Mathematics Area - SISSA - Mathematical Analysis, Modelling, and Applications

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Luca Giomi — Fri, 15 Jun 2012 12:23:47 +0000

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Research fields

geometry, in particular algebraic, differential, and noncommutative geometry, also with applications to quantum field and string theory
mathematical analysis, in particular calculus of variations, control theory, partial and ordinary differential equations
mathematical modelling, in particular mechanics of solids and fluids, modelling of complex and biological systems, multiscale analysis
mathematical physics, in particular integrable systems and their applications, nonlinear partial differential equations, mathematical aspects of quantum physics
numerical analysis and scientific computing, applied to partial differential equations and to control problems

PhD and MSC courses:

Geometry and Mathematical Physics
Mathematical Analysis, Modelling, and Applications
MSc in Mathematics (Laurea Magistrale in Matematica) in collaboration with the University of Trieste
Master Degree in Data Science and Scientific Computing in collaboration with the Universities of Trieste and Udine, and with ICTP
Joint SISSA-ICTP professional Master in High Performance Computing (MHPC)

Laboratories:

SISSA MathLab: a laboratory for mathematical modeling and scientific computing
SAMBA a laboratory in collaboration with the Cognitive Neuroscience Group

Area Coordinator

Antonio Lerario

Faculty

Massimiliano Berti, Professor
Stefano Bianchini, Professor
Antonio De Simone, Professor
Barbara Fantechi, Professor
Nicola Gigli, Professor
Antonio Lerario, Professor
Alberto Maspero, Professor
Giovanni Noselli, Professor
Marcello Porta, Professor
Luca Rizzi, Professor
Gianluigi Rozza, Professor
Jacopo Stoppa, Professor
Alessandro Tanzini, Professor
Gabriele Benedetti, Associate Professor
Mikhail Bershtein, Associate Professor
Andrea Cangiani, Associate Professor
Ilya Chevyrev, Associate Professor
Tamara Grava, Associate Professor
Davide Guzzetti, Associate Professor
Igor Krasovsky, Associate Professor
Gaultier Lambert, Associate Professor
Andrea Ricolfi, Associate Professor
Nicolò Sibilla, Associate Professor
Rafael Torres, Associate Professor
Pasquale Claudio Africa, Assistant Professor
Antonio Colanera, Assistant Professor
Pavlo Gavrylenko, Assistant Professor
Ricardo Grande Izquierdo, Assistant Professor
Federico Pichi, Assistant Professor
Davide Riccobelli, Assistant Professor
Niccolò Tonicello, Assistant Professor

Former Faculty Members

Former Professors

Visiting Professors

Andrei Agrachev, Visiting Professor
Andrea Braides, Visiting Professor
Gianfausto Dell'Antonio, Visiting Professor
Andrea Mola, IMT Lucca
Maria Giovanna Mora, Università di Pavia
Rodica Toader, Università di Udine

Area Secretariat

De Giorgi and Gromov working together

Marco Picerni — Wed, 24 Jun 2026 12:54:26 +0000

Lecturer:

Nicola Gigli

Course Type:

PhD Course

Academic Year:

2026-2027

Period:

October - November 2026

Duration:

20 h

Research Group:

Mathematical Analysis, Modelling, and Applications

Location:

A-133

Next Lectures:

Monday, October 5, 2026 - 09:00 to 11:00

Tuesday, October 6, 2026 - 09:00 to 11:00

Monday, October 12, 2026 - 09:00 to 11:00

Tuesday, October 13, 2026 - 09:00 to 11:00

Monday, October 19, 2026 - 09:00 to 11:00

Tuesday, October 20, 2026 - 09:00 to 11:00

Monday, October 26, 2026 - 09:00 to 11:00

Tuesday, October 27, 2026 - 09:00 to 11:00

Monday, November 9, 2026 - 09:00 to 11:00

Tuesday, November 10, 2026 - 09:00 to 11:00

Intermittent Nash iterations in PDEs

Guglielmo Padula — Thu, 18 Jun 2026 11:47:30 +0000

External Lecturer:

Vikram Giri (Zurich)

Course Type:

PhD Course

Academic Year:

2026-2027

Period:

May 2027

Duration:

20 h

Description:

In the first half of the course, we will develop the theory of Nash iterations (aka. convex integration) as discovered by Nash for the isometric embedding problem, apply it to the incompressible Euler equations as pioneered by De Lellis-Székelyhidi, and develop the theory of intermittent Nash iterations for the Navier-Stokes equations as pioneered by Buckmaster-Vicol. In the second half, we'll develop finer techniques used in recent applications to improve the regularity of the solutions constructed. For instance, we'll see the main ideas that go into the proof of the intermittent strong Onsager theorem. Prerequisites are minimal but include basic functional analysis (L^p spaces, Sobolev embedding) and some harmonic analysis (Calderón-Zygmund). There will be ample exercises through the course veering towards research problems near the end. The goal will be to get to grips with a robust toolkit that has potential applications to a wide class of PDEs.

