Talks

Monday, February 05, 2024.

14h - 14h40
Humberto Quoirin (Universidad Nacional de Córdoba/Argentina) Title: Critical points with prescribed energy for a class of functionals depending on a parameter.
PDF | Abstract

Given $c \in\mathbb{R}$ we look for couples $(\lambda, \mu) \in \mathbb{R}\times X$ solving the problem $$\phi_{\lambda} '(u)= 0, \,\, \phi_{\lambda} (u)= c$$ Here $\phi_{\lambda} = I_1 - \lambda_2$, , where $I_1$ and $I_2$ are $C^1$ even functionals on a Banach space X. Under further conditions on $I_1$ and$I_2$ we prove the existence of infinitely many couples $(\lambda_{n, c}, u_{n, c})$ solving this problem. More generally, we analyze the structure of the solution set of this problem with respect to $\lambda$ and $c$. In particular, we show that the maps $c\mapsto \lambda_{n, c}$ are continuous, which gives rise to a family of \emph{energy curves} for this problem. The analysis of these curves provide us with several bifurcation type results, which are then applied to some elliptic problems. Our approach is based on the \emph{nonlinear generalized Rayleigh quotient} method.

14h50 - 15h40
João Vitor da Silva (IMECC/UNICAMP - Brazil) Title: Sharp regularity for the obstacle problem for p-Laplacian type equations and applications.
PDF | Abstract

In this Lecture we show existence/uniqueness of weak solutions of an obstacle problem for a quasi-linear operator with unbounded source terms. In our results, we obtain sharp gradient estimates, namely, $C^{1, \alpha}_{loc}(B_1)$ for the solution to an explicit and universal regularity exponent. Our results are relevant even for the simplest model case governed by the $p$-Laplacian with Hölder continuous coeficients $$div( | \nabla_u |^{p-2} \mathfrak{U})\nabla_u = f(x) , \quad \mbox{in}\,\, \{u > \varphi \}\cap B_1\\ div( | \nabla_u |^{p-2} \mathfrak{U}(x)\nabla_u \leq f(x) , \quad \mbox{in}\,\, B_1\\ u(x) \geq \varphi(x), \quad \mbox{in}\,\, B_1\\ u(x) = \varphi(x), \quad \mbox{on}\,\, \partial B_1,\\ $$ where $f \in L^{q}(\Omega)$ for $q> n$ and $q \geq \frac{p}{p-1} \, (1 < p < \infty ), \mathfrak{U}\in C^{0, \sigma}(\Omega, \mathbb{R}^{n\times n})$ (for some $\sigma \in (0, 1]$) with $\mathfrak{U}$ a $(\lambda, \Lambda)$-uniformly elliptic matrix, and $\varphi\in C^{1, \beta}(\Omega)\cap {\mathfrak{X}}_{p,q}$, for some $\sigma \in(0, 1])$ with $\mathfrak{U}$ a $(\lambda, \Lambda)$- uniformly elliptic matrix, and $\varphi\in C^{1, \beta}(\Omega)\cap \mathfrak{X}_{p,q}$, for some $\beta \in (0,1]$ where $${\mathfrak{X}}_{p,q}:=\{ v\in W^{1, p}(\Omega); \quad div\,\, \mathfrak{a}(x, \nabla v) \in L^q(\Omega)\}.$$ For some specic scenarios, we show the non-degeneracy of solutions, which provides crucial information about the free boundary of solutions. Our regularity estimates improve and extend, to a certain extent, results previously obtained for the obstacle problem governed by the $p$-Laplacian with bounded source term (cf. [1] and [3]). Furthermore, we gave special emphasis to the study of the linear and non-homogeneous case, i.e., $p = 2$ and $f\ne 0$, which was not available in the literature and it plays a decisive role in analysing the non-linear case (cf. [2]). This is a joint work with Elzon C. Bezerra Júnior (UFCA) and Romário T. Frias (Unicamp).

16h30 - 17h10
Mayra Soares (MAT/UNB - Brazil) Title: Boundary Weak Harnack Estimates and Regularity for Elliptic Operators in Divergence Form and Applications in PDEs.
PDF | Abstract

We obtain a global extension of the classical Weak Harnack Inequality which extends and quanties the HopfOleinik boundary-point lemma, for uniformly elliptic equations in divergence form, under the weakest assumptions on the leading coecients and on the boundary of the domain. Our main tool is the use of suitable barrier functions, which are solutions of auxiliaries problems and the $C^1$-estimates up to the boundary.
Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the HopfOleinik lemma. Furthermore, we provide an application showing how to use this results in order to deduce a priori upper bounds and multiplicity of solutions for a class of quasilinear elliptic problems with quadratic growth on the gradient. \bigskip Joint work with Boyan Sirakov and Fiorella Rendón.

Tuesday, February 06, 2024.

14h - 14h40
Chao Ji (University of Science and Technology/China) Title: Some results on normalized solutions for the Schrödinger equations.
PDF | Abstract

In this talk, we will introduce some results on normalised solutions for the Schörodinger equations. First of all, we show the existence of normalised solutions for the Schrödinger equations with L^2-subcritical growth and different types of potentials. Then, existence and multiplicity of normalized solutions for a Schrodinger equation with critical growth in RN will be given. Finally, we investigate the existence of multiple normalized solutions to the logarithmic Schrodinger equation via the minimization techniques and the Lusternik-Schnirelmann category. These are joint works with C.O. Alves and O. H. Miyagaki.

14h50 - 15h40
Kaye Silva (IME/UFG - Brazil) Title: On the structure of the Nehari set associated to a Schrodinger-Poisson system with prescribed mass: old and new results
PDF | Abstract

Consider the Schrödinger-Poisson system, with prescribed $L^2$ norm, in the whole $mathbb{R}^3$
$$-\Delta u + q\phi_u u - \lambda |u|^{p-2}u = l lu, \quad \mbox{in}\,\, \mathbb{R}^3\\ \int u ^2 = r$$
By using the fibering method of Pohozaev and the notion of Extremal Parameters introduced by Il’yasov we show that many results in the literature concerning this system can be obtained in an unified way. We exhibit also new results.

16h30 - 17h10
Adilson E. Presoto (DM/UFSCar - Brazil) Title: A bit more of the classical Brezis-Nirenberg problem.
PDF | Abstract

In the well-known paper \cite{BN}, Brezis and Nirenberg have proved that if $0<\lambda<\lambda_1$ and \(N\geq 4\), then \begin{equation} \label{p}\tag{$P$} \left\{ \begin{array}[c]{ll} - \Delta u= \lambda u+|u|^{2^*-2}u \, \, \mbox{in} \, \, \Omega, & \\ u= 0 \, \, \mbox{on}\, \, \partial\Omega,&\\ \end{array} \right. \end{equation} where $ 2^{*}=2N/(N-2)$ and $\lambda$ is a real parameter, has a positive solution. In the following years, Capozzi, Fortunato and Palmieri extended the result for all $\lambda>0$ if $N\ge5$ or $\lambda>0$ and $\lambda\neq \lambda_k$ in the case $N=4$. Here $0<\lambda_{1}<\lambda_{2}\leq \cdots\leq\lambda_{j}\leq \cdots$ denote the eigenvalues of $-\Delta$ in $H_0^1(\Omega)$. In this talk, we will deal with the resonant case, i.e. when $\lambda$ an eigenvalue, in \(4\) dimensions. In this situation, a solution can no longer be obtained by applying the usual linking theorem. In \cite{CFP}, the authors use a version of the symmetric mountain pass theorem to solve the problem for $N\ge5$ and $\lambda$ being an eigenvalue, due to technicalities the proof does not work if $N=4$. It seems to us that the main difficult to apply a minimax theorem is to gettting the right estimates. For instance, we are not able to get a separation condition between the values over the boundary of a rectangle and a sphere in the finite-dimension subspace in order to apply the classical linking theorem. To overcome this, we use a variant of the linking theorem, where this condition is relaxed. We point out that this generalization was motivated by the orthogonalization technique introduced by Gazzolla and Ruf \cite{GR}, and some variant of the classical minimax theorems without the separation condition obtained by Silva \cite{Si}.

Wednesday, February 08, 2024.

14h - 14h40
João Marcos do Ó (DM/UFPB - Brazil) Title: $p$-Harmonic functions in the upper half-space.
PDF | Abstract

We investigate the existence, nonexistence, and qualitative properties of $p$-harmonic functions in the upper half-space $\mathbb{R}^N_+$ $(N\geq 3)$ satisfying nonlinear boundary conditions. Moreover, the symmetry of positive solutions is shown by using the method of moving planes.

14h50 - 15h40
João R. Santos Junior (UFPA - Brazil) Title: The method of the Nehari Manifold on cones.
PDF | Abstract

Inserir o resumo aqui.

14h50 - 15h40
Abiel Costa Macedo (IME/UFG - Brazil) Title: Estimate for concentration level of the Adams functional and extremals for Adams-type inequality
PDF | Abstract

Inserir o resumo aqui.

Thursday, February 07, 2024.

14h - 14h40
Lucas C. F. Ferreira (IMECC/UNICAMP - Brazil) Title: On the solvability with borderline regularity for the 2D inviscid Boussinesq equations.
PDF | Abstract

We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter $\kappa$ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large $\kappa$ regardless of the size of initial data.

14h50 - 15h40
Marcos L. M. Carvalho (IME/UFG - Brazil) Title: Comparison Theorem for the fractional Orlicz Sobolev space
PDF | Abstract

In this work, we obtain a new version of the comparison principle for sub and super solutions to an equation involving the fractional $\phi$-Laplacian operator. We consider a problem in a local sense, instead of zero-boundary conditions.

16h30 - 17h10
Edcarlos D. Silva (IME/UFG - Brazil) Title: Schrödinger-Poisson system with zero mass in R^2 involving (2, q)-Laplacian: existence, asymptotic behaviour and regularity of solutions.
PDF | Abstract

It is stablished the existence of positive least energy solution for the following class of planar ellipitic systems in the zero mass case
$$ - \Delta u - \Delta_q u + \phi |u|^{r-2}u = \lambda |u|^{p-2} u , \quad \mbox{in}\,\, \mathbb{R}^2,\\ \Delta \phi = 2 \pi |u|^r , \quad \mbox{in}\,\, \mathbb{R}^2, $$
where $\lambda \leq 0$, $1< q < 2$, $q^{\ast}:= 2q/(2- q) < r < \infty$ and $p\geq 2r$. Due to the nature of the problem, we deal with the logarithmic integral kernel. Our approach is based on Nehari manifold and a version of the Principle of Symmetric Criticality due to Palais. Futermore, we study the asymptotic behavior of the solutions wherenever the parameter $\lambda$ goes to zero or infinity. Finally, we study regularity of the solutions applying Moser interation scheme.
This is a join work with J. C. Albuquerque - UFPE and J. Carvalho - UFS.

17h20 - 18h00
Denilson Pereira (UAMAT/UFCG - Brazil) Title: SOME ELLIPTIC PROBLEMS OF CONCAVE AND CONVEX TYPE IN $\mathbb{R}^n$ WITH BOUNDED WEIGHT.
PDF | Abstract

In this talk we present recent results involving elliptic problems with sub-linear growth at the origin and super-linear behaviour at infinity with bounded weights in $\mathbb{R}^n$. Problems of this nature present several difficulties to deal with variational, once that the energy functional is not, a priori, well defined. We overcome this difficulty by obtaining a first solution, via the sub and super solution technique. Understanding the behaviour at infinity of this first solution, we are able to obtain a second solution via variational methods. Joint work with Pedro Ubilla (USACH), Juan Arratia(USACH) and Diego Ferraz (UFRN).

Talks

Tuesday February 06, 2024.

14h30 - 15h20
Eduardo Antônio (UNB - Brazil ) Title.
PDF | Abstract

In this talk, we utilize the framework of Markov processes to attain a more probabilistic perspective on the theory of transfer operators. In doing so, we establish a functional central limit theorem (FLCT) for an $O(N)$ model associated with Dyson potential on the one-dimensional lattice. We also proof a FLCT on a non-compact alphabet setting, for a model associated with a Dyson type potential on the one-dimensional lattice. A Breimann ergodic theorem for equilibrium measures arising from the Ruelle Operator Formalism is proved. Furthermore, we obtain a qualitative criterion (strong transitivity) to determine when the conformal measure has full support. As an application, we show how to connect the Ruelle operator framework with the perspective of the {\it Hopf theory of Markov Processes.}

15h20 - 16h10
Rémy Sanchis (UFMG - Brazil )Title. Percolação de Grau Restrito.
PDF | Abstract

No modelo de Percolação de Grau Restrito em um grafo, para cada aresta do grafo est á associada uma variável aleatória uniforme, sendo que essas variáveis aleatórias são independentes. Dado um inteiro $k$ (que é um parâmetro do modelo), cada aresta é declarada aberta no momento correspondente à sua variável uniforme se, e somente se, os graus dos vértices da aresta forem ambos estritamente menores do que $k$. Quando o parâmetro $k$ é igual ao grau dos vértices do grafo (no caso do grafo ser regular), recuperamos o modelo de percolção independente, mas quando $ k$ é menos do que esse grau, surgem novos problemas sobretudo ao fato do modelo se tornar altamente dependente. Nesta palestra discutiremos alguns aspectos relacionados à transição de fase neste modelo, à unicidade do aglomerado infinito e, se possível, aspectos relacionados ao decaimento da conectividade na fase subcrítica.

