[SAn] Nyström methods for Fredholm integral equations defined on planar domains | Maria Grazia Russo (Department of Engineering, University of Basilicata, Italy)
25 September 2024 2:15 pm - 3:15 pm
Room 1.6, building VII.
Title: Nyström methods for Fredholm integral equations defined on planar domains.
Speaker: Maria Grazia Russo (Department of Engineering, University of Basilicata, Italy).
Time: Wednesday, 25 September 2024, from 14:15 to 15:15.
Place: Room1.6, building VII.
Abstract: The talk is devoted to a general Nyström scheme for approximating bivariate linear integral equations of the second kind defined on suitable domains of the plane.
The basic idea of the numerical strategy is to use a global approach, that is to use cubature rules built starting from a polynomial approximation scheme.
In fact, the global approximation allows to obtain excellent convergence results, since the order of convergence of the corresponding Nyström methods, is the same as the best polynomial approximation of the solution in suitable selected spaces of functions. Furthermore, in this context it is quite easy to prove the stability of the methods and the good conditioning of the involved linear systems.
We will give an overview of several results obtained in the last few years, starting from rectangular bounded or unbounded domains, also in the case where the known functions of the equation may have singularities on the boundaries, and arriving at the case of equations defined on curvilinear domains.
[Mini-courseAn] Numerical methods for delay and fractional differential equations | Neville J. Ford (University of Chester, UK)
16 October 2024 2:15 pm - 4:15 pm
Room 1.4, building VII.
Title: Numerical methods for delay and fractional differential equations.
Speaker: Neville J. Ford* (University of Chester, UK).
Date| Time: Wednesday, 16 October 2024, from 14:15 to 16:15 - Room 1.4, building VII.
Abstract: We will consider how to solve delay differential equations and fractional differential equations using numerical schemes. To understand how to do this, we will begin by considering the fundamental theory associated with these equations. We look at the dimension of the underlying dynamical systems to explain how approximation schemes should be set up, and how their performance should be judged. In particular, we shall look at initial and boundary conditions needed to ensure a unique solution and we will see how these must be related to the numerical schemes. If time permits, we shall consider some fractional problems with delays and see how the insights from both types of problems combine in this case.
[SOR] Enhancing the Efficiency and Stability of Deep Neural Network Training through Controlled Mini-batch Algorithms | Corrado Coppola (Sapienza University of Rome, Italy)
16 October 2024 4:15 pm - 5:15 pm
Room 1.11 - VII
Enhancing the Efficiency and Stability of Deep Neural Network Training through
Controlled Mini-batch Algorithms
The exponential growth of trainable parameters in state-of-the-art deep neural networks
(DNNs), driven by innovations such as self-attention layers and over-parameterization, has led
to the development of models containing billions or even trillions of parameters. As training
datasets grow larger and tasks become more complex, the current challenge lies in balancing
convergence guarantees with the increasing need for efficient training. In this work, we focus on
supervised deep learning, where the training problem is formulated as the unconstrained
minimization of a smooth, potentially non-convex objective function with respect to network
weights.
We propose an approach based on Incremental Gradient (IG) and Random Reshuffling (RR)
algorithms, enhanced with derivative-free extrapolation line-search procedures. Specifically, we
present the Controlled Mini-batch Algorithm (CMA), proposed in [1], which incorporates
sufficient decrease conditions for the objective function and allows for line-search procedures to
ensure convergence, without assuming any further hypotheses on the search direction. We also
present computational results on large-scale regression problems.
We further introduce CMA Light, proposed in [2], an enhanced variant of CMA with
convergence guarantees within the IG framework. Using an approximation of the real objective
function to verify sufficient decrease, CMA Light drastically reduces the number of function
evaluations needed and achieves notable performance gains. We discuss computational results
both against CMA and against state-of-the-art optimizers for neural networks, showing a
significant advantage of CMA Light in large-scale classification tasks using residual
convolutional networks.
Finally, we present the Fast-Controlled Mini-batch Algorithm (F-CMA), extending the
convergence theory of CMA Light to the case where samples are reshuffled at each epoch. We
develop a new line-search procedure, and demonstrate F-CMA's superior performance when
training ultra-deep architectures, such as transformers SwinB and SwinT with up to 130 millions
of trainable parameters. Our results show significant advantages in both stability and
generalization compared to state-of-the-art deep learning optimizers.
[Mini-courseAn] Numerical methods for delay and fractional differential equations | Neville J. Ford (University of Chester, UK)
18 October 2024 2:15 pm - 4:15 pm
Room 3.2, building VII.
Title: Numerical methods for delay and fractional differential equations.
Speaker: Neville J. Ford* (University of Chester, UK).
Date| Time: Friday, 18 October 2024, from 14:15 to 16:15 - Room 3.2, building VII.
Abstract: We will consider how to solve delay differential equations and fractional differential equations using numerical schemes. To understand how to do this, we will begin by considering the fundamental theory associated with these equations. We look at the dimension of the underlying dynamical systems to explain how approximation schemes should be set up, and how their performance should be judged. In particular, we shall look at initial and boundary conditions needed to ensure a unique solution and we will see how these must be related to the numerical schemes. If time permits, we shall consider some fractional problems with delays and see how the insights from both types of problems combine in this case.
