Training Activity
In agreement with the tutor, the PhD student is required to present a study plan (template) consisting of advanced courses/activities for 24 credits (ECTS) and 3 complementary training activities, to be completed within the first two years of the PhD program. The study plan must be approved by the board. Advanced courses Type e) activities can contribute to the acquisition of no more than 6 credits. ECTS: for courses or activities that do not have an official assignment of credits, the number of corresponding credits is assigned by the coordinator, in agreement with the tutor, upon request from the student or the tutor. Usually for PhD courses 5 hours correspond to 1 ECT. These exams must be completed within the first two years of the PhD program. Complementary training PhD in Mathematics, University of Torino
Courses archive
The study plan must be presented, with at least part of the activities, within the end of January of the 1st year. It must be completed, and possibly modified, within the end of January of the 2nd year.
Students can choose among the following activities:
a) courses offered within the PhD program in Mathematics of the University of Turin or other universities, including foreign universities;
b) advanced courses offered within the Master degree in Mathematics of the University of Turin;
c) courses in the form of reading courses, where the PhD student studies independently a subject under the guidance of a teacher, followed by a final exam;
d) study activities in the form of a working group of advanced level, i.e. a group of professors, researchers, and PhD students who study together a specific research topic;
e) research schools and similar activities.
Exams and certification: for each course/activity it is required to pass a final exam, defined by the teacher or the person in charge of the activity, with the possible exception of type e) activities that do not have a final exam. For type e) activities that do not have a final exam, the tutor will verify that the student has positively carried out the activity, with an interview or a seminar.
The positive outcome of the exam/activity is certified by the person in charge with a message to dottorato.matematica@unito.it (no grade is required).
Besides their study plan, graduate students can follow courses or participate in research activities in all the years of their studies.
PhD students are also expected to actively participate in the seminars of our department; some of them give lectures within these seminars.
PhD students are required to participate in at least three courses/seminars offered for the Complementary Training by the Doctoral School of the University of Turin, within the first two years of the PhD program. Certificates of attendance must be sent to dottorato.matematica@unito.it
One of these activities can be replaced by an outreach activity (terza missione), in this case the attendance is certified by the person in charge of the activity.
Complementary training activities
Preliminary list of courses for a.a. 25/26
May 2025
Small points on subvarieties of algebraic tori
10 hours, 2 CFU
Period: November 2025
Let G=G_m^n be a power of the multiplicative group, V be a (geometrically irreducible) algebraic subvariety of G, defined over the field of algebraic numbers, not a translate of a proper subtorus by a torsion point of G. By the toric case of the former Manin-Mumford conjecture, the set of torsion points of G lying on V is not Zariski dense. Let now h be a normalized height on G(Q). Thus torsion points are points of zero height. In these lectures we focus on the distribution of points of small height (the so called small points) of V.
Hamiltonian and variational methods for Celestial Mechanics
30 hours, 6 CFU
Period: January/April 2026
The course aims to introduce students to some classical and more recent models and problems of Celestial Mechanics (Kepler problem, N-centre problem, N-body problem, galactic billiards...), to be investigated using tools of dynamical systems (Hamiltonian techniques) and nonlinear analysis (variational methods). Possible topics to be considered are the following:
- collision solutions, asymptotic estimates and regularization for perturbed Kepler problems
- periodic solutions for the relativistic Kepler problem
- symmetric periodic solutions for the N-body problem
- expanding solutions for the N-body problem
- chaotic behaviors for some models of Celestial Mechanics (anisotropic Kepler problem, N-centre problem, galactic billiards)
Numerical Optimization for Machine Learning
20 hours, 4 CFU
Period: April-May 2026
Program:
1) Nonlinear Systems
Newton’s Method and Its Variants, Modified Newton’s Methods, Quasi-Newton Methods, Secant-like Methods, Fixed-point Methods
2) Unconstrained Optimization
Direct Search Methods (Hooke and Jeeves Method, Nelder and Mead Method), Descent or Line Search Methods (Newton-like, Quasi-Newton, Secant-like, Gradient, Conjugate Gradient Directions), Trust Region Methods
3) Constrained Optimization
Penalty Method, Augmented Lagrangian Method
4) Applications to Machine Learning
Tropical Geometry
20 hours, 4 CFU
Period: end of 2025/beginning of 2026
Tropical geometry is a new approach in algebraic geometry that has been developed over the last twenty or so years. Working over a valued field, (some of) the geometry of the algebro-geometric varieties is encoded in their tropical counterparts. These are piecewise linear objects, often of a combinatorial flavour. The course will focus on the case of tropical curves, with topics including the relation to semistable models, moduli spaces, as well as liftability results. While the purely tropical statements require no prior knowledge, to follow the parts of the course that concern their translations into corresponding algebro-geometric results some familiarity with algebraic geometry will be needed.
