OVERVIEW
SUMMER INTERNSHIP PROGRAM
IMM Summer Internship Program
As part of the “Mathematics for Humanity” grant program by the International Centre for Mathematical Sciences (ICMS), the International Mathematics Master (IMM) is pleased to announce the 2026 edition of their Summer Internship Program (SIP). The programme is jointly organised with leading academic institutions, notably the Abdus Salam International Centre for Theoretical Physics (ICTP), the Scuola Matematica Interuniversitaria (SMI), and Consorzio Interuniversitario per l’Alta Formazione in Matematica (CIAFM) as affiliated partner.
The initiative aims to provide to outstanding IMM students currently enrolled either in the Algerian or in the Pakistani IMM Master programme with:- the possibility to strengthen their own skills in advanced mathematics;
- advanced training and exposure to interdisciplinary research of mathematics in science and technology;
- exposure to new learning methods and academic topics;
- direct mentorship from leading international scholars;
- increased opportunities to join a global academic network of researchers;
- engagement with internationally recognised professors, young scientists, and IMM alumni.
The SIP is structured in two distinct phases:
6 July – 18 July, 2026, at the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy. During the first 2-weeks period the participants will be attending mini-courses and advanced research talks on different fields of mathematics, gaining new perspectives on their research field of interest and more. Additionally, each student will have the opportunity to present a topic of their choice in an open poster session and a subsequent internal “Gong Talk” session. The preparation of such presentations will be guided by experienced mentors during the stay at ICTP. The didactic schedule will be accompanied by a series of scientific outreach and social activities.
20 July – 14 August, 2026, at the Scuola Matematica Interuniversitaria (SMI) in Perugia, Italy. The second part of the SIP offers a choice of basic courses consisting of 25 hours of lectures and 15 hours of problem-solving sessions. Students will have to choose and attend two among the following courses: Algebraic Geometry, Complex Analysis, Differential Geometry, Functional Analysis, Mathematical Statistics. At the end of each course the participants will undertake written examinations for which they will be given a final evaluation. Further information can be found here
Gallery of last year’s SIP
ICTP Courses and Seminars
During the first 2 weeks of programme, from the 6th until the 18th of July, the participants of the IMM SIP will be hosted at the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, and will be attending mini-courses and advanced research talks on different fields of mathematics, as well as scientific outreach seminars and activities
Mini-Courses
Gabriele Benedetti: “Four sides of convexity”
Convex sets and convex functions are used to model many phenomena in mathematics: from measuring distances to optimizing quantities, from interpolating values to clustering of data. In this mini-course we look at four sides of the multifaceted life of convexity and discuss its applications to combinatorics, algebra, analysis, and dynamics.Frank Neumann: “The Magic of Spectral Sequences: A Journey through Algebra, Topology and Geometry”
Spectral sequences are a fundamental tool in modern mathematics, providing a powerful framework for computing algebraic invariants that are otherwise difficult or impossible to determine directly. They arise naturally in a wide range of contexts, including algebraic topology, homological algebra, algebraic geometry, basically wherever homology, cohomology or homotopy groups need to be calculated. Introduced by J. Leray in the 1940s in order to compute sheaf cohomology, they were used later by J. P. Serre and A. Borel to compute cohomology groups of fibrations and fiber bundles, while A. Grothendieck used them to compute the derived functors of a composition of functors. These lectures will offer a rapid introduction into spectral sequences and their use in algebra, topology and geometry. Beginning with filtered complexes and exact couples, we will develop the basic machinery of spectral sequences, discuss convergence issues, and learn how to interpret differentials and extension problems. We will then examine several fundamental examples, including the Serre, Leray, and Grothendieck spectral sequences and how to use them in concrete calculations. Throughout the lectures, applications from algebra, geometry, and topology will serve as the motivating examples.Stefano Luzzatto & AliTahzibi: “Dynamical Systems”
The theory of Dynamical Systems starts with the study of deterministic processes, such as those which happen in nature under the effect of physical laws. It aims to describe the way a system evolves in time or, more technically, the future orbit of a given initial condition under the effect of a given “physical law”. In this mini-course we will give a simple introduction to the main concepts in the theory.Research Talks
Peter Stevenhagen: “Categories and functors”
In the mid 20th century, people realized that many constructions in algebra and other parts of mathematics share common features. The language of categories and functors that developed out of this (“abstract nonsense”) has become commonplace in modern mathematics. We give an introduction and many examples.
