## Presentation

### THE DEPARTMENT: STATE OF THE ART

The Department of Mathematics of the University of Rome Tor Vergata is distinguished by first class research, often motivated by applications from theoretical physics, astronomy, aerospace, finance, technology and medical science, a high level educational system, and the organisation of events in the context of the so-called third mission of the University. For details we refer to the Department's website, http://www.mat.uniroma2.it.

The Department has obtained important results in numerous research fields, such as representation theory, geometry of manifolds, varieties and schemes, quantum field theory, operator algebras, variational methods (interfaces, homogenisation), PDE (gauge theory, Hamilton-Jacobi equations, mean field games, degenerate equations), control theory, probability theory and statistics (random fields, mathematical finance), statistical mechanics (equilibrium and non-equilibrium; classical and quantum), dynamical systems (real and holomorphic), celestial mechanics (normal forms, KAM theory) and numerical analysis (spline theory, numerical linear algebra).

Currently the projects with external funding are an ERC Advanced Grant (R. Longo), FARE-MIUR (R. Longo), FIR-MIUR (H. Speleers), "Young reseachers programme Levi Montalcini"-MIUR (Y. Tanimoto), SIR-MIUR (L. Arosio) and ASI-JUICE (A. Celletti). From 2012 to 2016 there were two additional ERC Advanced Grants (C. Liverani, R. Longo), two ERC Starting Grants (F. Bracci, D. Marinucci), three Marie Curie-ITN grants (A. Celletti, R. Schoof), and two FIRB-MIUR projects (F. Bracci, G. Tarantello).

The Department has an intensive programme of seminars and visiting professors. It hosts the Centre of Mathematics e Theoretical Physics (director R. Longo), which frequently organises workshops with internationally renowned speakers, and the Italian Society for Celestial Mechanics and Astrodynamics.

The Department offers Bachelor Programmes in "Mathematics" and "Science and Technology for the Media", a Master Programme in "Pure and applied mathematics", a second level Master Programme in "Space science and technology" (Master STS), and a Graduate School in "Mathematics". Since a decade the Department awards every year the "Premio Cuozzo" for the best Italian PhD-thesis in Mathematics discussed in that year.

The activities in the context of the third mission include numerous individual and departmental initiatives, such as SCIENZAORIENTA, the Math Olympics and educational events for high school teachers. A network of industrial partners, in particular through the Master STS, fosters mutual knowledge transfer. The Math Culture is spread by means of videoregistration and streaming of seminars and colloquia, also through YouTube and a Facebook site of the Department.

### FUTURE DEVELOPMENT OF THE DEPARTMENT: OBJECTIVES

The Department aims to increase its leading role in research, math education and math culture. The recently awarded national Excellence Project 2018-2022, denoted by MATH@TOV, offers the opportunity to face new challenges, and its main objectives are

- foster new collaborations between staff members on advanced research themes
- hire excellent staff members, able to participate in multiple research projects
- stimulate the interaction with excellent math groups, both in public research institutions and industry, and transform the Department into a strategic asset for the development of highly advanced mathematics and its application to specific problems
- increase the international visibility of the Department
- improve the Master and PhD Programmes in Mathematics
- intensify the spreading of Math Culture.

To stimulate simultaneously the progress of pure and applied mathematics and its interactions, with long term benefits for the scientific development of the Department, MATH@TOV distinguishes two research areas. Below we describe some of their principal research themes.

#### The Pure Math Area

The Pure Math Area has an outstanding know-how in various fields. Through MATH@TOV we plan to develop major synergy between different fields, foster a dense interaction with the Applied Math Area, and create an international centre of excellence focussed, though not exclusively, on the following research themes.

**Dynamical systems and statistical mechanics**. Dynamical systems with few degrees of freedom are important for both applied (detection of areas of chaotic motion, particle flows, systems with
small dissipation) and pure research. We intend to investigate connections between geometric and dynamical objects on a given variety, determining e.g. the existence of invariant or asymptotically
stable subvarieties as well as the global/local dynamics of dynamical systems and their possible normal forms. Recently the importance of studying the induced dynamics on the space of distributions
(rather than measures) has emerged, which opens up connections with semiclassical analysis and quantum mechanics. In this context we shall study dynamical systems with many weakly interacting degrees
of freedom, which are fundamental in nonequilibirum statistical mechanics or in variational problems with percolation. In the context of probability theory we intend to study the geometry of level
sets of random fields on varieties, emphasizing relationships with quantum mechanics, analysis of cosmological data and Markov chains on infinite trees (harmonic analysis).

**Operator algebras, quantum field theory (QFT) and representation theory**. QFT is rich of connections with several areas of physics and mathematics. Part of the structure is already present at
a classical level. The theory of operator algebras has recently established new connections of QFT, for example with tensor categories, knot theory, free probability and quantum groups; in this
perspective, we intend to tackle new problems in conformal QFT and quantum statistical mechanics. Another important line of research in this framework is given Connes' non-commutative geometry,
which deals with spectral theory of operators, index theorems, number theory, orbits of foliations and singular quotients. Collaborations with experts in representation theory will also be important,
introducing new approaches to Kac-Moody algebras, for which there exist large classes of representations with unknown characters.

**Algebraic geometry**. Algebraic Geometry has many interactions with several other disciplines such as Theoretical Physics, cyber security (Cryptography) and Telecommunication.
We intend to develop innovative approaches for the birational, modular and enumerative study of families of curves on algebraic varieties as well as for the study of Hilbert schemes and moduli
spaces of algebraic varieties or of vector bundles on them. These subjects also have deep connections with the use of vector bundles in dynamics and its applications, in cosmology and statistical
analysis of data or with topics in Computer Aided Design and Computer Graphics.