Responsible teacher Andrea Seppi
Homogeneous spaces and (G,X) geometric structures on manifolds
Hours: 15/20, 3/4 CFU
Period: April - June 2026
The notion of a (G,X) structure, introduced by Thurston in the second half of the 20th century, provides a unifying language for describing a wide range of differential geometric structures on manifolds. The idea is simple: given a "model" manifold X and a nice Lie group G acting transitively on X (possibly preserving some geometric structure), one can study manifolds M that are "locally modeled" on X with transition maps corresponding to elements in G. This approach includes hyperbolic manifolds, flat manifolds, affine manifolds, constant curvature spacetimes, conformally flat structures, and many others.
The course will provide a general introduction to the theory of Lie groups and homogeneous spaces, with a focus on Riemannian and pseudo-Riemannian symmetric spaces. Then, we introduce the notion of (G,X)-manifold, discuss the general theory and the interactions with representation varieties. We will then specialize to certain classes of (G,X)-structures, with selected topics in hyperbolic geometry, real and complex projective structures, and conformally flat geometry.
Going beyond formal teaching and learning: theoretical and methodological issues
20 hours, 4 CFU
February – June 2025
Topics
Multimodality: Focuses on phenomenological and multimodal perspectives in teaching, emphasizing the role of signs and representations.
Laboratory Didactics: Explores mathematical tools like tracers, covariation, and integrating formal/informal learning environments.
Task Design: Covers inquiry-based game design in Dynamic Geometry Software (DGS) and classroom resources.
Theories and Methodologies: Examines educational theories, networking, and academic writing.
K3 surfaces
30 hours, 6 CFU
Period: February - April 2026
Introduction to Lattice Theory and to the K3 lattice. Examples and applications to the theory of 4-manifolds and of complex surfaces. Introduction to K3 surfaces. Examples. Classification. Period Map. Torelli Theorem. Time allowing we will cover some special families of K3's : Kummer surfaces, complete intersections, elliptic K3.
Planar Graphs and Polyhedra
10 hours, 2 CFU
Period: January-April 2026
A graph is the data of vertices (points) and edges (lines connecting the points). This course will focus on planar graphs, i.e. those that can be drawn in the plane (or on the surface of a sphere) without any edge crossings, except at vertices. A special type of planar graphs are the 3-polytopes (polyhedra), that are the planar, 3-connected graphs (planar, and however we remove 0, 1 or 2 vertices, the resulting graph is connected).
We will cover the following topics: characterisation of maximal planar graphs (by the way these are a subclass of polyhedra), overview of Kuratowski's Theorem, statement of the Rademacher-Steinitz Theorem (characterising polyhedra), enumeration of polyhedra, Tutte's and related algorithms, graphical degree sequences, unigraphicity, characterisations of special subclasses of polyhedra.
A few of the topics covered are at the cutting edge of research, and are related to interesting open problems in this area.
Geometric aspects of the AdS/CFT correspondence
10 hours, 2 CFU
Period: June or Semptember 2026
The aim of this course is to introduce the students to a selection of contemporary topics of the AdS/CFT correspondence, where supersymmetry and geometry play a crucial role. Possible topics include: advanced geometric toolkits for the study of supergravity solutions; the interplay of extremization principles between (Riemannian) geometry and superconformal field theories; the holographic approach to the microscopic structure of black holes. A detailed program will be coordinated with the interested students.
Topics in Mathematical Logic: set theory, model theory, and beyond - Reading course
30 hours, 6 CFU
Period: from November 2025 to May 2026
This is a reading course organized in collaboration with the Universities of Genova, Losanna, Udine. The course is particularly relevant to PhD students at the beginning of their studies. It will cover a variety of basic topics in logic that are not included in the master courses of the University of Turin, ranging from set theory and model theory to computability theory and category theory.
From Classical Approximation Theory to Machine Learning
20 hours, 4 CFU
Period: January-March 2026
This course offers a journey through approximation theory, from its classical foundations to recent advances, including its role in machine learning. It begins with the study of trigonometric and algebraic approximation on compact and non-compact domains, introducing core concepts such as moduli of continuity, functional spaces, best approximation estimates, embedding theorems, and polynomial inequalities. Particular attention is given to the convergence of concrete approximation processes based on orthogonal polynomials in Sobolev spaces. Subsequently, the course focuses on the construction of stable and convergent numerical methods, with applications to functional equations and machine learning. So, it provides both a rigorous theoretical framework and practical tools for advanced research in analysis and applied mathematics. Prior knowledge of real and functional analysis is recommended.