[SAn] A mathematical framework for dynamical social interactions with dissimulation | Max O. Souza (Center for Mathematics and Applications (NOVA Math), NOVA FCT, Universidade NOVA de Lisboa, Portugal)
23 October 2024 2:15 pm - 3:15 pm
Room 1.6, building VII.
Title: A mathematical framework for dynamical social interactions with dissimulation.
Speaker: Max O. Souza (Center for Mathematics and Applications (NOVA Math), NOVA FCT, Universidade NOVA de Lisboa, Portugal).
Time: Wednesday, 23 October 2024, from 14:15 to 15:15.
Place: Room 1.6, building VII.
Abstract: Modeling social interactions is a challenging task that requires flexible frameworks. For instance, dissimulation and externalities are relevant features influencing such systems --- elements that are often neglected in popular models. This paper is devoted to investigating general mathematical frameworks for understanding social situations where agents dissimulate, and may be sensitive to exogenous objective information. Our model comprises a population where the participants can be honest, persuasive, or conforming. Firstly, we consider a non-cooperative setting, where we establish existence, uniqueness and some properties of the Nash equilibria of the game. Secondly, we analyze a cooperative setting, identifying optimal strategies within the Pareto front. In both cases, we develop numerical algorithms allowing us to computationally assess the behavior of our models under various settings. Joint work with Y Saporito and Y Thamsten.
[SAL] When a Ring meets a Lattice | João Dias (CIMA, Universidade de Évora)
28 October 2024 2:00 pm - 3:00 pm
Seminar room, building VII
Abstract:
The rational numbers have been used to measure quantities since ancient times; however, their implementation in computer languages raises a significant problem: zero has no inverse. To address this issue, J. Bergstra and J. Tucker introduced an algebraic structure called a meadow, which allows for the inversion of zero.
In this talk, I will introduce meadows and their various classes, demonstrating that they correspond to labelled lattices, where the rings label the vertices. We will explore how concepts from ring theory, such as Artinian rings and decomposition theorems, can be adapted to this new context. Finally, I will present a connection between meadows and sheaves over a topological space, highlighting the implications of this relationship.
- J. Dias and B. Dinis. "Strolling through common meadows." Communications in Algebra, 2024, 1–28.
- J. Dias and B. Dinis. "Towards an enumeration of finite common meadows." International Journal of Algebra and Computation, 2024, 1-19.
- J. Dias, B. Dinis and P. Marques. Bridging Meadows and Sheaves. arXiv:2410.05921
[SAL] Square root crystals and Grothendieck positivity | Eric Marberg (Hong Kong University of Science and Technology)
4 November 2024 2:00 pm - 3:00 pm
Zoom link: https://videoconf-colibri.zoom.us/j/91864016501?pwd=J0qrDNWOva0DeM1XTOuUiDN3ToRd3b.1
The classical theory of type A crystals provides a graphical framework for proving Schur positivity results. In this talk we will discuss a new category of "square root crystals" (introduced implicitly in work of Yu) that can be used to establish instances of Grothendieck positivity. For example, Buch's combinatorial interpretation of the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the tensor product for this category. We will also discuss some shifted analogues and applications to a conjectural formula of Cho-Ikeda for K-theoretic Schur P-functions.
[SAL] Zeros of homogeneous polynomials, linear sections of Veroneseans, and projective Reed-Muller codes | Sudhir R. Ghorpade (Indian Institute of Technology Bombay)
11 November 2024 2:00 pm - 3:00 pm
Let F be a finite field and let m, d and r be positive integers. Consider the following
question: What is the maximum number of common zeros over F that a system of r linearly
independent homogeneous polynomials of degree d in m + 1 variables can have?
Because of homogeneity, we will disregard the trivial zero (viz., the origin) and regard two
zeros as equivalent if they are proportional to each other, i.e., if one is obtained from another
upon multiplying all coordinates with a nonzero scalar. In other words, we look for zeros in
the m -dimensional projective space over the field F.
This question was first raised by M. Tsfasman in the case of a single homogeneous
polynomial, i.e., when r = 1. It was then settled by J.-P. Serre (1991). Later Tsfasman
together with M. Boguslavsky formulated a remarkable conjecture in the general case, and
this was shown to hold in the affirmative in the next case of r = 2 by Boguslavsky (1997).
Then about two decades later, it was shown that the conjecture is valid conjecture is valid if
the number of polynomials is at most the number of variables, i.e., r ≤ m + 1, but the
conjecture can be false in general. Newer conjectures were then formulated and although
there has been considerable progress concerning them, the general case is still open.
These questions are intimately related to the study of maximal sections of Veronese varieties
by linear subvarieties of the ambient projective case, and also to the study of the an important
class of linear error correcting codes, called projective Reed-Muller codes.
In this talk, we will outline these developments and explain the above connections.
An attempt will be made to keep the prerequisites at a minimum.