Research Group:

Mathematical Analysis, Modelling, and Applications

Location:

TBC(to be checked)

Next Lectures:

Existence and localization of periodic magnetic trajectories with low energy

Marco Picerni — Fri, 22 May 2026 12:44:25 +0000

Research Group:

Mathematical Analysis, Modelling, and Applications

Valerio Assenza

Institution:

IMPA

Schedule:

Monday, June 8, 2026 - 14:00

Abstract:

One of the classical and most celebrated problems in Hamiltonian dynamics is the existence of periodic orbits on a prescribed energy level. In this talk, we focus on the setting of magnetic dynamics. After providing an introduction to the subject, we present a variational approach to the study of periodic magnetic trajectories and discuss recent results concerning both the existence and the localization of low-energy periodic magnetic orbits. This work is part of a joint project with Leonardo Macarini and Gabriele Benedetti.

Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology

Marco Picerni — Fri, 22 May 2026 12:42:19 +0000

Research Group:

Mathematical Analysis, Modelling, and Applications

Leonardo Macarini

Institution:

IMPA

Schedule:

Friday, May 29, 2026 - 15:00

Abstract:

We consider closed contact manifolds $(M,\xi)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which equivariant symplectic homology is a well defined invariant. Under weak index assumptions on a non-degenerate contact form $\alpha$ on $M$, we establish a sharp lower bound $r_M$ for the number of closed Reeb orbits of $\alpha$, and show that equality holds if and only if $\alpha$ is lacunary. The bound $r_M$ is essentially determined by the deviation of a suitable finite sum of the contact Betti numbers of $M$ from its asymptotic average. We also compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim H_*(M/S^1;\Q)$. This is joint work with Miguel Abreu.

SISSA Women in Mathematics 2026

Guglielmo Padula — Thu, 14 May 2026 11:18:51 +0000

Research Group:

Mathematical Analysis, Modelling, and Applications

Sara Daneri, Harini Desiraju, Giovanna Marcelli

Schedule:

Friday, June 12, 2026 - 14:00 to 15:45

Abstract:

Sara Daneri

"At the roots of self-assembled patterns: intrinsic nonlocal curvatures"

Patterns, meant as periodically repeated geometric structures, emerge in several physical and biological systems both at small and very large scales. Such patterns are often rather simple and common to diverse physical systems: stripes/lamellae, checkerboards, periodically ordered spherical droplets, etc. It is universally believed that in many of such systems the emergence of patterns is due to the competition between short range attractive forces (which would prefer to form large droplets and aggregate the mass together) and long range repulsive forces (which would prefer to split the mass in smaller and smaller droplets). This mechanism is very important both from the theoretical and the applied point of view, because it allows microstructures to form without the aid of external forces (self-assembly).
In this talk we are going to present a recent approach which is able to give a rigorous explanation of such a phenomena, in some physical settings. The focus will be on the role played by newly found intrinsic nonlocal curvatures, which provide the link between the energy of admissible configurations and their spatial organization into patterns.

Giovanna Marcelli

"Adiabatic perturbation theory in quantum dynamics and its applications to the integer quantum Hall effect"

First, I will review the standard adiabatic theorem in quantum dynamics for gapped fermionic systems (GFS). Then, I will discuss the concepts of super-adiabaticity and higher-order response, from which the validity of the linear response for GFS can be deduced.
Finally, I will report recent results on the integer quantum Hall effect obtained in joint works, based on adiabatic perturbation theory.

Harini Desiraju

"Random curves and isomonodromy"

Schramm-Lowner Evolutions are a special class of random curves that are central to the study of 2D lattice models. In this talk, I will introduce the connection between the observables known as SLE martingales, and special transcendental functions from integrable systems known as isomonodromic tau-functions.

Prescribing the Lorentzian mean curvature of a spacelike hypersurface, and the Born-Infeld model

Marco Picerni — Fri, 27 Mar 2026 09:06:11 +0000

Research Group:

Mathematical Analysis, Modelling, and Applications

Luciano Mari

Institution:

University of Milan

Schedule:

Wednesday, April 29, 2026 - 14:00

Abstract:

