Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 29-04-2024 al 05-05-2024
Lunedì 29 aprile 2024
Ore 12:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Dipartimento
Alessandro Alla
A computational framework for large-scale dynamics: model reduction, data-driven modeling and optimal control
This talk provides an overview of my research interests which rely on the theoretical and computational aspects of optimal control problems, with a particular emphasis on the Hamilton-Jacobi-Bellman(HJB) equations, model order reduction, and data-driven modeling. Many complex mathematical models encountered in real-world scenarios pose challenges in numerical simulations due to their complexity and large scale. To tackle this, model order reduction replaces the original problem with a surrogate model, identifying a subspace that captures the essential dynamics of the underlying nonlinear Partial Differential Equations (PDEs) and projecting these PDEs onto that subspace. By reducing the problem dimension, the original nonlinear PDEs can be replaced by smaller systems of ordinary differential equations, enabling efficient and accurate solution of the approximate problem. Applications to Turing patterns and fluid dynamics will be shown. I have also engaged in data-driven modeling using classical data science techniques such as data mining, machine learning and bigdata. My focus is on discovering rigorous mathematical models behind experimental data such that we can use it to make predictions or reconstruct solutions for missing data within a required time frame. Nowadays, we can deal with a huge amount of data that describe unknown dynamical systems. Dynamic Mode Decomposition (DMD) is an example of data-driven modeling. Sparse optimization methods can also be employed to reconstruct nonlinear differential equations. Applications to TIM data will be provided. Furthermore, I have delved into both infinite and finite horizon optimal control problems for nonlinear high-dimensional dynamical systems. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation, stemming from the dynamic programming (DP) approach. However, the primary challenge lies in the curse of dimensionality due to its exponential growth in computational cost, making HJB equations solvable only in a relatively small dimension. To address this challenge, my contributions focus on the stationary HJB approach for high dimensional problems. I developed an accelerated method, coupling value iteration and policy iteration to enhance the computational efficiency of the numerical scheme. Subsequently, I shifted my focus to finite horizon optimal control problems and computed the value function using a DP algorithm on a tree structure algorithm (TSA) constructed by the time-discrete dynamics. In this way, there was no need to build a fixed space triangulation and to project on it. Applications to the control of PDEs will be presented. To conclude, I will provide insights into my current works and outline future research directions.
Lunedì 29 aprile 2024
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Analisi Matematica
Carlo Mantegazza (Università degli Studi di Napoli Federico II)
Stability for the surface diffusion flow
We study the surface diffusion flow in the flat torus, that is, smooth hypersurfaces moving with the outer normal velocity given by the Laplacian of their mean curvature. This model describes the evolution in time of interfaces between solid phases of a system, driven by the surface diffusion of atoms under the action of a chemical potential. We show that if the initial set is sufficiently ``close'' to a strictly stable critical set for the Area functional under a volume constraint, then the flow actually exists for all times and asymptotically converges to a ``translated'' of the critical set. This generalizes the analogous result in dimension three, by Acerbi, Fusco, Julin and Morini. Joint work with Antonia Diana e Nicola Fusco. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.
Per informazioni, rivolgersi a: azahara.delatorrepedraza@uniroma1.it
Martedì 30 aprile 2024
Ore 11:00, aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
corso di dottorato
Martina Lanini (Tor Vergata)
Algebre di Hecke
Martedì 30 aprile 2024
Ore 14:30, aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
seminario di Geometria
Zhi Jiang (Fudan University (Shanghai))
Irregular surfaces of general type with minimal holomorphic Euler characteristic
We explain our recent work on the classification of surfaces of general type with p_g=q=2 or p_g=q=1. Our approach is based on cohomological rank functions, the Chen-Jiang decomposition/Fujita decomposition and Severi type inequalities. This talk is based on a joint work with Jiabin Du and Guoyun Zhang and a joint work in progress with Hsueh-Yung Lin.
Per informazioni, rivolgersi a: guidomaria.lido@gmail.com
Martedì 30 aprile 2024
Ore 15:00, Sala Cappelletta, Villa Mirafiori, Sapienza Università di Roma
Seminari di ricerca in Didattica della Matematica
Enrico Rogora (Sapienza Università di Roma)
Attività interdisciplinari per stimolare lo sviluppo di attitudini fondamentali per l'apprendimento della matematica
Per informazioni, rivolgersi a: annalisa.cusi@uniroma1.it
Martedì 30 aprile 2024
Ore 16:00, aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
seminario di Geometria
Gianluca Pacienza (Institut Élie Cartan de Lorraine - Nancy)
Regenerations and applications
Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective K3 surfaces. In the talk I will present a joint work with G. Mongardi in which we show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results.
Per informazioni, rivolgersi a: guidomaria.lido@gmail.com
Giovedì 02 maggio 2024
Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica, Università Roma Tre
Seminario di Geometria
Francesco Veneziano (Genova)
Rational angles in plane lattices
Generalising classical questions about regular polygons with vertices on a plane lattice, we are interested in pairs of points A,B on a lattice such that the angle \(\hat{AOB}\) is a rational multiple of \(\pi\). This problem leads to diophantine-trigonometric equations that in turn involve the study of rational points on curves of genus 0,1,2,3,5. I will present the full classification of plane lattices according to how many independent rational angles they contain and in which configurations they appear. This is a joint work with R. Dvornicich, D. Lombardo and U. Zannier
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it
Giovedì 02 maggio 2024
Ore 14:30, Aula piano terra e canale youtube dell'IAC https://www.youtube.com/watch?v=vTSwHfdbyt0, Istituto per le Applicazioni del Calcolo, Cnr, via dei Taurini 19, Roma
Premio internazionale "Tullio Levi-Civita"
Benjamin Schlein (University of Zurich)
Bogoliubov theory for dilute quantum systems
In this talk, I am going to present a mathematically rigorous version of Bogoliubov theory that has been developed in the last years and I am going to explain how it can be used to determine with high accuracy the low-energy spectrum of dilute Bose gases.
Per informazioni, rivolgersi a: roberto.natalini@cnr.it
Giovedì 02 maggio 2024
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Shuhei Kitano (Waseda University (Japan))
Calderón–Zygmund estimates for fully nonlinear nonlocal equations
In this talk, we will consider several regularity results for viscosity solutions of fully nonlinear nonlocal equations. For fully nonlinear second order partial differential equations, the regularity theory made a remarkable progress in the late 20th century, and especially the following three important estimates were established: Schauder estimate, Cordes--Nirenberg estimate and Calderón-Zygmund estimate. In the case of nonlocal equations, the analogs of the first two estimates have already been proved: Schauder estimates by Serra (2015) and Jin--Xiong (2015) independently and Cordes-Nirenberg estimate Caffarelli--Silvestre (2011), whereas Calderón-Zygmund estimate for integral equations is our new result. The key is to analyze Riesz potentials of solutions for integral equations, which are also solutions of certain second-order partial differential equations.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it
Giovedì 02 maggio 2024
Ore 16:00, Room 1B (Pal. RM002), Dipartimento di Scienze di Base e Applicate per l'Ingegneria - SBAI
PhD Course
Stefano Buccheri (Università di Napoli - Federico II)
A crash course on regularity theory for elliptic PDEs: when coefficients get rough
Abstract: (i) A powerful tool in regularity theory for PDEs is given by representation formulas: your solution is expressed by the convolution of the data of the equation with a kernel with known (good) properties. Regularity properties are therefore deduced by such an "explicit" formula. However, if the coefficients of the considered differential operator are not smooth enough, such an approach may fail. (ii) An alternative is provided by a test-function based strategy, that provides the decay of some integral quantities related to truncations of the solution, that in turn is connected with its regularity. (iii) Combining this idea with Steiner symmetrization, one can obtain pointwise estimates on the decreasing rearrangement of both the solution and its gradient. As a by-product of these estimates one can obtain sharp results for the summability of the solution in Lorentz spaces and more in general in rearrangement invariant spaces. In the course we will recall some classical results connected to (i) for the Poisson equation, apply the strategies outlined in (ii) and (iii) to a large class of equations in divergence form and merely measurable coefficients, and finally try to generalise the obtained results for nonlocal operators.
Per informazioni, rivolgersi a: stefano.buccheri@unina.it
Venerdì 03 maggio 2024
Ore 11:00, Room 1B (Pal. RM002), Dipartimento di Scienze di Base e Applicate per l'Ingegneria - SBAI
PhD Course
Stefano Buccheri (Università di Napoli Federico II)
A crash course on regularity theory for elliptic PDEs: when coefficients get rough
Abstract: (i) A powerful tool in regularity theory for PDEs is given by representation formulas: your solution is expressed by the convolution of the data of the equation with a kernel with known (good) properties. Regularity properties are therefore deduced by such an "explicit" formula. However, if the coefficients of the considered differential operator are not smooth enough, such an approach may fail. (ii) An alternative is provided by a test-function based strategy, that provides the decay of some integral quantities related to truncations of the solution, that in turn is connected with its regularity. (iii) Combining this idea with Steiner symmetrization, one can obtain pointwise estimates on the decreasing rearrangement of both the solution and its gradient. As a by-product of these estimates one can obtain sharp results for the summability of the solution in Lorentz spaces and more in general in rearrangement invariant spaces. In the course we will recall some classical results connected to (i) for the Poisson equation, apply the strategies outlined in (ii) and (iii) to a large class of equations in divergence form and merely measurable coefficients, and finally try to generalise the obtained results for nonlocal operators.
Per informazioni, rivolgersi a: stefano.buccheri@unina.it
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