**PDE**. The description of certain relevant states in the theories of Ginzburg-Landau and gauge fields, as well as some monodromy problems for the Fuchs equation, are all examples of topics
which have lead to the solution of Liouville type equations. In this context we shall intensify research activities concerning the classification theory of conformal geometries in relation with
hyperelliptic curves, integrable Hamiltonian systems (Toda), electroweak theory, interface dynamics and the evolution of microstructures.

#### The Applied Math Area

Among the many experiences in applied problems, we have chosen to focus on modelling and analysis problems from cosmological and aerospace applications, where Department staff members have reached important results (ERC, MC-ITN). MATH@TOV foresees the foundation of a Centre of Excellence for Space Mathematics, an international novelty with strong academic and industrial impact, also in view of the vicinity of institutions and industries dedicated to aerospace research. The major themes to be developed are described below. A strong interaction with the pure math area will be required.

**Celestial mechanics - space debris**. The dynamics of satellite rests abandoned in space is one of the key research themes of the Space Agencies. We plan to analyse the dynamics of orbital debris,
identifying chaotic and regular regions and possible orbital deflections. The results of original studies based on chaos theory, KAM theory and normal forms, can lead to innovative techniques for the
control of space debris with important technological applications.

**Dynamical systems - stability of planets and satellites**. Up to date, we do not have a rigorous proof of the stability of a realistic model of the Solar system. This can be obtained merging
KAM and Nekhoroshev theories. Through MATH@TOV we intend to develop dynamical system's theories to study topics like the dynamics of extrasolar planetary systems and station-keeping orbits near
irregularly shaped asteroids, also using optimal control theory and methods of numerical analysis.

**Probability theory and statistics - data analysis in cosmology**. The collection and analysis of cosmological data has seen a big expansion in the last 15 years; for example, we recall the datasets
on the cosmic background radiation (WMAP and Planck missions) and next ESA Euclid mission on gravitational lensing. We intend to develop techniques which involve several different aspects
related to pure and applied mathematics: spectral analysis of random fields on the sphere, wavelets/needlets systems on spherical fields, geometrical properties of random fields in cosmology,
Minkowski functionals for the excursion sets.

**Numerical analysis - aeronautic and aerospace design**. Aeronautic and aerospace design requires advanced numerical simulation and computational geometry techniques. Recently, isogeometric
analysis (IgA) connected the so far disjoined areas of numerical treatment of PDEs and geometric modeling. MATH @ TOV will tackle difficult problems related to the aforesaid aspects such as: extension
of the IgA approach to complex geometries of arbitrary topology and large dimensions, local refinement, and fast solvers.

**Statistical mechanics - air traffic**. We intend to study the congestion processes observed in air traffic in the neighborhood of large airports. The topic is very sensitive from a sustainability
point of view, but it is also mathematically complex, since the process of arrivals is self-correlated.

**PDEs - climatology**. We intend to apply analytical methods for controllability and observability of degenerate parabolic equations to the solution of inverse problems for differential energy
balance models. These models describe the contribution of ice masses to climate evolution. In the context of extreme atmospheric events, a preliminary study on the effect of suspended particles on
turbulent flows needs to be extended to more realistic flows (hurricanes, tropical cyclones, dust and fire storms).

MATH@TOV will intensify collaborations with excellence centres and first-rank scientists, also through the programme of visiting professors and the organisation of schools and workshops, and with research institutes and industry, in particular the Italian Space Agency, Thales Alenia Space Italia, Telespazio, Deep Blue, and 2 Institutes of the National Research Council (Istituto per le Applicazioni del Calcolo "M. Picone" and Istituto di Geoscienze e Georisorse).

### IMPLEMENTATION OF THE PROJECT

To achieve its objectives, the strategy of implementation of MATH@TOV concerns various issues.

**Research**. The research activities will be supported by a programme of visiting professors, schools, workshops, and thematic seminar series. Formal agreements of collaboration with excellence
centres of mathematics will concern both math research and teaching.

**Hiring policy**. Excellent young scientists will be hired, also from abroad, and their research activities will be integrated in existing research groups of the Department.

**PhD and Master Programme in Mathematics**. MATH@TOV offers a unique opportunity to improve substantially the PhD and Master Programme in Mathematics and make them internationally competitive.
Concerning the Graduate School, part of the budget of MATH@TOV will be used to increase the financial entity of PhD scholarships to attract the best students, special courses dedicated to the
research themes of MATH@TOV will be offered, highly qualified teachers will be attracted through the visiting programme, and stages in prestigious research in institutions will be funded.
Intensive cycles of dedicated seminars will be offered to the students of both the PhD and Master Programme. For the Master Programme a new curriculum of excellence will be created and scholarships
will be offered.

**Networking**. The interaction with the Italian Space Agency, the National Research Council and aerospace industries will stimulate knowledge transfer and form a bridge between research and
industrial innovation. Multi-disciplinary working groups will be created and stimulated to participate in calls for research projects and additional funding.

**Third Mission**. We shall organise events for the spreading of Math Culture and meetings with high school students and teachers, also through lectures and experiments in a new lecture
room/laboratory with modern equipment and a permanent exhibition of mathematics and interactive experiments.

MATH@TOV imposes a strict government, and its implementation will be monitored from inside and outside the Department. The Advisory Board of MATH@TOV consists of the following distinguished professors:

E. Trelat (Universitè Pierre et Marie Curie, Paris)

T. Rivière (ETH, Zurich)

C. Voisin (Collège de France, Paris)

T.J.R. Hughes (ICES, Austin).

Proceedings and lecture notes of workshops and schools will be provided, possibly in special issues of first-rank journals. Major events will be registered and broadcasted in streaming.