[SAL] Algebraic Machine Learning | Fernando Martin-Maroto (Champalimaud Foundation, Algebraic AI)
18 November 2024 2:00 pm - 3:00 pm
I will give a brief introduction to Algebraic Machine Learning (AML), a purely algebraic method that does not use statistics, search or optimization. Instead, AML relies on semantic embeddings in semilattices and subdirect decompositions. AML is capable of learning from data and also from just a problem statement in the form of a set of formulas; for example, AML can learn to find Hamiltonian Cycles from the problem statement. With the same algorithm, AML can learn from data (e.g. classifying medical images) with a test accuracy that rivals that of deep multilayer perceptrons. I will also give an introduction to Atomized Semilatices, a mathematical method developed to operate and compute AML models.
[Bio Short-course] Modelling and application of bifurcation theory for epidemiological models , Bob W. Kooi (VU Amsterdam & BCAM)
19 November 2024 10:00 am - 12:00 pm
Lab 2.2
[SOR] Integrating public transport in sustainable last-mile delivery | Claudia Archetti (University of Brescia, Italy)
20 November 2024 2:00 pm - 3:00 pm
Sala 1.4 - VII
Abstract: Integrating public transport in sustainable last-mile delivery
We consider a delivery system for last-mile deliveries in urban areas based on the use of Public Transport Service. The idea is to exploit the spare capacity of public transport means to transport parcels within urban areas, thus reducing externalities caused by commercial delivery vans. Specifically, the system is such that parcels are first transported from origins to drop-in stations on public vehicles itineraries. Then, they are transported through public vehicles to drop-out stations, from where they are delivered to destination by freighters using green vehicles. The system is known as Freight-On-Transit (FOT). We present the optimization problem related with and operational decisions, as well as ad-hoc solution methodologies and simulations on synthetic data.
SHORT BIO
ARCHETTI CLAUDIA
Claudia Archetti is Associate Professor of Operations Research at University of Brescia. From September 2021 to September 2024, she was Full Professor in Operations Research at ESSEC Business School in Paris. The main areas of the scientific activity are: models and algorithms for vehicle routing problems; mixed integer mathematical programming models for the minimization of the sum of inventory and transportation costs in logistic networks; exact and heuristic algorithms for supply-chain management; reoptimization of combinatorial optimization problems.
She is author of more than 100 papers in international journals. She is co-Editor in Chief of Networks. She was VIP3 of EURO, the Association of European Operational Research Societies, in charge of publications and communication.
[SBio] Bifurcation analysis of epidemiological models for Dengue fever, Bob W. Kooi (VU Amsterdam & BCAM)
20 November 2024 3:15 pm - 4:15 pm
Lab 2.2
Joint with Analysis Seminar
[Mini-courseBio] Modelling and application of bifurcation theory for epidemiological models , Bob W. Kooi (VU Amsterdam & BCAM)
21 November 2024 2:00 pm - 4:00 pm
Lab 2.2
[SAL] Transformation Semigroups and their Groups of Permutations: Transversal Properties | Wolfram Bentz (Universidade Aberta and NOVA Math)
25 November 2024 2:00 pm - 3:00 pm
Laboratory 2.2, Building VII
Abstract:
In recent years, the interplay between permutations groups and transformations semigroups has lead to many interesting new results. In particular, the application of the classification of finite simple groups to problems in transformation semigroups paved the way for the exploration of new and interesting ideas, and it led to the production of impressive mathematics involving combinatorics, automata theory, and algebra.
Underlying this connection is the realization that unit groups profoundly shape the structure of any containing monoids. For example, the endomorphism monoid of a mathematical object is restricted by its isomorphism group. Conversely, important properties of monoids of transformations can often be translated into properties of permutation groups and lead to natural questions about group actions on partitions and sections.
In this seminar, we will look at several cases of transversal properties.
This is joint work with João Araújo (Universidade Nova de Lisboa), João Pedro Araújo (Stanford University), Peter J. Cameron (University of St Andrews), and Pablo Spiga (University of Milano-Bicocca).
[SDataScience] Optimizing the Present and Future of Smart Electric Power Grids | Miguel F. Anjos (University of Edinburgh)
27 November 2024 2:00 pm - 3:00 pm
Room 1.5, Building VII, NOVA FCT
[CourseDataScience] Optimization Models for Unit Commitment in Electric Energy Systems | Miguel F. Anjos (University of Edinburgh)
27 November 2024 3:00 pm - 6:00 pm
NOVA FCT, Lab 2.2, Building VII
Title: Optimization Models for Unit Commitment in Electric EnergySystems
Speaker: Miguel F. Anjos (University of Edinburgh)
Description: The unitcommitment (UC) problem addresses a fundamental decision that is taken whenoperating a power system, namely to set the schedule of power production foreach generating unit in the system so that the demand for electricity is met atminimum cost. The schedule must also ensure that each unit operates within itstechnical limits; these typically include ramping constraints and minimumuptime/downtime constraints. Units that are scheduled to produce electricityduring a given time period are said to be committed for that period. Variousjurisdictions solve UC on a daily basis. In particular, it is the standard toolto clear spot markets, and particularly the day-ahead markets in the USA. InNorth American jurisdictions without markets, the system operators use UC todetermine the day-ahead commitments and dispatches. This mini-course will coversome of the most relevant mathematical optimization models for UC and lead upto open research problems.
Part I, 27thNovember, 15:00 – 18:00 - Basics of Unit Commitment and Modern Electric Energy Systems
Part II, 28th November, 14:00-17:00 - Unit Commitment Under Uncertainty & Additional Topics
Registrationcan be completed here.
