# Introduction

The scientific activity of the Department of Mathematics ‘G. Peano’ (DMGP) aims at reinforcing and enriching the breadth of research topics and the diversity of approaches that have always characterised the Mathematics Area. Special attention is devoted to the international reach of the main contemporary research and the inauguration of novel lines of research. At the same time we wish to deepen prior knowledge, critically analyse it and re-elaborate it under a modern light.

The main research areas are listed below, together with the scholars involved and the topics currently under investigation.

The main research output of the Department are published on major international and national journals, as attested by UGOV.

Several members of the DMGP sit on the editorial boards of international and national journals, and a number of ongoing international and national collaborations has been established.

The DMGP hosts the ‘Mathematical Seminar of the University of Turin and the Polytechnic of Turin’, which consists of the members of the maths’ Departments of the University and the Polytechnic. Its mission is to promote research in all areas of mathematics and applications and its spread.

It publishes the official journal of the Mathematical Seminar *“Rendiconti del Seminario Matematico”*.

To guarantee the quick divulgation of its research, the Department maintains the series *“Quaderni Scientifici del Dipartimento di Matematica”*. The list of all issues with pdf files can be accessed freely on the University catalogue.

The DMGP organises regular research seminars, with the participation of distinguished scholars from all over the world. These touch upon a wide range of topics and contribute significantly to the development of the various subjects, both nationally and internationally. For this reason it also organises a series of special events, called ‘*Lagrange Lectures’*, whose scope is to explore cutting-edge progress in the field of Complexity in relation to other areas. These seminars are held by mathematicians of renowned international fame, who should also be effective in popularising the subject. The text to each seminar is published in an issue of the *“Rendiconti del Seminario Matematico”*.

# Research areas

### Mathematical Logic (MAT/01)

— Set theory -- forcing axioms, large cardinals and omega-logic. Infinite combinatorics. Determinacy and internal models. Descriptive set theory and applications.

— Model theory -- Higher-order amalgamation. Applications to permutation groups. Definable closure vs. algebraic closure.

### Algebra and Geometry (MAT/02, MAT/03)

— Number theory -- algebraic number theory: quaternionic modular forms. Transformations of integer sequences and fixed points. Diophantine approximations of quadratic and cubic irrationals. Operations induced by pencils of conics. Universal magic squares. Discrete mathematics and elementary number theory with applications.

— Algebraic groups -- Algebraic groups and Chevalley groups over rings, representations and homomorphisms. Kac-Moody groups and algebras, representations.

— Computational and combinatoric algebra -- Characterisation of toric varieties. Hilbert schemes of subschemes of projective space using computational algebra techniques. Algebraic methods in statistics and applications to medicine. Implementation of computer programs for the computation of invariants of singularities.

— Hopf Algebras and category theory -- Classification of Hopf algebras satisfying the dual Chevalley property. Cocyclic deformation of Hopf algebras. Generalised Lie algebras. Monadic decomposition of an adjunction between functions.

— Differential and complex geometry -- Special structures on smooth manifolds: Hermitian, contact, symplectic structures on Lie groups and compact quotients; generalised complex structures and deformations. H-structures and special holonomy. Topology and geometry of homogeneous spaces: deRham and Dolbeault cohomology of compact quotient of Lie groups in relation to deformations of complex structures. Flat Riemannian manifolds. Orbits of actions of Lie groups and submanifolds, in relation to normal holonomy.

— Algebraic geometry -- toric varieties. Higher-dimensional varieties: Fano varieties, Mori theory, rational curves. Geometric transitions and their analytic equivalence in Algebraic Geometry and Superstring Theories. Relationship between Milnor numbers of deformations and birational invariants of desingularisations for special transitions. Gopakumar-Vafa conjecture and M-theory lift. Aspects of duality in theoretical physics. Aspects of Mirror Symmetry. Hodge theory, variations of Hodge structure, algebraic cycles.

### Didactics and History of Mathematics (MAT/04)

— Didactics —

Research in the didactics of mathematics aims at innovating and making teaching better at all educational levels. In particular:

- Processes and output of teaching and learning from nursery school to university; general aspects and specific themes of mathematical areas (algebra, arithmetic, geometry, analysis, probability, statistics, logic, numerical analysis, modelling et c.), relations to other disciplines (physics, economy, informatics, biology et c.).

- Theoretical frameworks and concrete projects to interpret/steer classroom activities and educate maths teachers along the lines set out by research, didactics, institutions and regulations: in particular semiotics, embodiment, multi-modality and teaching/learning.