The talk aims to discuss the existence and regularity problem for spacelike hypersurfaces $M$ in Lorentzian manifolds whose mean curvature is a prescribed, possibly singular distribution. When considering ambient Minkowski space, this leads to study the following Dirichlet problem for the function that (locally) describes $M$ as a graph, which we call $(\mathcal{BI})$: $$ \left\{ \begin{array}{ll} -{\rm div} \left( \frac{Du}{\sqrt{1-|Du|^2}}\right) = \rho & \quad \text{in a domain } \, \Omega \Subset \mathbb R^n, \\[0.5cm] u = \phi & \quad \text{on } \, \partial \Omega, \end{array} \right.$$for a given measure $\rho$ (the prescribed mean curvature) and boundary data $\phi$. Problem $(\mathcal{BI})$ also appears in Born-Infeld's theory of electrostatics, according to which $u$ describes the electric potential generated by the charge $\rho$. Even though $(\mathcal{BI})$ is formally the Euler-Lagrange equation of a nice convex functional $I_\rho$, the lack of smoothness of $I_\rho$ where $|Du|=1$ (i.e. where the graph of $u$ becomes lighlike) may prevent the unique variational minimizer $u_\rho$ to solve $(\mathcal{BI})$. As we shall see, this possibility actually occurs. We shall describe both existence and non-existence results helping to guess the possible sharp thresholds on $\rho$. A chief difficulty comes from the possible presence of ``light segments" in the graph of $u_\rho$, a fact that we will investigate in detail. Various open problems and research directions will be discussed.The talk is based on joint works with J. Byeon, N. Ikoma, A. Malchiodi and L. Maniscalco, available at arXiv:2112.11283 and arXiv:2512.17670.

On the topological invariance of helicity

Marco Picerni — Fri, 27 Mar 2026 09:00:19 +0000

Research Group:

Mathematical Analysis, Modelling, and Applications

Oliver Edtmair

Institution:

ETH Zurich

Schedule:

Thursday, April 16, 2026 - 14:00

Abstract:

Helicity is an invariant of divergence free vector fields on a three-manifold. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic Hopf invariant, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows. In this talk, I will present an affirmative answer to both questions for non-singular flows. The proof draws on recent advances in C^0 symplectic geometry, in particular regarding the algebraic structure of the group of area preserving homeomorphisms, which I will also survey. This is based on joint work with Sobhan Seyfaddini

Fast dynamo action on the 3-torus for pulsed-diffusions

Marco Picerni — Fri, 27 Mar 2026 08:56:49 +0000

Research Group:

Mathematical Analysis, Modelling, and Applications

Massimo Sorella

Schedule:

Friday, April 10, 2026 - 11:00

Abstract:

For the passive vector equation, the fast dynamo conjecture predicts exponential-in-time growth of the L^2 norm of the solution under a Lipschitz flow of a vector field, at a rate independent of the resistivity. We establish this conjecture for the pulsed diffusion model with a time-periodic stretch-fold-shear (SFS) vector field. Our approach uses anisotropic Banach spaces adapted to the dynamics of the underlying flow to prove the existence of a discrete eigenvalue with positive real part in this distributional setting.

The Diffeomorphism Group of Manifolds

Guglielmo Padula — Mon, 16 Mar 2026 13:25:34 +0000

Lecturer:

Gabriele Benedetti

Course Type:

PhD Course

Academic Year:

2025-2026

Period:

April-May

Duration:

20 h

Description:

Course Content:

In life as well as in mathematics, it is not important if you come back where you started, but it is the journey you followed that makes all the difference. Thus, this 20-hour course is dedicated to the study of the group G of (volume-preserving/symplectic) diffeomorphisms of a manifold M. The milestone of the theory, and our goal together, is the Epstein-Thurston-Mather-Banyaga Theorem: the group G is, up to the natural Flux Homomorphisms, simple, that is, it has no non-trivial normal subgroups. As a corollary, two diffeomorphism groups G_1 and G_2 are isomorphic if and only if the associated manifolds M_1 and M_2 are isomorphic. The proof relies on many beautiful ideas in analysis and dynamics such as Herman's theorem on the conjugacy of torus diffeomorphisms and the fragmentation of diffeomorphisms. Our main source for the course is the book The Structure of Classical Diffeomorphism Groups by Augustin Banyaga.

Lectures:

All lectures are in room 133 except for the ones of 15/04 and 22/04 which are in room 136.

Wed Apr 8 11:00-13:00. Introduction. Diffeomorphism groups and vector fields
Fri Apr 10 9:00-11:00. Volume-preserving and symplectic diffeomorphisms
Wed Apr 15 11:00-13:00. Hamiltonian diffeomorphisms.
Fri Apr 17 9:00-11:00. The linearized Epstein-Thurston-Mather-Banyaga Theorem.
Wed Apr 22 11:00-13:00. From perfectness to simplicity.
Fri Apr 24 9:00-11:00. The fragmentation property of diffeomorphisms.
Wed May 20 11:00-13:00. Herman's theorem on the conjugacy of torus diffeomorphisms.
Fri May 22 9:00-11:00. The Epstein-Thurston-Mather-Banyaga Theorem.
Wed May 27 11:00-13:00. The groups are isomorphic if and only if the manifolds are.
Fri May 29 9:00-11:00. Outlook: regularity issues, quasi-morphisms, open problems.

Research Group:

Mathematical Analysis, Modelling, and Applications

Next Lectures: