematica. Lettura critica di articoli e prodotti di ricerca; significatività degli studi all’interno della letteratura internazionale; coerenza tra progettazioni e metodologie con i quadri teorici.

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Venue: TBA

Syllabus: The course will present classical and modern methods for studying Hölder regularity properties of solutions to elliptic partial differential equations. We will analyze, under a common geometric framework, two different approaches: a linear approach based on integral representations of the solutions, as well as a nonlinear approach mainly based on maximum principles. We will focus on 2nd order linear PDEs of uniformly elliptic type, and we will discuss the respective assumptions on the diffusion coefficients in order to trigger the two approaches.

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Venue: TBA

Syllabus: The course aims to describe some methods of research in the history of mathematics, showing how the study of different themes and historical sources requires different approaches and tools of inquiry.

In the first part of the course (10 hours) we will present elementary methods of historical research that can be applied in the critical examination of printed mathematical texts of the past. We will provide examples of critical reading taken from some important Italian mathematical works of the 18th and 19th century devoted to the foundations of infinitesimal calculus. In the second part of the course (10 hours) we will introduce some unpublished original sources in the history of mathematics in order to explain how to approach the critical reading, transcription and analysis of them.

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Syllabus: The course will consider some selected topics in the theory of algebraic curves over finite fields. Useful previous knowledge: elementary theory of algebraic curves. The topics will be selected among the
following ones.

Maximal curves over finite fields: properties, classical examples (Hermitian, Suzuki and Ree curves), recent families (GK curve, GGS curve, BM curve, Skabelund curves).

Automorphism groups of curves, and quotient curves: bounds on the size, examples.
Automorphism and quotients of the Hermitian curve: classification.

Rational points of curves over finite fields: criteria and methods for the analysis of absolutely irreducible rational components of curves, in particular for what concerns plane curves.

Applications of the study of rational points to some remarkable families of polynomials over finite fields
which are of interest in cryptography.

Dates/Schedule: TBA

Venue: TBA

Syllabus: In this course we present the theory of semigroups of bounded operators in Banach spaces, paying particular attention to analytic semigroups. Applications are given to the analysis of partial differential equations of parabolic type.

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Venue: TBA

Syllabus: Theory of several complex variables. Hartogs theorem, Cartan-Thullen theorem, Kontinuitatsatz. Domains of holomorphy, Levi convexity and plurisubharmonic functions. Cauchy-Riemann equation. Sheaves and cohomology (Cech cohomology).

Dates/Schedule: TBA

Venue: TBA

Syllabus: Introduction to Deep Learning, basic steps of the algorithm analogies with the human visual systems and its mathematical models.

– The geometry of the space of data and the space of parameters; KL divergence and its information geometry interpretation.

– Geometric Deep Learning: the algorithm of Deep Learning on Graphs.

– Message passing and GATs: a geometrical modeling via heat equation and Laplacian on graphs.

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Venue: TBA

Syllabus: The course will introduce an advanced topic in Representation Theory such as

1) Hopf algebras and quantum groups.

2) Yangians and quantum affine algebras.

3) Categorified quantum groups and KLR algebras.

4) Cluster algebras and quantum groups.

5) Geometric approach to quantum loop algebras.

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Venue: TBA

Syllabus: The course deals with some important methods for the study of boundary value problems to ordinary and partial differential equations. The Leray-Schauder topological degree will be briefly introduced, and its applications discussed in the study of periodic solutions and solutions satisfying Cauchy multi-point conditions in parabolic equations. The upper and lower solutions technique for ordinary differential equations will be then proposed and its application given to the study of traveling wave solutions of reaction-diffusion equations with degenerate diffusivities.

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Venue: TBA

Syllabus: The course aims at introducing classical and advanced methods for solving large-scale optimization problems arising in imaging applications. In the first part, we present first order iterative methods suited for minimizing the sum of a differentiable function plus a convex term, with application to image restoration problems arising in astronomy and microscopy. The second part of the course is devoted to presenting data-driven approaches for solving imaging problems: under the Deep Image Prior (DIP) framework, we employ neural network architectures for solving image restoration problems and image segmentation tasks, both in a supervised and unsupervised fashion.

Dates/Schedule: TBA

Venue: TBA