The Haldane model is a standard tight binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar, magnetic field. We consider its interacting version and study the critical case at the transition between the trivial and the "topological" insulating phases. In previous works, we proved the quantization of the critical longitudinal conductivity for weak enough interaction strength. We now report a recent extension of the result to the critical transverse conductivity, which turns out to be quantized at half-integer values, irrespective of the interaction strength. Proofs are based on a combination of constructive Renormalization Group methods and exact lattice Ward Identities. Joint works with S. Fabbri, V. Mastropietro, M. Porta, R. Reuvers.
The talk will be divided in two parts: in part 1, motivations, main results and main ideas of the proof will be stated and explained. Part 2 will be more technical and will discuss in more detail some selected aspects of the proof.
This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.
The stunning success of deep learning (DL) over the past decade is a puzzling natural phenomenon of great scientific interest. This success has been demonstrated through a plethora of convincing experiments, and yet it is crying out for a thorough scientific understanding through foundational research. I believe that there must exist theoretical principles of learning and intelligence underlying those network architectures, that perform so well in learning tasks
I will review some of the main recent results in understanding the three parts of modern deep learning: approximation, generalization and optimization. In particular, I will focus on the new principles of compositional sparsity that explains much of approximation and generalization.
This initiative is part of the “Ph.D. Lectures” activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.
Contatto: paolo.zunino@polimi.it
Tomaso A. Poggio, is the Eugene McDermott Professor at the Department of Brain and Cognitive Sciences; Co-Director, Center for Biological and Computational Learning; Member of the Computer Science and Artificial Intelligence Laboratory at MIT; since 2000, member of the faculty of the McGovern Institute for Brain Research. Born in Genoa, Italy, he received his Doctor in Theoretical Physics from the University of Genoa in 1971 and was a Wissenschaftlicher Assistant, Max Planck Institut fuer Biologische Kybernetik, Tuebingen, Germany from 1972 until 1981 when he became Associate Professor at MIT. He is an honorary member of the Neuroscience Research Program, a member of the American Academy of Arts and Sciences and a Founding Fellow of AAAI. He received several awards such as the Otto-Hahn-Medaille Award of the Max-Planck-Society, the Max Planck Research Award (with M. Fahle), from the Alexander von Humboldt Foundation, the MIT 50K Entrepreneurship Competition Award, the Laurea Honoris Causa from the University of Pavia in 2000 (Volta Bicentennial), the 2003 Gabor Award, the 2009 Okawa prize and 2009 Okawa prize and the American Association for the Advancement of Science (AAAS) Fellowship (2009). He is one of the most cited computational neuroscientists (with a h-index greater than 149 – based on GoogleScholar).
The Haldane model is a standard tight binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar, magnetic field. We consider its interacting version and study the critical case at the transition between the trivial and the "topological" insulating phases. In previous works, we proved the quantization of the critical longitudinal conductivity for weak enough interaction strength. We now report a recent extension of the result to the critical transverse conductivity, which turns out to be quantized at half-integer values, irrespective of the interaction strength. Proofs are based on a combination of constructive Renormalization Group methods and exact lattice Ward Identities. Joint works with S. Fabbri, V. Mastropietro, M. Porta, R. Reuvers.
The talk will be divided in two parts: in part 1, motivations, main results and main ideas of the proof will be stated and explained. Part 2 will be more technical and will discuss in more detail some selected aspects of the proof
This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.
Tomaso A. Poggio, is the Eugene McDermott Professor at the Department of Brain and Cognitive Sciences; Co-Director, Center for Biological and Computational Learning; Member of the Computer Science and Artificial Intelligence Laboratory at MIT; since 2000, member of the faculty of the McGovern Institute for Brain Research. Born in Genoa, Italy, he received his Doctor in Theoretical Physics from the University of Genoa in 1971 and was a Wissenschaftlicher Assistant, Max Planck Institut fuer Biologische Kybernetik, Tuebingen, Germany from 1972 until 1981 when he became Associate Professor at MIT. He is an honorary member of the Neuroscience Research Program, a member of the American Academy of Arts and Sciences and a Founding Fellow of AAAI. He received several awards such as the Otto-Hahn-Medaille Award of the Max-Planck-Society, the Max Planck Research Award (with M. Fahle), from the Alexander von Humboldt Foundation, the MIT 50K Entrepreneurship Competition Award, the Laurea Honoris Causa from the University of Pavia in 2000 (Volta Bicentennial), the 2003 Gabor Award, the 2009 Okawa prize and 2009 Okawa prize and the American Association for the Advancement of Science (AAAS) Fellowship (2009). He is one of the most cited computational neuroscientists (with a h-index greater than 149 – based on GoogleScholar).