Other info:
The lectures will be based on the tutorial:
M.F. Anjos and A.J. Conejo. Unit Commitmentin Electric Energy Systems, Now Foundations and Trends, 2017 (ISBN978-1-68083-370-6). http://dx.doi.org/10.1561/3100000014
Participants should have a laptop computer with access to theinternet. No specific software is required. Knowledge of AMPL or a similar optimizationmodelling language will help but is not essential as the mini-course will beself-contained in this regard.
[CourseDataScience] Optimization Models for Unit Commitment in Electric Energy Systems | Miguel F. Anjos (University of Edinburgh)
28 November 2024 2:00 pm - 5:00 pm
NOVA FCT, Lab 2.2, Building VII
Title: Optimization Models for Unit Commitment in Electric EnergySystems
Speaker: Miguel F. Anjos (University of Edinburgh)
Description: The unitcommitment (UC) problem addresses a fundamental decision that is taken whenoperating a power system, namely to set the schedule of power production foreach generating unit in the system so that the demand for electricity is met atminimum cost. The schedule must also ensure that each unit operates within itstechnical limits; these typically include ramping constraints and minimumuptime/downtime constraints. Units that are scheduled to produce electricityduring a given time period are said to be committed for that period. Variousjurisdictions solve UC on a daily basis. In particular, it is the standard toolto clear spot markets, and particularly the day-ahead markets in the USA. InNorth American jurisdictions without markets, the system operators use UC todetermine the day-ahead commitments and dispatches. This mini-course will coversome of the most relevant mathematical optimization models for UC and lead upto open research problems.
Part I, 27thNovember, 15:00 – 18:00 - Basics of Unit Commitment and Modern Electric Energy Systems
Part II,28th November, 14:00-17:00 - Unit Commitment Under Uncertainty & Additional Topics
Registrationcan be completed here.
Other info:
The lectures will be based on the tutorial:
M.F. Anjos and A.J. Conejo. Unit Commitmentin Electric Energy Systems, Now Foundations and Trends, 2017 (ISBN978-1-68083-370-6). http://dx.doi.org/10.1561/3100000014
Participants should have a laptop computer with access to theinternet. No specific software is required. Knowledge of AMPL or a similar optimizationmodelling language will help but is not essential as the mini-course will beself-contained in this regard.
[SAL] Keys and evacuation via virtualization | Olga Azenhas (CMUC, Universidade de Coimbra)
2 December 2024 2:00 pm - 3:00 pm
Room 2.3, building VII
Abstract:
We show that the key map on crystals of any classical type can be reduced to a key map on simply-laced types by using virtualization of crystals. Our proofs are type-free and crystal model independent. The virtualization map also induces an embedding on the Weyl groups by the diagram folding. Thus our results also apply to uniform models like the alcove and the Lakshmibai–Seshadri paths.
As a direct application we obtain new algorithms to compute evacuation, keys, and Demazure atoms in type B_n in terms of Kashiwara–Nakashima tableaux. In particular, we are able to use type A_n and C_n methods. For type C_n, we apply results obtained by Azenhas–Tarighat Feller–Torres, Azenhas–Santos and Santos. This is a joint work with González, Huang and Torres.
[SAn] Pulse vaccination in a SIR model: Global dynamics, bifurcations and seasonality | Alexandre Rodrigues (ISEG - Universidade de Lisboa, Portugal)
4 December 2024 2:15 pm - 3:15 pm
Room 1.6, building VII.
Title: Pulse vaccination in a SIR model: Global dynamics, bifurcations and seasonality.
Speaker: Alexandre Rodrigues (ISEG - Universidade de Lisboa, Portugal).
Time: Wednesday, 4 December 2024,from 14:15 to 15:15.
Place: Room 1.6, building VII.
Abstract: In this talk, I analyze a periodically forced dynamical system inspired by the SIR model with impulsive vaccination. I characterize its dynamics according to the proportion of vaccinated individuals and the time between doses. I draw the associated bifurcation diagram. I also explore analytically and numerically chaotic dynamics by adding seasonality to the disease transmission rate. This is a joint work with João Maurício de Carvalho (University of Porto).
[SDataScience] Mechanistic mathematical models of harmful algal species | Ming Li (University of Maryland Center for Environmental Science)
4 December 2024 3:30 pm - 4:30 pm
Room 1.5, Building VII, NOVA FCT
[SAL] The stylic monoid (and other power quotients) | Antoine Abram (Université du Québec à Montréal)
9 December 2024 2:00 pm - 3:00 pm
Zoom link: https://videoconf-colibri.zoom.us/j/97149261405?pwd=dwPsbvzqf72uSGBWpW0Y11AtJXT6Kn.1
The plactic monoid is a central object in algebraic combinatorics. Generally represented with Young tableaux, Cain, Gray and Malheiro proved that it has a confluent presentation with column tableaux as generators.
We will look at a left action of the free monoid A^* on this set of generators by Schensted left insertion. This action defines a monoid, called the stylic monoid, which happens to be a non-trivial finite quotient of the plactic monoid. It has a nice presentation by simply quotienting the plactic monoid by the relations x^2=x for all letters x in A. We will look at some interesting properties using a set of representants, the N-tableaux, paired with insertion algorithms.
