• Topology optimization of structures
and metamaterials

(courtesy of Proff. Simona Perotto, Stefano Micheletti, Dr. Nicola Ferro)
Topology optimization of structures
and metamaterials
(courtesy of Proff. Simona Perotto, Stefano Micheletti, Dr. Nicola Ferro)

Additive Manufacturing has changed the way to conceive the production of structures and materials. Topology optimization offers us a rigorous tool for increasingly efficient and new designs

General information

The Department of Mathematics of the Politecnico di Milano is hosting a PhD program in Mathematical Models and Methods in Engineering, aiming at training high level researchers in advanced areas of Pure and Applied Mathematics. The teaching activities include a broad range of possibilities designed for the PhD program. As a successful approach to practical applications depends on a deep understanding of real-world phenomena and knowledge of mathematical tools for their description and design, both modelling, methodological and theoretical aspects are included. Courses are offered in various areas of mathematics, and may vary every year.

Geometrindi e Matematindi (Luigi Serafini 2002)

Numerical simulation of blood flow in the heart
(courtesy of Prof. Alfio Quarteroni)

Objectives

The development of advanced technologies in Science and Engineering brings an increasing demand of advanced mathematical theories and methods, which in turn fosters the demand for education and training of skilled mathematicians in pure and applied research. The main scope of the Ph.D. course “Mathematical Models and Methods in Engineering” is to train high-level researchers in various fields of pure and applied mathematics.

Professional Opportunities

Expected professional placements for Ph.D. doctorates are: academic research in Italian or International Universities and Research Institutions, R & D divisions of private companies, study and research centers of public Agencies, financial and insurance Institutions.

Upcoming events
•  mar 28 tue 2023 Seminar Patrick Dondl, Relaxation, Existence, and Numerical Implementation for Models of Single-Crystal Elastoplasticity,  03-28-2023, 17:00

• Seminar
• Patrick Dondl
• Albert Ludwigs University of Freiburg
• Relaxation, Existence, and Numerical Implementation for Models of Single-Crystal Elastoplasticity
• Tuesday, 28 March 2023 at 17:00
• Sala Consiglio 7 piano, Edificio La Nave
• Abstract
We will consider non-convex models for single-crystal elasto-plasticity, where the non-convexity arises through so-called cross-hardening, i.e., deformation in one slip system is prohibiting deformation in a different slip system at the same position. Such models are typically ill-posed and require relaxation. A different avenue to pursue is the addition of a strain-gradient term which regularizes the model by introducing a length scale on which the plastic strain may oscillate. We will see that physically reasonable strain gradient penalizations, which only assign an energy to geometrically necessary dislocations - penalizing only the curl of the plastic strain - still require some relaxation. Further issues arise regarding numerical implementation of the strict conditions of infinite cross-hardening. We will thus regularize the side-condition by introducing a large cross-hardening penalty into the plastic energy. The regularized model is then amenable to implementation with finite-element methods, and, with the aid of div-curl arguments, one can show that it Gamma-converges to the single-plane model for large penalization. Finally we show some microstructures arising in the numerical implementation of this model.
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  mar 29 wed 2023 Seminar Foresta Martin, Giacomelli, Scandone, La Penisola dei Vulcani,  03-29-2023, 12:15

• Seminar
• Foresta Martin, Giacomelli, Scandone
• Giornalista Scientifico, Vulcanologa, Università Roma Tre
• La Penisola dei Vulcani
• Wednesday, 29 March 2023 at 12:15
• Sala Consiglio VII piano - tiny.cc/CM2023
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  mar 30 thu 2023 Seminar Gianmarco Sperone, Steady-state Navier-Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate,  03-30-2023, 15:15

• Seminar
• Gianmarco Sperone
• Politecnico di Milano
• Steady-state Navier-Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate
• Thursday, 30 March 2023 at 15:15
• Aula seminari III piano
• Abstract
The steady motion of a viscous incompressible fluid in an obstructed finite pipe is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while a transversal flux rate F is prescribed along the pipe. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument based on Bernoulli’s law. Through variational techniques and with the use of an exact flux carrier, an explicit upper bound on F (in terms of the viscosity, diameter and length of the tube) ensuring the uniqueness of such weak solution is given. This upper bound is shown to converge to zero at a given rate as the length of the pipe goes to infinity. In an axially symmetric framework, we also prove the existence of a weak solution displaying rotational symmetry.
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  apr 05 wed 2023 Seminar Roberta Fulci, Video killed the radio math?,  04-05-2023, 12:15

