Seminars

Metaverse / Social Networks Novels

Abstract: In this talk we will discuss the three (r)evolutions that lead to the current state-of-the-art in new media. We will present the concepts behind these societal, scientific and cultural innovations. Furthermore, we will also show a vision of what is possible in the foreseeable future, along with its inherent mechanisms and challenges. We will close by revealing the interconnections between recent trends in this scenario.

Live on YouTube: https://www.youtube.com/live/UHMnDtNfIXA?si=WDsGxI4ria5NivS-

Conjuntos invariantes para homeomorfismos de $3$-variedades hiperbólicas com velocidade de escape positiva

Abstract: O objetivo desta palestra é apresentar o problema de entender homeomorfismos minimais de $3$-variedades. Vamos nos concentrar em $3$-variedades hiperbólicas, onde, através do uso de algumas propriedades geométricas, podemos obter alguns resultados de existência de compactos invariantes e formular algumas perguntas precisas. É um trabalho conjunto com Elena Gomes e Santiago Martinchich.

Blown-up toric surfaces with non-polyhedral effective cone

Abstract: A variety can fail to be a Mori Dream Space in several ways, such as non-polyhedrality of the pseudo-effective cone, existence of nef but not semi-ample line bundles, existence of irrational walls, and so on. I will survey what is known about these matters for projective toric surfaces blown up at one point. In particular, in joint work with Castravet, Laface and Ugaglia, we construct blown-up toric surfaces with a non-polyhedral pseudo-effective cone. As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space of stable rational curves with n marked points is not polyhedral for n at least 10.

Counting points of elliptic curves over finite fields

Abstract: This is an expository talk about counting points of elliptic curves $$E: y^2=f(x),\ \ f=4x^3-t_2x-t_3, \ \ t_2,t_3\in{\mathbb F}_p.$$ Instead of brute force substitution of elements of a finite field in the equation of elliptic curve, I will explain how computation of the Hasse-Witt invariant of elliptic curves do this counting job! A variant of this method (due to N. Katz and B. Mazur) uses vector fields on elliptic curves. This will be reduced to compute the following recurion of polynomials in $x$ $$V_{2n+2}=fV_{2n}''+\frac{1}{2}f'V_{2n}',\ \ V_0=x, \ '=\frac{\partial}{\partial x}.$$ For a prime $p\not=2$, it turns out that $V_{p-1}$ modulo $p$ is a degree one polynomial in $x$ and $p+1-\#E( {\mathbb F}_p)$ modulo $p$ is the coefficient of $x$ in $V_{p-1}$.

Investigations of SL(4)-structures.

Abstract: The studies of convex-cocompact $SO(3,1),SO(2,2),$ and $SO(2,1)\times{\Bbb R}^{2,1}$ structures are the Lie-theoretic analogues of the studies of de Sitter, anti de Sitter, and Minkowski spacetimes respectively. The analytic treatments of these three theories are almost identical, suggesting a unified treatment involving Anosov SL(4)-structures. The main challenge in constructing such a unification is the absence of any invariant metric. It is thus necessary to enquire what structures may take their place. The purpose of this talk is to present some preliminary investigations into the theory of SL(4) structures aimed at resolving this problem.