Wednesday, February 07, 2024.

16h30 - 17h10
Leandro Cioletti (UNB - Brazil ) Title: The Universal Space, Independence and the Wiener-Lévy-Ciesielski Construction of the Brownian Motion.
PDF | Abstract

In this expository lecture we show how to construct a countable infinite sequence of independent univariate Gaussian random variables on the Universal Space. Afterwards, we use this sequence to construct a simple (separable) Hilbert space supporting the Wiener-Lévy-Ciesielski Construction of the Brownian Motion. At the end we will briefly comment on how to apply the presented ideas to construct a Gaussian Processes with a prescribed Hölder covariance function not requiring at any step the use of the Kolmogorov Existence theorem.

Talks

link Zoom

https://us02web.zoom.us/j/85426285138?pwd=UUg0L2gxcXZYQjJxamprbHlWVGVpdz09

ID of the meeting

854 2628 5138

Password

0607080224

Tuesday, February 06, 2024 .

14h - 14h40
Maria Bras-Amorós (UNIVERSITAT ROVIRA I VIRGILI TARRAGONA, CATALONIA, SPAIN) Title: ON THE SEEDS AND THE GREAT-GRANDCHILDREN OF A NUMERICAL SEMIGROUP.
PDF | Abstract

We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The algorithm has been used to prove that there are no Eliahou semigroups of genus 66, hence proving the Wilf conjecture for genus up to 66. We also found three Eliahou semigroups of genus 67. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado’s type. However, it is a member of a new family suggested by Shalom Eliahou.
Keywords: numerical semigroup, exploring algorithm, seeds of a numerical semigroup.

14h50 - 15h30
Mariana Rosas (URV - Spain) Title: INFINITE DESCENDANT NODES IN A NUMERICAL SEMIGROUP TREE.
PDF | Abstract

Motivated by studying the growth of the sequence $n_g$ that counts the number of numerical semigroups of genus $g$, in this talk we will see how to organize such semigroups in a tree so that at each level $g$ of the tree there are numerical semigroups of genus $g$. In this tree - which seems to only grow but the proof of this is an open problem - we will characterize the nodes with infinite descendants, which appear to be rare but support the tree.
Keywords: numerical semigroup, semigroup tree, infinite chains.

14h50 - 15h30
Sávio Ribas (UFOP - Brazil) Title: FACTORIZATION THEORY VIA ADDITIVE COMBINATORICS
PDF | Abstract

We will introduce the relationship between factorization theory (focused on non-uniqueness) and zero-sum problems in additive combinatorics. Often the elements of a ring (such as the ring of integers of a number field) or of a monoid $H$ can be written in several different ways as a finite product of irreducibles. The {\emph{ set of lengths}} of $ a \in H$, $L(a) =\{k \in\mathbb{ N} \,|\, \exists \, u_1, \cdots,u_k \,\, \mbox{irreducible s.t.}\,\, a = u_1\cdots u_k\}$, and the system of {\emph{ set of lengths}}, ${\cal L}(H) = \{L(a)\, |\, a \in H\}$, are means of describing the non-uniqueness of factorization in $H$. It is conjectured that ${\cal L}({\cal B}(G))$, where ${\cal B}(G)$ is the monoid of zero-sum sequences over an abelian group $G$, completely determines $G$ except for a few small cases. Furthermore, for $ k \in \mathbb{N}$, let ${\cal U}_k(H)$ be the set of all $m \in\mathbb{ N}$ such that there are irreducible $u_1,\cdots,u_k,v_1,\cdots,v_m$ satisfying $u_1 \cdots u_k = v_1\cdots v_m$. Define $\lambda_k(H) = min\, {\cal U}_k(H), \rho_k(H) = sup\, {\cal U}_k(H) $ ($k$-th elasticity of $H$) and $\rho(H) = sup\,\displaystyle\frac{ \rho_k(H)}{k}$ (elasticity of $H$). It is known that $\lambda_k(H)$ can be written in terms of $ \rho_k(H)$, which in turn can be lower and upper bounded by the Davenport constant of $G$, which is defined as the maximum length of a minimal zero-sum sequence. The sets ${\cal U}_k(H)$ are usually well structured (they look like arithmetic progressions). For example, $ {\cal U}_k(H) = [\lambda_k(H),\rho_k(H)]$ if and only if the ideal class group $G$ of $H$ is finite abelian. This makes the $k$-th elasticity one of the most important invariants to describe the non-uniqueness of the factorization in $H$. We will present the main results and conjectures about $\rho_k(H)$.
Keywords: Factorization theory, Krull monoids, Davenport constant.