• Seminar
• Roberta Fulci
• Giornalista Scientifico Rai3
• Video killed the radio math?
• Wednesday, 5 April 2023 at 12:15
• Sala Consiglio VII piano - tiny.cc/CM2023
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  apr 13 thu 2023 Seminar Giovanni Siclari, Unique continuation for the fractional heat operator,  04-13-2023, 15:15

• Seminar
• Giovanni Siclari
• Università degli Studi di Milano-Bicocca
• Unique continuation for the fractional heat operator
• Thursday, 13 April 2023 at 15:15
• Aula Seminari III piano
• Abstract
We study unique continuation properties and the asymptotic behaviour for a class of equations
involving the fractional heat operator with an Hardy-type potential. Our methods are based on
a Almgren-Poon monotonicity formula combined with a blow-up argument. Since the operator
has a global nature we will also need suitable extension results in the spirit of Caffarelli-Silvestre extension.
Key words: Parabolic partial differential equations, unique continuation, blow-up, asymptotics,
monotonicity formula, Hardy potential.
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  apr 13 thu 2023 Seminar Francesco Esposito, A classification result for a Gross-Pitaevskii type system,  04-13-2023, 16:15

• Seminar
• Francesco Esposito
• Università della Calabria
• A classification result for a Gross-Pitaevskii type system
• Thursday, 13 April 2023 at 16:15
• Aula seminari III piano
• Abstract
This talk will be focused on the study of a family of semilinear elliptic systems defined in $R^n$, which is doubly critical since it involves Sobolev critical exponents and Hardy-type potentials. We aim to provide qualitative properties of positive solutions for these Gross-Pitaevskii type systems. In particular, we shall deduce that solutions are symmetric about the origin. In order to do it, we apply a suitable version of the moving planes technique for cooperative singular systems. Finally, we are able to provide a classification result for these kind of problems.
This is based on a joint work with Rafael López-Soriano (University of Granada, Spain) and Berardino Sciunzi (University of Calabria, Italy).
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  apr 14 fri 2023 Seminar Chiara Andrà, Aspetti affettivi nei processi di modellizzazione matematica - evitare o sfruttare?,  04-14-2023, 14:30

• Seminar
• Chiara Andrà
• Università del Piemonte Orientale
• Aspetti affettivi nei processi di modellizzazione matematica - evitare o sfruttare?
• Friday, 14 April 2023 at 14:30
• Aula didattica FDS - IV piano
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  apr 18 tue 2023 MOX Seminar Sangeeta Yadav, Data-Driven Stabilization Schemes for Singularly Perturbed Partial Differential Equations,  04-18-2023, 14:00

• MOX Seminar
• Indian Institute of Science
• Data-Driven Stabilization Schemes for Singularly Perturbed Partial Differential Equations
• Tuesday, 18 April 2023 at 14:00
• Online
• Abstract
In this talk, the speaker will introduce innovative ways to use Artificial Neural Networks (ANN) in conjunction with conventional numerical techniques to solve challenging Singularly Perturbed Differential Equations (SPDEs). SPDEs often exhibit spurious oscillations in the standard numerical solution due to the presence of boundary and interior layers. Stabilization techniques are employed to eliminate these oscillations, but the accuracy of these techniques depends on a user-selected stabilization parameter that is difficult to determine optimally. To address this challenge, the authors have developed several ANN-based techniques to predict the optimal stabilization parameter. The first method, called SPDE-Net, uses ANN to predict the stabilization parameter for the streamline upwind/Petrov-Galerkin (SUPG) stabilization technique for solving one-dimensional SPDEs. The second method, AI-stab FEM, is an optimization scheme that employs an unsupervised ANN to minimize the residual along with the crosswind derivative term to solve two-dimensional SPDEs. The third method is called the SPDE-ConvNet, a convolutional neural network that predicts the local (cell-wise) stabilization parameter. The proposed techniques have outperformed other state-of-the-art ANNbased partial differential equation (PDE) solvers such as Physics Informed Neural Network (PINN) and VarNet for several benchmark problems. In conclusion, the speaker will compare the performance of these data-driven stabilization schemes and discuss potential future extensions.

