Colloquium, Dept. of Mathematics and Applied Math. -- Γενικό Σεμινάριο

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Colloquium

Abstract: This talk will be about the representation theory of finite roups. Having the symmetric group as our starting point, we will move to discussing certain families of finite groups, such as Weyl groups, reflection groups and finite groups of Lie type. A wrong conjecture and the island of Spetses may come up during this discussion, as well as finite Hecke algebras and their applications to other mathematical theories.

Universal Taylor series of one variable, an open question, and multi-variable approximation

Abstract: The existence of universal Taylor series of one variable shows that the partial sums of a power series $f$ can approximate a function $g$, without $g$ being a continuation of $f$. It is an open question whether this can happen with the additional requirement that $g$ is defined in an open subset of the domain of holomorphy of $f$.

Extensions of universal Taylor series are possible in products of simply connected domains of the complex plane and the question of whether they enjoy equally wild properties is posed.

Abstract: The Prime Number Theorem describes the asymptotic behavior of prime numbers, while the Prime Geodesic Theorem says that the lengths of prime geodesics on a Riemann surface have a similar asymptotic behavior.

We will present the historical evolution of these two central Theorems, as well as some additional similarities between prime numbers and geodesics on Riemann surfaces. If time permits, we will also discuss some more recent results.

Abstract: This talk will describe the key elements of the analysis of the SARS-CoV2 virus which causes the Covid-19 disease. The first part will give a short account of the accumulation of the key evidence during the early stages of the pandemic. This will be followed by the main results on deterministic and stochastic epidemic models. The third part of this lecture is concerned with the statistical analysis of the pandemic data and the insights gained via the use of appropriate epidemic models.

Abstract: Many fundamental results in geometry and topology have been established through the development of geometric flow techniques. In this lecture we will show how the mean curvature flow can be used to obtain results concerning the topological type of smooth maps between compact Riemannian manifolds. More precisely, we will see how the flow can be employed to provide some answers to the following problems:

Problem 1 (Gromov): Is it true that smooth maps between spheres with small $k$-dilation are null-homotopic?

Problem 2 (Smale): Is it true that a diffeomorphism of the sphere can be smoothly deformed into an isometry?

Problem 3 (Gromov): Is it true that a symplectomorphism of $CP^n$ can be deformed into a bi-holomorphic isometry of $CP^n$?

Η Μελέτη της Διδασκαλίας των Μαθηματικών στο Πανεπιστήμιο

Abstract: Στο σεμινάριο αρχικά θα συνοψίσω αποτελέσματα ερευνών που αφορούν στη μελέτη της διδασκαλίας των μαθηματικών στο Πανεπιστήμιο. Στη συνέχεια θα αναφερθώ σε έρευνα που κάνουμε με άλλους συναδέλφους στην οποία μελετάμε τη διδασκαλία στο Πανεπιστήμιο παίρνοντας υπόψη θεσμικούς και κοινωνικούς παράγοντες που τη διαμορφώνουν. Τέλος θα δώσω κάποια παραδείγματα από την προσπάθεια μας να προωθήσουμε διερευνητικούς τρόπους διδασκαλίας και μάθησης στα Θεμέλια της Μαθηματικής Ανάλυσης, ένα μάθημα μετάβασης από το σχολείο στο Πανεπιστήμιο.

Abstract: An integer distance set is a set in the Euclidean plane with the property that all pairwise distances between its points are integers. In this talk we will show that any integer distance set can be covered by a small number of lines and circles. This helps us address some questions by Erdős on the size of integer distance sets. In particular, we deduce that integer distance sets in general position (no 3 on a line, no 4 on a circle) are very sparse, and we derive a near-optimal lower bound on the diameter of any non-collinear integer distance set of a given size. The main ingredients of our proof are known refinements of the Bombieri-Pila determinant method.

Abstract: In Einstein’s theory of General Relativity the dynamics of the gravitational universe are determined by the so-called Einstein Equations. Mathematically, these equations form a coupled system of nonlinear geometric PDEs of hyperbolic type. As is the case for Maxwell’s equations of electro-dynamics, initial data sets for the Einstein Equations need to satisfy constraint equations. The study of these constraint equations can already provide fundamental insights into the rigidity and flexibility features of the Einstein Equations. In this talk I will first introduce the Einstein Equations, present a novel gluing problem for initial data, the so-called "null gluing problem”, and explain some of the new constructions it allows.

Abstract: Let $E$ be an elliptic curve over a number field $K$ and $\ell$ a prime number. Suppose $E$ has a $K$-rational $\ell$-isogeny, then the reduction of $\tilde{E}_{\mathfrak{p}}/\mathbb{F}_{\mathfrak{p}}$ also has a $\mathbb{F}_{\mathfrak{p}}$-rational $\ell$-isogeny for almost all primes $\mathfrak{p}$. When does the converse hold?

Abstract: Using computational multiphysics models, we developed an innovative numerical frame to investigate airborne virus transmission. The above includes the accurate modelling and simulation of different multiscale phenomena that involve particles, such as transport, dispersion, breakup, coalescence, filtration and evaporation. Using the above models, we study the advanced physics of virus-contaminated saliva droplets emitted by a human, i.e., from a cough or a sneeze, subject to different conditions of the surrounding environment (temperature, relative humidity and wind speed). We shed light on i- the social distance in the presence of wind, ii- the efficiency of face masks, iii- the weather effects on airborne virus survival and transmission, iv- the transmission in confined spaces and v- how computational fluid dynamics can be employed in the pandemic, seasonal forecasting by its coupling with epidemiological models. We will present the challenges, uncertainties, and future perspectives for interdisciplinary research, considering that pandemics will likely happen again.

Αξιοποίηση της ανώτερης μαθηματικής γνώσης στη διδασκαλία των Μαθηματικών στο σχολείο

Abstract: Τα τελευταία χρόνια αρκετές έρευνες εστιάζουν στην αξιοποίηση της ανώτερης μαθηματικής γνώσης, δηλαδή της μαθηματικής γνώσης που αποκτούν οι εκπαιδευτικοί στο Πανεπιστήμιο, στη διδασκαλία των Μαθηματικών στο σχολείο. Στην ομιλία θα παρουσιαστούν σχετικές έρευνες που εστιάζουν στις αντιλήψεις εκπαιδευτικών για την χρησιμότητα αυτής της γνώσης στη διδασκαλία τους και άλλες που αναδεικνύουν την αξία αυτής της γνώσης στην αναγνώριση και στην ερμηνεία λανθασμένων απαντήσεων μαθητών.

05 Jun 2024, 12:15:00 PM, Room: A303
Speaker: Roberto Paroni (Università di Pisa)

Abstract: In 1930 Sadowsky gave a constructive proof for the existence of a developable Möbius band and posed the problem of determining the equilibrium configuration of a Möbius strip formed from an unstretchable material. He tackled this latter problem variationally and he deduced the bending energy for a strip whose width is much smaller than the length. This energy, now known as Sadowsky's energy, depends on the curvature and torsion of the centerline of the band and it is singular at the points where the curvature vanishes.

In this talk, we re-examine the derivation of the Sadowsky's energy by means of the theory of Gamma-convergence of functionals in the Calculus of Variations.. We obtain an energy that is never singular and agrees with the classical Sadowsky functional only for "large" curvature of the centerline of the strip. The derived energy is then used to show that the centerline of a developable Möbius band at equilibrium cannot be a planar curve.

The talk is based on several joint works with L. Freddi, P. Hornung, and M.G. Mora.