Chiara Franceschini: “Markov Duality for Interacting Particle Systems”
In this seminar, I will present an introduction to the theory of Markov duality from an algebraic perspective, showing how representations of Lie algebras naturally lead to the construction and description of Markov processes. I will introduce several interacting particle systems, including both particle and energy transport models. As an application, I will show how duality can be used to obtain information on the non-equilibrium stationary states of these systems when they are coupled to stochastic reservoirs. In this setting, the reservoirs break the conservation law of the bulk dynamics (either particle number or energy) and generate a non-vanishing flux through the system.Sophie Dabo-Niang: “Infinite-Dimensional Data Analysis”
In this talk, we introduce the concept of learning from data from functional spaces. Functional data, originate from random variables valued in a functional space, present significant challenges when it comes to modeling curves, patterns, images, and other intricate structures. We will give an overview of functional data, emphasizing their frequent presence in several domains. A key emphasis will be placed on the theoretical aspects of learning from non-random samples. We’ll explore some practical implementations aimed at capturing variability, dimensionality reduction and supervised learning.Andrea Di Lorenzo: “Elliptic curves and their moduli”
This talk aims at introducing a first example of moduli space of smooth curves, focusing on the genus one case. I will adopt both an algebraic and an analytic perspective. Time permitting, I will also explain how is possible to compactify this moduli space by adding one point at the boundary representing a singular curve, and give an idea of how this construction works for moduli spaces of higher genus curves.Matias Luis del Hoyo:“On the Riemann-Hilbert correspondence”
The Riemann–Hilbert correspondence is a beautiful instance of a general principle in geometry: local infinitesimal data can sometimes be integrated into global topological data. In this talk, we will introduce vector bundles and connections, and see how a flat connection gives rise, by parallel transport, to a representation of the fundamental group. We will discuss the converse construction and the resulting correspondence between flat vector bundles and representations. Along the way, we will emphasize the local-to-global nature of the result and explain how Lie groupoids provide a natural framework for its formulation. We will conclude with a glimpse of higher versions of this correspondence and some questions arising in current research.
ICTP Schedule
WEEK 1 
WEEK 2 The Abdus Salam International Centre For Theoretical Physics
ICTP supports the International Mathematics Master by providing guidance for the scientific and academic programme and by supporting international, faculty and international students in the framework of the ICTP Affiliated Centre Programme.
In August 2025 IMM and ICTP have signed a Memorandum of Understanding. With this MoU we’re moving from shared intentions to concrete steps pillared on:
Co-design and implement Master and PhD programs in the Global South: Opening advanced academic pathways and ensuring access to high-level education for talented students worldwide, with a special focus on underrepresented regions.Organize Scientific Schools and Workshops: Creating vibrant hubs for collaboration through ICTP’s affiliated centers and IMM host institutions, where students and researchers can exchange ideas, gain training and build communities.Strengthen our Networks: Expanding ICTP and IMM connections to foster interdisciplinary exchanges and joint projects that connect across cultures and regions.Encourage Scientific Research, Technology Development and Capacity Building: Supporting initiatives that not only advance knowledge but also address the need for stronger educational and scientific infrastructures in diverse contexts.Scuola Matematica Interuniversitaria (SMI)
The Scuola Matematica Interuniversitaria (SMI) is a long-established institution in Perugia, Italy, developing an internationally recognised summer schools bridging mathematicians from the world over since 1971.The IMM has a partnership with CIAFM and SMI to increase inclusion in access and quality of mathematical courses of renowned professors joining for a three-week full and examinated programme. Through this partnership the IMM enables IMM Pakistan and IMM Algerian young scientists to participate wiuth reduced costs, thereby helping bridge the gap between the global mathematics community and talents in Mathematical sciences from the global South.