References:
[1] S. Yadav, S. Ganesan, SPDE-Net: Neural Network based prediction of stabilization parameter for SUPG technique Proceedings of the 13th Asian Conference on Machine Learning, PMLR 157:268-283, (2021)
[2] S. Yadav, S. Ganesan, AI-augmented stabilized finite element method. arXiv:2211.13418 (2022).
[3] S. Yadav, S. Ganesan, SPDE-ConvNet: Solve Singularly Perturbed Partial Differential Equation with deep learning. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS CONGRESS (2022).
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  may 11 thu 2023 Seminar Paolo Luzzini, Shape sensitivity and optimization of Grushin eigenvalues,  05-11-2023, 15:15

• Seminar
• Paolo Luzzini
• Università degli Studi di Padova
• Shape sensitivity and optimization of Grushin eigenvalues
• Thursday, 11 May 2023 at 15:15
• Aula Seminari III piano
• Abstract
It is well known that among all domains of a fixed volume, the ball minimizes the first eigenvalue of the Dirichlet Laplacian. The counterpart of this result for the degenerate operator known as the Grushin Laplacian is instead an open problem.
In this talk I will present some results in the direction of understanding such a problem. I will first consider the shape sensitivity of Grushin eigenvalues on general domains with the aim of characterizing critical domains under isovolumetric perturbations.
Next I will pass to the simplified case of cartesian product domains, showing that in this class the first eigenvalue admits a unique minimizer and providing some estimates on the minimum. Finally I will discuss some numerical experiments and some open problems.

The talk is based on joint works with Pier Domenico Lamberti (Università degli Studi di Padova), Paolo Musolino (Università Ca' Foscari Venezia) Luigi Provenzano (Sapienza Università di Roma), and Joachim Stubbe (EPFL).
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  may 18 thu 2023 Seminar Claudio Barbieri, Synchronization and divergence: implications for central bank policies,  05-18-2023, 11:30 precise

• Seminar
• Claudio Barbieri
• European Central Bank
• Synchronization and divergence: implications for central bank policies
• Thursday, 18 May 2023 at 11:30 right
• Aula IV - Edificio 11
• Abstract
The talk discusses the implications of the synchronization of interest rates and credit growth for central bank policy in the Euro Area. The bedrock that sustains the importance of such an issue is found in the growing evidence that monetary policy in the Euro Area has heterogeneous impacts across several dimensions, e.g. across countries, within countries, and across different bank market segments. Given policy asymmetries, it becomes essential for policy makers to quantify the extent of synchronization and divergence across interest rates and credit growth and, accordingly, tailor central bank policy to ensure its effectiveness. We investigate the synchronization of the Eurozone’s government bond yields at different maturities. For this purpose, we combine principal component analysis with random matrix theory. We describe the dynamic of credit granted across Italian provinces over the last two decades. We focus on the synchronization and divergence of the credit growth to identify for which provinces credit grows or decreases similarly and which provinces show a diverging behavior. Moreover, by decomposing credit growth into a common component and components capturing heterogeneity in supply and demand we introduce a framework to support policymaker’s choice between different types of macroprudential instruments.
• Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568

•  aug 13 sun 2023 aug 18 fri 2023 WorkShop Summer school on higher structures in algebra and geometry 08/13/2023 - 08/18/2023

• WORKSHOP
• organizers
Andreas Hochenegger, Karin M. Jacobsen, David Ploog
• This school links three different areas, namely representation theory, symplectic geometry and algebraic geometry, using categorical approaches and higher structures in algebra. The target audience are categorically inclined PhD students and post-docs familiar with at least one of these fields, who want to broaden their vision of neighbouring areas.
• Sunday, 13 August 2023 - Friday, 18 August 2023
Sophus Lie Conference Center, Nordfjordeid, Norway
Politecnico di Milano, Dipartimento di Matematica ed. 14 "Nave", Piazza Leonardo da Vinci, 32, 20133 Milano, Telefono: +39 0223994505 - Fax: +39 0223994568