PhD in MATHEMATICS

Educational goals and objectives

MATHEMATICS

The course aims to train students capable of carrying out highly qualified mathematical research, both at universities and other public or private bodies, using mathematical research in any form. The training can occur either in pure mathematics, i.e. fundamental research and with outlets mainly in the academic sphere, or in applied mathematics, with outlets both in applied research in the academic world and in the industrial, financial, commercial, and public domain. The central part of the course consists of the dissertation work, during which the students develop cognitive tools that allow them to deal autonomously, originally, and innovatively with the theoretical and practical problems they will have to face in their future work environments. To provide a solid knowledge base on which to develop these skills, during the initial two years, Ph.D. students attend specialized courses held by internal and external lecturers, usually in English. In addition, Ph.D. students attend seminar series within the department and participate in conferences and workshops in Italy and abroad. Ph.D. students are also encouraged to spend training periods abroad for at least three months at centers where the Ph.D. program has scientific links. This achieves another objective, essential for the modern labor market, to train individuals capable of operating effectively and smoothly in an international environment.

Available Positions and Scholarships

University scholarship	5	University scholarship reserved to foreign applicants	0	Department scholarship	0
Scholarship funded by private research institution	0	Scholarship funded by private non-research institution	0	Scholarship funded by public research institution	0
Scholarship funded by public non-research institution	0	PhD position under national research programs	1	PhD position under EU research programs	1
Foreign independent candidates	0	Employee of private research institution/organisation under a cooperation agreement	0	Employee of private non-research institution/organisation under a cooperation agreement	0
Employee of public research institution/organisation under a cooperation agreement	0	Employee of public non-research institution/organisation under a cooperation agreement	1	PhD under higher education and research apprenticeship contract at a private research institution/organisation	0
PhD under higher education and research apprenticeship contract at a private non-research institution/organisation	0	PhD under higher education and research apprenticeship contract at a public research institution/organisation	0	PhD under higher education and research apprenticeship contract at a public non-research institution/organisation	0
No scholarship	0

Research topics

Founded by: European Research Council Executive Agency (ERCEA)

Research topic: Probabilistic approach in Kähler geometry
Description: This ERC project investigates singular Kähler spaces, with an emphasis on their special geometric structures and their interactions with various areas of analysis. More specifically, the project focuses on the existence and properties of special (possibly singular) Kähler metrics with desirable curvature features, such as Kähler–Einstein (KE) and constant scalar curvature Kähler (cscK) metrics. The problem of their existence can be reformulated as a complex Monge-Ampère equation, a fully nonlinear partial differential equation. In the smooth setting, the KE case was solved by Aubin and Yau, who established the Calabi conjecture, and later by Chen–Donaldson–Sun, who established the Yau–Tian–Donaldson conjecture. More recently, the existence theory for cscK metrics has been completed by Chen and Cheng, resolving a conjecture of Tian. However, these results are restricted to smooth Kähler manifolds, and the extension to singular varieties remains a major challenge. This is where pluripotential theory plays a central role. Recent advances by Boucksom, Eyssidieux, Guedj, and Zeriahi, as well as by the author in collaboration with Darvas and Lu, have demonstrated the flexibility of pluripotential methods in handling degenerate complex Monge-Ampère equations in singular settings. The proposed PhD project aims to contribute to this line of research, with particular emphasis on the probabilistic approach to the construction of canonical metrics, initiated by Berman.
Scholarship type: DOTTORANDO SU PROGRAMMA DI RICERCA EUROPEO

CUP : E53C25001150006
Principal Investigator: Eleonora Di Nezza
Program: HORIZON ERC Grants
Project: SinGular Monge-Ampère equations (SiGMA)
Grant Agreement: Grant Agreement n. 101125012 (amendment AMD-101125012-2)
UpB: DiNezzaE25_SIGMA
Lenght: 36
Amount: € 68754,00
Founded by: Ministero dell’Università e della Ricerca (MUR)

Research topic: Random matrices and logarithmically correlated fields
Description: This FIS project investigates connections between Random Matrix Theory and the universality class of logarithmically correlated fields, such as the Gaussian Free Field and the Branching Brownian motion. In 2012 Fyodorov, Hiary, and Keating formulated a striking conjecture: the maxima of log-determinants of large random matrices behave like those of logarithmically correlated fields. Surprisingly, they conjectured that also the Riemann zeta function belongs to this log-correlated group. While there has been huge progress in proving this conjecture, most of the current results are for integrable models. The goal of this project is to establish new analytical techniques to show that this connection holds for a large class of random matrix models, and more. In general, the broad interest of our research group lies at the interface of mathematical physics and probability, with connections to quantum mechanics and theoretical machine learning. The proposed PhD project aims to contribute to strengthening and building new connections between Random Matrix Theory and logarithmically correlated fields, with the possibility of additional projects in our group's research interests, depending on the candidate's interest.
Scholarship type: DOTTORANDO SU PROGRAMMA DI RICERCA NAZIONALE

CUP : F53C25000940001
Principal Investigator: Giorgio Cipolloni
Program: FIS 3
Project: The ubiquity of logarithmically correlated fields
UpB: CipolloniG26_FIS2024
Lenght: 36
Amount: € 68754,00

Self-certification of the academic qualification obtained through Model A:

• Master’s Degree
• Single-cycle Degree
• Specialist Degree
• Old-system Degree
• Degree obtained abroad corresponding to EQF Level 7
• provided that they belong to the following classes: 45/S, 20/S, LM-40, LM-17, LM-82, 92/S, 91/S, 90/S, LM-83, 23/S, LM-18, LM-Data Science