If time permits, we will conclude with some generalization and analogous quotients called power quotients.
Based on joint work with C. Reutenauer for the stylic monoid, and with F. Hivert, J. Mitchell, M. Tsalakou, and J.-C. Novelli for the power quotients.
[Mini-courseAn] Introduction to Dyadic Analysis | George Tephnadze (University of Georgia, Tbilisi, Georgia)
10 December 2024 10:00 am - 12:00 pm
Room 1.6, building VII
Title: Introduction to Dyadic Analysis.
Speaker: George Tephnadze (University of Georgia, Tbilisi, Georgia).
Lecture 1: Tuesday, 10 December 2024, from 10:00 to 12:00.
Lecture 2: Thursday, 12 December 2024, from 10:00 to 12:00.
Lecture 3: Monday, 16 December 2024, from 10:00 to 12:00.
Lecture 4: Wednesday, 18 December 2024, from 10:00 to 12:00.
Place: Room 1.6, building VII.
Abstract: The fact that the Walsh system is the group of characters of a compact Abelian group connects dyadic analysis with abstract harmonic analysis. Later on, in 1947 Vilenkin introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters, which include the dyadic group and the Walsh system as a special case. Pontryagin, Rudin, Hewitt and Ross investigated such problems of harmonic analysis on groups.
Unlike the classical theory of the Fourier series, which deals with decomposing a function into continuous waves, the Walsh (Vilenkin) functions are rectangular waves. There are many similarities between these theories, but there are also differences. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. This point of view leads naturally to a new domain of considering Fourier Analysis on locally compact Abelian groups and dyadic (Walsh) group provides an important model on which one can verify and illustrate many questions from abstract harmonic analysis.
This introduction consists of 4 lectures and is aimed at Ph.D. students and researchers without an initial background on the subject.
Lecture 1: We define the Walsh group and functions and equip this group with the topology and Haar measure. Moreover, we investigate the character functions of the Walsh group, and the representation of the Walsh group on the interval [0,1). We also investigate some rearmament of the Walsh system, which is called the Kaczmarz system, and some generalizations, which are called Vilenkin groups and zero-dimensional groups.
Lecture 2: We define and investigate Dirichlet kernels, Lebesgue constants and partial sums with respect to the Walsh system and show that the localization principle holds for the Walsh-Fourier series and it is not true for the Walsh-Kaczmarz Fourier series. We define Lebesgue points and investigate almost everywhere convergence of subsequences of partial sums of the Walsh-Fourier series of integrable functions.
Lecture 3: We define and discuss Walsh-Fejér kernels and means, Walsh-Lebesgue points and investigate approximation properties and almost everywhere convergence of Fejér means in Lebesgue spaces.
Lecture 4: We define and discuss conditional expectation operators, martingales and martingale Hardy spaces. We also state several interesting open problems in this theory.
This introduction to dyadic analysis is based on the following recent book (where complementary information and several open problems can be found in more general case):
L. E. Persson, G. Tephnadze and F. Weisz, Martingale Hardy Spaces and Summability of one-dimensional Vilenkin-Fourier Series, Birkhäuser/Springer, 2022.
[Mini-courseAn] Introduction to Dyadic Analysis | George Tephnadze (University of Georgia, Tbilisi, Georgia)
12 December 2024 10:00 am - 12:00 pm
Room 1.6, building VII.
Title: Introduction to Dyadic Analysis.
Speaker: George Tephnadze (University of Georgia, Tbilisi, Georgia).
Lecture 1: Tuesday, 10 December 2024, from 10:00 to 12:00.
Lecture 2: Thursday, 12 December 2024, from 10:00 to 12:00.
Lecture 3: Monday, 16 December 2024, from 10:00 to 12:00.
Lecture 4: Wednesday, 18 December 2024, from 10:00 to 12:00.
Place: Room 1.6, building VII.
Abstract: The fact that the Walsh system is the group of characters of a compact Abelian group connects dyadic analysis with abstract harmonic analysis. Later on, in 1947 Vilenkin introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters, which include the dyadic group and the Walsh system as a special case. Pontryagin, Rudin, Hewitt and Ross investigated such problems of harmonic analysis on groups.
Unlike the classical theory of the Fourier series, which deals with decomposing a function into continuous waves, the Walsh (Vilenkin) functions are rectangular waves. There are many similarities between these theories, but there are also differences. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. This point of view leads naturally to a new domain of considering Fourier Analysis on locally compact Abelian groups and dyadic (Walsh) group provides an important model on which one can verify and illustrate many questions from abstract harmonic analysis.
This introduction consists of 4 lectures and is aimed at Ph.D. students and researchers without an initial background on the subject.
Lecture 1: We define the Walsh group and functions and equip this group with the topology and Haar measure. Moreover, we investigate the character functions of the Walsh group, and the representation of the Walsh group on the interval [0,1). We also investigate some rearmament of the Walsh system, which is called the Kaczmarz system, and some generalizations, which are called Vilenkin groups and zero-dimensional groups.