Random graphs
10 hours, 2 CFU
Period: second semester
The course introduces the theory of random graphs. We will focus on structural properties and limiting behaviours of some of the best-known models, such as the Erdős-Renyi, the configuration model and preferential attachment models. Selected results will be discussed in detail.
Comparison Geometry
20 hours, 4 CFU
Period: March 2026
The concept of curvature is an important invariant of Riemannian manifolds, which is however often hard to understand geometrically, with few results like Gauss-Bonnet, or Bonnet-Myers to provide some intuition for its meaning. Comparison geometry denotes a collection of results in Riemannian geometry which improve this understanding, by comparing the geometry of a Riemannian manifold with curvature bounded above by k with that of a space of constant curvature k. This theory allows to translate information about the curvature of a manifold into topological and metric information about distances between points. The end goal of the course is to see and prove important results in global differential Geometry, such as Toponogov Theorem, Bishop-Gromov Theorem, the Splitting Theorem, and the Soul Theorem. This theory has opened the doors to the creation of synthetic geometry which is a central topic of research nowadays. If time allows, we will dip our toes into this theory.
Learning and teaching mathematics with technologies
20 hours, 4 CFU
Period: January – May 2025
Topics
Theoretical frameworks to study educational processes with the use of technologies (instrumental approach, TPACK, humans-with-media).
Embodied and material approaches in mathematics education to study the teaching and learning of mathematics with technologies.
Cognitive analysis of students’ processes when learning mathematics in technological environments.
Educational and theoretical challenges offered by new emerging technologies.
Responsible teacher: Andrea Seppi
Higgs bundles and non-abelian Hodge correspondence
20 hours, 4 CFU
Period: mid January - end of March 2026
The aim of this course is to introduce the analytic and holomorphic aspects of Higgs bundles defined over a compact Riemann surface, and to explore their connection with the representation theory of the fundamental group of the surface into SL(n, ℂ). We will begin by reviewing key concepts from complex differential geometry, such as holomorphic vector bundles, Hermitian metrics, and Chern connections. The core of the course will be devoted to a detailed proof of the correspondence between polystable Higgs bundles and reductive representations—known as the non-abelian Hodge correspondence—based on the theory of harmonic maps from a Riemann surface into a non-compact symmetric space. Depending on students’ interests, we may delve deeper into the notion of stability for vector bundles, possibly extending the discussion to the setting where the Riemann surface is replaced by a compact Kähler manifold. Alternatively, we may explore how more general versions of this correspondence underpin current developments in higher Teichmüller theory, which studies representations of surface groups into real semisimple Lie groups.
An introduction to partial representation theory
15 hours, 3 CFU
Period: February - May 2026
The study of partial symmetries is a relatively recent research area whose origins can be traced back to the study of C*-algebras generated by partial isometries. Here we aim at offering an introduction to the topic. We start with partial actions of groups on sets, topological spaces, algebras, their connections with inverse semigroups and groupoids, and we discuss the globalization question and the partial crossed product construction. Then, we shift to the linear setting by studying partial representations of groups and the associated partial group algebra, Maschke's Theorem, and the question of distinguishing finite abelian groups via their partial group algebras. Finally, we will broaden the course’s horizons by discussing partial modules of Hopf algebras and, time permitting, their partial comodules, the associated universal Hopf algebroid, and possibly the more categorical geometric case.
Stochastic approach to non-local PDEs
10 hours, 2 CFU
Period: January - February 2026
In this course we will introduce random processes governed by integro-differential equations. The involved equations are evolution equations that are non-local (also) in the time variable; the prototypes are fractional kinetic equations. The corresponding processes are non-markovian processes obtained as scaling limit of continuous time random walks. These equations and processes are suitable to model anomalous diffusion, i.e., motion of particles subject to trapping or sticking effects. Computational aspects (Monte Carlo methods) for pointwise approximation can be also included in the course, if the students are interested in these aspects.
An introduction to Shape Optimization
20 hours, 4 CFU
Period: December 2025 - January 2026
Shape Optimization is a modern branch of research within the Calculus of Variations, focusing on minimization problems where the unknown is a domain (the shape) in the Euclidean space. The course explores various facets of Shape Optimization by addressing common questions, approaches, and open problems. Topics will include historical, geometrical, and physical motivations; the existence, regularity, and geometric properties of optimal shapes; as well as numerical methods relevant to the field.
Every academic year the research groups of the Math Department organize a few working groups on specific research topics. Participation to such a working group can be part of the study plan of a graduate student.