- Representation and communication technologies in maths’ teaching/learning and the eduction of maths teachers: theoretical frameworks and concrete projects for schools.

- Tools for summative and formative assessment in mathematical learning.

— History of mathematics —

The research aims at exploiting the rich historical heritage of our archives in order to foster learning processes and stimulate the spread of mathematical knowledge in the society. Topics:

- Italian mathematical Schools in modern and contemporary times (17th-20th centuries), which strived for the international recognition of their research work and its diffusion in various areas of knowledge.

- Critical editions of letters, manuscripts and sources for historical studies.

- History of mathematics teaching in Italy (17th-20th centuries) and comparison with foreign institutions.

- History of mathematics at all levels of school teaching.

### Mathematical Analysis (MAT/05)

**-** Time-Frequency Analysis: localisation operators; continuity properties for Fourier and Gabor-Fourier integral operators in modulation spaces; Banach algebras of Fourier integral operators; solving hyperbolic equations by Gabor frames. Time-frequency representations within Cohen classes; Strichartz estimates.

- Gevrey micro-local analysis for linear and nonlinear PDEs: decay and regularity for semilinear elliptic equations; wavefronts and Fourier integral operators.

- Well-posedness of the Cauchy problem for hyperbolic systems: global solutions in space variables.

- Micro-local analysis on smooth manifolds and manifolds with conical singularities: hyperbolic operators on manifolds with ends; Fourier integral operators on manifolds with ends; Fourier integral operators on manifolds with boundary.

- Global regularity for second-order parabolic equations with non-regular coefficients; stochastic equations with non-locally Lipschitz coefficients.

- Nonlinear ODEs: boundary-value problems associated to ODEs on unbounded intervals; eigenvalues and bifurcations for Dirac-type systems; quasi-periodic solutions of oscillators in resonance; multiplicity for asymptotically linear second-order systems.

- Calculus of variations: mean curvature problems; singularities and scattering for the n-body problem; variational approach to nonlinear elliptic and hyperbolic equations; radially-symmetric solutions to the Dirichlet problem with concave/convex nonlinear terms.

**Probability and Statistics (MAT/06)**

- Analytical and numerical methods, simulations for first-pass problems in diffusion processes: with and without jumps, univariate or multivariate.

- Development of an R-package to compute first-pass times and the relative functionals.

- Parameter estimators for diffusion processes and properties.

- Fractional processes and properties.

- Estimators for information measures and properties.

- Probabilistic and statistical study of dependencies of point processes.

- Employ of copulas to study dependencies of stochastic processes.

- Stochastic models for the study of neural codes, in particular single neurons and small neural networks.

- Stochastic models to describe the propagation of internet data.

- Stochastic models to describe the error of atomic clocks.

Mathematical Physics (MAT/07)

- Analytical mechanics, dynamical systems and complex systems.

- Symmetries and separation of variables for the equations of mathematical physics.

- Geometric and global methods in physics.

- Variational structure of field theories.

- Conservation laws and entropy of singular solutions to gravity.

- Applications of gravity theory and cosmology.

- Popularisation of physics.

- Mechanics of elastic continua, both constrained and not (equations of state, material symmetries, static problems, wave propagation).

- Core properties and variational theorems for thin thermoelastic continua.

- Modelling of response processes and evaluation of multiple-choice tests in education/psychometrics.

- Evolutive game theory and of networks.

- Changes in conformation and stability of complex biological structures.

### Numerical Analysis (MAT/08)

- Numerical methods for singular integral equations and relative quadrature formulas.

- Numerical methods for ODEs and PDEs.

- Multivariate spline approximation operators for Computer Aided Geometric Design and isogeometrical analysis of PDEs.

- Numerical methods for info-security problems.

- Hermite-Birkhoff interpolation methods on the sphere and Banach spaces, obtained by bi-orthonormal functions and operators. Lagrange interpolation in metric spaces and applications. Methods and algorithms to interpolate sparse data in the plane and the sphere by radial-basis functions.

- Spectral analysis and study of preconditioners for interpolation and collocation matrices generated by radial-basis functions.

- Transformations by radial-basis functions for image registration.

- Mathematical modelling of several problems arising, e.g., in biomedicine, socio-economics and technology.

- Models for biomathematics, in particular systems for the dynamics of interacting populations, possibly equipped with age structure. Eco-epidemiology models, that simulate the spread of epidemics among interacting populations. Meta-ecoepidemic models (collections of different eco-epidemic models at different places that allow for migrations.) Mathematical epidemiology models, esp. to study epidemics in cattle, pig and goat farms. Models for biological control in agriculture, applied to orchards and vineyards in particular.