Lecture 2: We define and investigate Dirichlet kernels, Lebesgue constants and partial sums with respect to the Walsh system and show that the localization principle holds for the Walsh-Fourier series and it is not true for the Walsh-Kaczmarz Fourier series. We define Lebesgue points and investigate almost everywhere convergence of subsequences of partial sums of the Walsh-Fourier series of integrable functions.
Lecture 3: We define and discuss Walsh-Fejér kernels and means, Walsh-Lebesgue points and investigate approximation properties and almost everywhere convergence of Fejér means in Lebesgue spaces.
Lecture 4: We define and discuss conditional expectation operators, martingales and martingale Hardy spaces. We also state several interesting open problems in this theory.
This introduction to dyadic analysis is based on the following recent book (where complementary information and several open problems can be found in more general case):
L. E. Persson, G. Tephnadze and F. Weisz, Martingale Hardy Spaces and Summability of one-dimensional Vilenkin-Fourier Series, Birkhäuser/Springer, 2022.
[Mini-courseAn] Introduction to Dyadic Analysis | George Tephnadze (University of Georgia, Tbilisi, Georgia)
16 December 2024 10:00 am - 12:00 pm
Room 1.6, building VII.
Title: Introduction to Dyadic Analysis.
Speaker: George Tephnadze (University of Georgia, Tbilisi, Georgia).
Lecture 1: Tuesday, 10 December 2024, from 10:00 to 12:00.
Lecture 2: Thursday, 12 December 2024, from 10:00 to 12:00.
Lecture 3: Monday, 16 December 2024, from 10:00 to 12:00.
Lecture 4: Wednesday, 18 December 2024, from 10:00 to 12:00.
Place: Room 1.6, building VII.
Abstract: The fact that the Walsh system is the group of characters of a compact Abelian group connects dyadic analysis with abstract harmonic analysis. Later on, in 1947 Vilenkin introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters, which include the dyadic group and the Walsh system as a special case. Pontryagin, Rudin, Hewitt and Ross investigated such problems of harmonic analysis on groups.
Unlike the classical theory of the Fourier series, which deals with decomposing a function into continuous waves, the Walsh (Vilenkin) functions are rectangular waves. There are many similarities between these theories, but there are also differences. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. This point of view leads naturally to a new domain of considering Fourier Analysis on locally compact Abelian groups and dyadic (Walsh) group provides an important model on which one can verify and illustrate many questions from abstract harmonic analysis.
This introduction consists of 4 lectures and is aimed at Ph.D. students and researchers without an initial background on the subject.
Lecture 1: We define the Walsh group and functions and equip this group with the topology and Haar measure. Moreover, we investigate the character functions of the Walsh group, and the representation of the Walsh group on the interval [0,1). We also investigate some rearmament of the Walsh system, which is called the Kaczmarz system, and some generalizations, which are called Vilenkin groups and zero-dimensional groups.
Lecture 2: We define and investigate Dirichlet kernels, Lebesgue constants and partial sums with respect to the Walsh system and show that the localization principle holds for the Walsh-Fourier series and it is not true for the Walsh-Kaczmarz Fourier series. We define Lebesgue points and investigate almost everywhere convergence of subsequences of partial sums of the Walsh-Fourier series of integrable functions.
Lecture 3: We define and discuss Walsh-Fejér kernels and means, Walsh-Lebesgue points and investigate approximation properties and almost everywhere convergence of Fejér means in Lebesgue spaces.
Lecture 4: We define and discuss conditional expectation operators, martingales and martingale Hardy spaces. We also state several interesting open problems in this theory.
This introduction to dyadic analysis is based on the following recent book (where complementary information and several open problems can be found in more general case):
L. E. Persson, G. Tephnadze and F. Weisz, Martingale Hardy Spaces and Summability of one-dimensional Vilenkin-Fourier Series, Birkhäuser/Springer, 2022.
[SAL] The Bright Side of AITP | João Araújo (NOVA Math)
17 December 2024 2:00 pm - 3:00 pm
Laboratory 2.2, building VII.
Abstract:
Significant efforts have been made to have LLMs prove theorems, but their reasoning power is very limited. Their problem-solving depends on how problems are phrased, struggling with variations from known patterns. They handle "Do X" better than "Don't do X", as they focus on "X" regardless. Their performance drops in long conversations due to attention mechanism limits. Neural networks aren't built for true reasoning. The current belief claims that symbolic AI offers reasoning but is rarely practical. ProverX is symbolic AI Theorem Proving and during this talk I will show how completely wrong is the current belief.
[Mini-courseAn] Introduction to Dyadic Analysis | George Tephnadze (University of Georgia, Tbilisi, Georgia)
18 December 2024 10:00 am - 12:00 pm
Room 1.6, building VII.
Title: Introduction to Dyadic Analysis.
Speaker: George Tephnadze (University of Georgia, Tbilisi, Georgia).
Lecture 1: Tuesday, 10 December 2024, from 10:00 to 12:00.
Lecture 2: Thursday, 12 December 2024, from 10:00 to 12:00.
Lecture 3: Monday, 16 December 2024, from 10:00 to 12:00.
Lecture 4: Wednesday, 18 December 2024, from 10:00 to 12:00.
Place: Room 1.6, building VII.