In this talk we consider the finite element approximation of some mixed formulations of linear elasticity, namely, the displacement-pressure, the displacement-stress and the displacement-pressure-stress approaches. As for any mixed formulation, involving unknowns belonging to different functional spaces, the global inf-sup stability that ensures well posedness of the problem is a consequence of “little” inf-sup conditions that need to be satisfied between the interpolating spaces of the different unknowns. Approximations satisfying these conditions are sometimes difficult to implement, and even very rare, as in the case of the displacement-pressure-stress approach. The alternative is to modify the discrete problem by adding stabilisation terms to the Galerkin equations that yield stable approximations for any choice of the interpolating spaces. We discuss this possibility and show that one may prove stability and convergence in the same norms as for the continuous problem.
Contatto: simona.perotto@polimi.it
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
The talk is based on the recent accepted paper: Fractional slice regular functions of a quaternionic variable. José Oscar González-Cervantes, Juan Bory-Reyes, Irene Sabadini. Results in Mathematics (2023), in which the theory of quaternionic fractional slice regular functions, in Riemann-Liouville and Caputo senses, as null-solutions of a fractional Cauchy-Riemann operator is presented.
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
The seminar will discuss the correct modeling of the fully nonlinear free surface boundary conditions to be prescribed in ship hydrodynamics simulations based on potential flow theory. The main goal of such a discussion is that of identifying a mathematical formulation and a numerical treatment that can be used both to carry out transient simulations, and to compute steady solutions - for any flow admitting them. In the literature on numerical towing tank in fact, steady and unsteady fully nonlinear potential flow solvers are historically characterized by radically different mathematical models. We will analyze the reasons why this happened, and propose a formulation that can be successfully used both to carry out transient simulations and - upon elimination of time derivatives - to compute steady solutions.
Contatti:
edie.miglio@polimi.it
gabriele.ciaramella@polimi.it
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
The seminar will discuss the correct modeling of the fully nonlinear free surface boundary conditions to be prescribed in ship hydrodynamics simulations based on potential flow theory. The main goal of such a discussion is that of identifying a mathematical formulation and a numerical treatment that can be used both to carry out transient simulations, and to compute steady solutions - for any flow admitting them. In the literature on numerical towing tank in fact, steady and unsteady fully nonlinear potential flow solvers are historically characterized by radically different mathematical models. We will analyze the reasons why this happened, and propose a formulation that can be successfully used both to carry out transient simulations and - upon elimination of time derivatives - to compute steady solutions.
Contatti:
edie.miglio@polimi.it
gabriele.ciaramella@polimi.it
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
The classification of minimal surfaces of general type is a classical and long-standing research topic. In this context fixing invariants turns out to be fundamental.
In this talk we will review some recent results on the case pg=q=2, which is still widely open in spite of several contributions by many authors over the last two decades. More specifically, given a minimal surface S of general type with pg=q=2, it turns out that the self-intersection K^2 of the canonical divisor K is between 4 and 9. We will focus on the cases K^2=5,6, describing some constructions (endowed with explicit and global equations) developed in a joint work with Fabrizio Catanese.
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
The classification of minimal surfaces of general type is a classical and long-standing research topic. In this context fixing invariants turns out to be fundamental.
In this talk we will review some recent results on the case pg=q=2, which is still widely open in spite of several contributions by many authors over the last two decades. More specifically, given a minimal surface S of general type with pg=q=2, it turns out that the self-intersection K^2 of the canonical divisor K is between 4 and 9. We will focus on the cases K^2=5,6, describing some constructions (endowed with explicit and global equations) developed in a joint work with Fabrizio Catanese.
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
La ricerca operativa (RO) è una branca della matematica applicata solitamente insegnata a livello universitario. Tuttavia, negli ultimi anni sono state sviluppate diverse iniziative per introdurre la RO agli studenti delle scuole secondarie. A queste iniziative si aggiunge ROAR (Ricerca Operativa Applicazioni Reali), un corso triennale composto da tre unità didattiche, pensato per gli studenti del triennio delle scuole secondarie. In questo seminario ci si concentrerà sulla prima unità didattica, mettendo in luce sia gli aspetti contenutistici sia le metodologie utilizzate per la sua implementazione con gli studenti. In particolare, si delineeranno gli aspetti teorici che hanno guidato la progettazione e poi si mostreranno esempi di problemi di ottimizzazione proposti agli studenti, che mirano a sviluppare le loro capacità di modellizzazione e problem-solving.
Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.
His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).
In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).
Seminari Matematici al
Politecnico di Milano
- Analisi
- Cultura Matematica
- Seminari FDS
- Geometria e Algebra
- Probabilità e Statistica Matematica
- Probabilità Quantistica