Modena-Reggio Emilia 2023/24

Title and Credits: Duality Theory of Markov Processes, 3 CFU
Teacher: Cristian Giardinà, Gioia Carinci

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: 2023/24 the interested. students are asked to contact the teachers in advance.

Title and credits: Topological and comparison-type methods for the study of boundary value problems in differential equations,  4CFU

Teacher: Luisa Malaguti

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.

Dates February-March 2024 for an amount of 18 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.

Title and credits: Selected topics on algebraic curves over finite fields (20 hours)
Teacher: Giovanni Zini

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: Lectures will take place from  March 2024 to May 2024; the interested students, are encouraged to contact the teacher in advance.

Title and credits: Hodge Theory (20hours, 6CFU)
Teacher: Camilla Felisetti

Syllabus: Hodge 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

Dates: Lectures will take place from 15 November 2023 to 15 December 2023 at the University of Modena and Reggio Emilia for an amount of 20 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.

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Syllabus: Design Theory is a rich branch of Combinatorics that deals with the existence and construction of discrete structures

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Title: Variational Methods and Data-Driven Approaches for Imaging (6 CFU 16h)

Teachers: Alessandro Benfenati, Simone Rebegoldi

Syllabus: In the first part of the course, we introduce first order iterative methods suited for the minimization of the sum of a differentiable (possibly non-convex) function plus a convex (possibly non-differentiable) term, with particular attention to forward-backward methods. Convergence of these methods will be analyzed, both in the convex and non-convex setting. Furthermore, we discuss some popular acceleration strategies, such as the introduction of variable metrics, or the use of inertial steps in the iterative procedure. Finally, we present numerical results obtained on some image restoration problems, assessing the impact of acceleration techniques and hinting at possible future developments in this field.

The second part of the course is devoted to presenting data-driven approaches for solving imaging problems; we work in the MatLab environment, using the Deep Learning Toolbox. We address classification problems using deep learning techniques under supervised frameworks, i.e. when large datasets containing the ground truths for neural network training are available. Under the Deep Image Prior (DIP) framework we employ neural network architectures for solving image restoration problems, where one has to recover images corrupted by linear operators (e.g. Gaussian or motion blur) and by the presence of noise (e.g. Poisson noise). We also address semantic segmentation tasks, both in a supervised and unsupervised fashion.

Course plan:

1. The image formation model: forward and inverse problems.
2. Statistical approach for Gaussian and Poisson noisy data.
3. Forward-backward methods in differentiable and non-differentiable optimization.
4. Applications in image restoration problems arising from astronomy and microscopy.
5. Task 1: classification problem of MNIST database.
6. Task 2: image restoration under Deep Image Prior framework for Poisson data. Creation of custom functions for the neural network training in the MatLab environment.
7. Task 3: supervised semantic segmentation of retina images.
8. Task 4: unsupervised semantic segmentation of natural images.

References:

1. M. Bertero, P. Boccacci, V. Ruggiero, Inverse Imaging with Poisson Data, IOP Publishing, 2053—2563, 2018.

1. S. Bonettini, F. Porta, M. Prato, S. Rebegoldi, V. Ruggiero, L. Zanni, Recent Advances in Variable Metric First-Order Methods, in Computational Methods for Inverse Problems in Imaging, Springer INdAM Series 36, 1—31, 2019.
2. A. Chambolle, T. Pock, An Introduction to Continuous Optimization for Imaging, Acta Numer. 25, 161—319, 2016.
3. P. Combettes, J.-C. Pesquet, Proximal Splitting Methods in Signal Processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl., 185—212, 2011.
4. Y. Nesterov, Introductory lectures on convex optimization: a basic course, Applied Optimization, Kluwer Academic Publ., Boston, Dordrecht, London, 2004.
5. Ulyanov, D., Vedaldi, A. & Lempitsky, V. Deep Image Prior. Int J Comput Vis 128, 1867–1888 (2020).
6. A. Benfenati, A. Catozzi, G. Franchini and F. Porta, Piece-wise Constant Image Segmentation with a Deep Image Prior Approach, Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009.
7. S. Minaee, Y. Boykov, F. Porikli, A. Plaza, N. Kehtarnavaz and D. Terzopoulos, "Image Segmentation Using Deep Learning: A Survey," in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 7, pp. 3523-3542, 1 July 2022

Dates: February 2024 ; 16 h lectures; the interested. students are asked to contact the teachers in advance.

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Title: Inexact and stochastic optimization methods for big data applications (6 CFU, 16h)

Teachers: Giorgia Franchini, Federica Porta

Syllabus: Over the past few years, machine learning and deep learning techniques have emerged as cutting-edge methodologies in several domains, representing promising alternatives to traditional approaches. Learning techniques usually require solving minimization problems which are characterized by both large scale datasets and many parameters to be optimized. Due to their very low cost per iteration, inexact and stochastic gradient-based methods represent effective tools to address these problems. 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. Convergence results and practical implementation aspects of such algorithms will be provided.

Course plan:

1. Formal optimization problem statements in machine and deep learning applications
2. Stochastic gradient methods: standard and advanced approaches
3. Neural architecture search techniques
4. Algorithms unrolling
5. Inexact gradient methods in a data driven framework
6. Implementation issues of stochastic and inexact optimization methods for large-scale training

References:

1. L. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning. SIAM Review, 60(2):223–311, 2018
2. G. Franchini, F. Porta, V. Ruggiero, and I. Trombini. A line search based proximal stochastic gradient algorithm with dynamical variance reduction. Journal of Scientific Computing, 94:23, 2023
3. G. Garrigos, and R. M. Gower, Handbook of convergence theorems for (stochastic) gradient methods, arXiv:2301.11235, 2023
4. V. Monga, Y. Li, and Y. C. Eldar, Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing, IEEE Signal Processing Magazine, 38(2):18-44, 2021
5. S. J. Wright,  and B. Recht, Optimization for data analysis, Cambridge University Press, 2022

Dates: March 2024; 16 h lectures; the interested. students are asked to contact the teachers in advance

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Title: Hypoelliptic Partial Differential Equations (6 CFU)
Teachers: Sergio Polidoro, Maria Manfredini

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. Some open research problems will be described. The course will focus on the following topics:
-Bony's maximum principle for degenerate second order PDEs, propagation set and Hormander's hypoellipticity condition.
-Perron method for the boundary value problem in a bounded open set of the Euclidean space.
-Boundary regularity, barrier functions. Boundary measure, Green function.
-Fundamental solution. Mean value formulas. Harnack inequalities.
-Degenerate Kolmogorov equations. Applications to some financial problems.
The program may be modified in accordance with the requirements of the students.

Reference text: A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians (Springer Monographs in Mathematics - 2007). Further references and lecture notes will be given during the course..

Dates: May-June 2024, 15h lectures + 3 h of  autonomous work;  the interested. students are asked to contact the teachers in advance