Abstract: The fact that the Walsh system is the group of characters of a compact Abelian group connects dyadic analysis with abstract harmonic analysis. Later on, in 1947 Vilenkin introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters, which include the dyadic group and the Walsh system as a special case. Pontryagin, Rudin, Hewitt and Ross investigated such problems of harmonic analysis on groups.
Unlike the classical theory of the Fourier series, which deals with decomposing a function into continuous waves, the Walsh (Vilenkin) functions are rectangular waves. There are many similarities between these theories, but there are also differences. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. This point of view leads naturally to a new domain of considering Fourier Analysis on locally compact Abelian groups and dyadic (Walsh) group provides an important model on which one can verify and illustrate many questions from abstract harmonic analysis.
This introduction consists of 4 lectures and is aimed at Ph.D. students and researchers without an initial background on the subject.
Lecture 1: We define the Walsh group and functions and equip this group with the topology and Haar measure. Moreover, we investigate the character functions of the Walsh group, and the representation of the Walsh group on the interval [0,1). We also investigate some rearmament of the Walsh system, which is called the Kaczmarz system, and some generalizations, which are called Vilenkin groups and zero-dimensional groups.
Lecture 2: We define and investigate Dirichlet kernels, Lebesgue constants and partial sums with respect to the Walsh system and show that the localization principle holds for the Walsh-Fourier series and it is not true for the Walsh-Kaczmarz Fourier series. We define Lebesgue points and investigate almost everywhere convergence of subsequences of partial sums of the Walsh-Fourier series of integrable functions.
Lecture 3: We define and discuss Walsh-Fejér kernels and means, Walsh-Lebesgue points and investigate approximation properties and almost everywhere convergence of Fejér means in Lebesgue spaces.
Lecture 4: We define and discuss conditional expectation operators, martingales and martingale Hardy spaces. We also state several interesting open problems in this theory.
This introduction to dyadic analysis is based on the following recent book (where complementary information and several open problems can be found in more general case):
L. E. Persson, G. Tephnadze and F. Weisz, Martingale Hardy Spaces and Summability of one-dimensional Vilenkin-Fourier Series, Birkhäuser/Springer, 2022.
[SAn] Non-uniqueness of Hölder continuous solutions to 3D stochastic Euler equations on torus | Kush Kinra (NOVA Math, NOVA FCT)
8 January 2025 2:15 pm - 3:15 pm
Room 1.6, building VII
In this talk, we shalldiscuss the construction of infinitely many global-in-time Höldercontinuous analytically weak solutions to stochastic Euler equations inthree-dimensional periodic domain.
The proof is based on astochastic convex integration technique using Beltrami waves asbuilding blocks.
[SAL] Transformation representations of diagram monoids | James East (Western Sydney University)
13 January 2025 11:00 am - 12:00 pm
Zoom link: https://videoconf-colibri.zoom.us/j/92565611443?pwd=diOblm3Pu1aC5vYRMoN1BKd0DyktmB.1
Cayley's Theorem states that any finite monoid can be faithfully represented as a semigroup of transformations (self-maps) of a finite set. The minimum size of such a set is the (minimum transformation) degree of the monoid.
We obtain formulae for the degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley-Lieb and Motzkin monoids. For example, the partition monoid Pn has degree 1 + ( B(n + 2) - B(n + 1) + B(n) ) / 2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.
This is joint work with Reinis Cirpons and James Mitchell, both at Univ St Andrews.
[SAn] On the DiPerna-Majda gap problem for 2D Euler equations | Óscar Domínguez Bonilla (UNEF University-Madrid, Spain)
14 January 2025 2:15 pm - 3:15 pm
Room 1.6, building VII
A famous result ofDelort (1991) establishes the concentration-
cancellationphenomenon for approximating solutions of 2D Euler equations
with a vortex sheetwhose vorticity maximal function has a log-decay of order
1/2 . On the otherhand, DiPerna and Majda (1987) showed that if the log-
decay assumption isstrictly larger than 1 then the lack of concentration
(and hence energyconservation) holds. Then the so-called DiPerna-Majda
gap problem asks:concentration-cancellation vs. energy conservation in the
remaining log-range(1/2,1]?
In this talk, afterreviewing earlier contributions to the DiPerna-Majda
gap problem, I willpresent a new approach to this question based on sparse-
ness. This is based onjoint projects with Mario Milman and Daniel Spector.
The talk will beself-contained, and no additional prerequisites are needed.
[SOR] Worst-Case Complexity in Single-Objective and Multi-Objective Optimization | Rohollah (Nima) Garmanjani (NOVA Math)
29 January 2025 2:00 pm - 3:00 pm
NOVA FCT, VII-Lab 2.2
Abstract:
This talk examines the worst-case complexity in continuous optimization, defined as the computational effort required by an algorithm, in the worst-case scenario, to reduce a stationarity measure below a given positive threshold. We begin with an overview of foundational concepts and key results in worst-case complexity. Next, we delve into recent findings on the complexity of directional direct-search methods for nonsmooth unconstrained problems. Moving to the domain of multi-objective optimization, we highlight its distinctive challenges and recent advances. Finally, we present the worst-case complexity of a trust-region algorithm for solving (strongly) convex smooth unconstrained problems.
[SAn] Toepliz operators on Hardy spaces and their abstract generalizations | Oleksiy Karlovych (NOVA Math and Department of Mathematics, NOVA FCT, Portugal)
29 January 2025 2:15 pm - 3:15 pm
Room 3.2, Building VII, FCT UNL, Portugal
This talk is anontechnical overview of some topics of spectral theory of Toeplitz operatorson classical Hardy spaces and their abstract generalizations built upon Banachfunction spaces. We pay special attention to Toeplitz operators with continuousand piecewise continuous symbols.
[SAL] Translations between logics: a unified view | Gilda Ferreira (Universidade Aberta and CEMS.UL/CMAFcIO)
10 February 2025 2:00 pm - 3:00 pm
Lab. 2.2, building VII.
Abstract:
We begin with a very introductory overview of classical, intuitionistic, and linear logics. Several proof translations exist between classical and intuitionistic logic (negative translations) [1, 2, 3], as well as between intuitionistic and linear logic (Girard translations) [4, 5]. These translations serve various purposes, including transferring properties between systems, simplifying proofs, facilitating the extraction of constructive computational content from proofs, and controlling the use of logical resources.
We will show that all these systems can be expressed as extensions of a basic logical system (essentially, intuitionistic linear logic). By establishing a common logical basis, we are able to formalize a unified approach to devising and simplifying such proof translations [6]. This approach clarifies the relationships between different logical systems, and reveals the underlying structure that connects them. Through this simplification process, we obtain the most well-known translations in the literature.
This is joint work with Paulo Oliva and Clarence Protin.
References:
[1] A.S. Troelstra, D. van Dalen, Constructivism in mathematics: An introduction. In Studies in Logic and the Foundations of Mathematics, volume 1, 1988.
[2] M. Heine Sørensen, P. Urzyczyn, Lectures on the Curry-Howard Isomorphism, volume 149. Elsevier, 2006.
[3] G. Ferreira, P. Oliva, On various negative translations, Electronic Proceedings in Theoretical Computer Science, 47:21-33, 2011.
[4] J.-Y. Girard, Linear logic, Theoretical Computer Science, 50:1–101, 1987.
[5] J.-Y. Girard, A tutorial on linear logic. In Substructural Logics, Studies in Logic and Computation 2, pages 327–355. Oxford Science Publications, 1994.
[6] G. Ferreira, P. Oliva, C. Protin, On the various translations between classical, intuitionistic and linear logic. (subm.)
https://arxiv.org/abs/2409.02249
[SSRM] On Generalized Mean Reverting Processes with Possible Structural Change | Yunhong Lyu (Trent University, Peterborough, Canada)
11 February 2025 2:00 pm - 3:00 pm
Statistics and Risk Management Seminar
Department of Mathematics, NOVA MATH/FCT NOVA
Title: On Generalized Mean Reverting Processes with Possible Structural Change
Speaker: Yunhong Lyu,Trent University, Peterborough, Canada
Date | Time: February 11, 2025 | 14h00
Zoom: https://videoconf-colibri.zoom.us/j/88333359956
Abstract: In this presentation, we address the inference problem for the drift parameter in a generalized mean-reverting process, which is well-suited for modeling data exhibiting periodic characteristics. Additionally, we examine the case where linear constraints may be imposed on the drift parameters. We introduce three estimators: the unrestricted estimator, the restricted estimator, and the shrinkage estimator, and evaluate their relative efficiency. Furthermore, we explore change-point detection within this framework. Simulation studies validate the effectiveness of the proposed methods, which are subsequently applied to real-world environmental data. The results highlight the significance of accurate inference and timely change detection in environmental processes. Finally, the proposed methodology is expected to enhance the understanding and management of environmental systems.
Short Bio: Yunhong Lyu is an Assistant Professor in the Department of Mathematics & Statistics at Trent University, specializing in statistical modeling and structural change detection. Her research focuses on the inference problems in mean-reverting processes, such as parameter estimation, hypothesis testing and change-point detection, with applications in economics, health, and environmental studies. She holds a PhD from the University of Windsor and has published work on financial modeling, healthcare economics, and education policy. Beyond her research, she enjoys mentoring and teaching both undergraduate and graduate students.
Organizers: Mina Norouzirad & Isabel Natário
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This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications)
[SOR] A specialized second-order augmented Lagrangian method for mathematical programs with cardinality constraints | Mariana da Rosa (UNICAMP, Brazil)
12 February 2025 3:00 pm - 4:00 pm
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Seminar of Operations Research
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Title: A specialized second-order augmented Lagrangian method for mathematical programs with cardinality constraints
Speaker: Mariana da Rosa, UNICAMP, Brasil
Date | Time: February 12, 2025 | 15h00
Place: FCT NOVA, VII-Second Floor, Seminar room
Abstract:
In this talk, a practical and specialized second-order augmented Lagrangian method for solving mathematical programs with cardinality constraints (MPCaC) is presented. The seminar begins with a review of the stationarity conditions for MPCaC and a discussion of some augmented Lagrangian methods proposed in the literature. The presentation then considers a new approach that incorporates a second-order refinement step tailored to the structure of MPCaC. Under reasonable assumptions, the method ensures convergence to second-order stationary points, making it a potential tool for tackling this class of optimization problems.