16h40 - 17h20
Drew Duncan (John Carroll University - USA) Title: ADDITIVE FORMS OVER TOTALLY RAMIFIED EXTENSIONS OF $\mathbb{Q}_2.$
PDF | Abstract

Let $Gamma^{\ast}(d,k)$ be the least number $s$ so that any additive form $a_1x^d_1+ \cdots + a_sx^d_s$ over the field $k$ has a nontrivial zero. We would like to know that $Gamma^{\ast}(d,k) \leq d_2+ 1$ holds for every $p$-adic field $k$. Thus far, progress on this has been restricted to degrees of ramification at most $2$. In this talk we will see that, in the case of $d = 2m$, $m$ odd, and $k$ a totally ramified extension of $\mathbb{Q}_2$ of arbitrarily high degree, with relatively little work we can obtain this bound, and even improve on it.
Keywords: Equations in many variables, $p$-adic fields, Forms of degree higher than two.

Wednesday, February 07, 2024 .

14h - 14h40
Lucas Reis (UFMG , BRAZIL ) Title: THE DENSITY OF SPECIAL ELEMENTS IN FINITE FIELDS, ON AVERAGE.
PDF | Abstract

In this talk we discuss the behaviour of some arithmetic functions related to the density of special elements in finite field extensions (normal, $k$-normal and primitive elements). Namely, if $q$ is a fixed prime power and $f(n)$ denotes the proportion of elements in the finite field $\mathbb{F}_{q^n}$ with one of those specified properties, we discuss the behaviour of the arithmetic function f(n), on average. In all cases, the function $f(n)$ has a positive mean value, in the sense that the limit
$$\lim_{x\arrowvert +\infty}\frac{1}{x}\sum_{n\leq x}f(n),$$
exists and it is positive. We provide further information on this mean value and propose some possible questions for future research. The techniques are simple and rely on basic rudiments of the theory of arithmetic functions.
\noindent {\bf Keywords:} normal elements, mean values, arithmetic functions.

14h50 - 15h30
Cicero Carvalho (UFU - Brazil) Title: ON SOME MONOMIAL EVALUATION CODES AND THEIR DUALS
PDF | Abstract

In this talk we would like to present some results on the parameters of a family of evaluation codes defined over the points of the so-called extended norm-trace curve, and their duals. We will also present some facts from Gröbner basis theory which, together with the concept of indicator functions, will be useful in studying this new class of codes.

15h50 - 16h30
Fabio Brochero (UFMG - Brazil) Title:
PDF | Abstract

Inserir o resumo aqui.

16h40 - 17h20
David Leep (University of Kentucky - USA) Title: IMPROVEMENTS TO WARNING’S THEOREM
PDF | Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a power of the characteristic $p$, and let $f = \{f_1, \cdots ,f_r\}$ be a set of polynomials in $\mathbb{F}_q[x_1,\cdots ,x_n]$ of degrees $d_1,\cdots ,d_r$, respectively. Let $N(f, \mathbb{F}_q)$ denote the number of common zeros of $f_1,\cdots ,f_r$ defined over $\mathbb{F}_q$.
The theorems due to Chevalley and Warning state that if $N(f,\mathbb{F}_q) > 0$ and if $n > d_1 +\cdots +d_r$, then $N(f,\mathbb{F}_q)$ is divisible by $p$ and $N(f,\mathbb{F}_q) \geq q^{n-(d_1+ \cdots + d_r)}$.
There are examples where $ N(f,\mathbb{F}_q) = q^n-(d_1+\cdots+d_r)$ under the stated hypotheses, showing that this bound is best possible.
Heath-Brown proved that if $N(f,\mathbb{F}_q) = q^n-(d_1+\cdots +d_r)$, then the set of common zeros of $f$ over $\mathbb{F}_q$ form an affine linear space (a subspace or a translate of a subspace).
Heath-Brown studied systems $f$ where the set of common zeros of $f$ over $\mathbb{F}_q$ does not form an affine linear space. In these cases, then subject to the above hypotheses, he showed that $N(f,\mathbb{F}_q) > q^n-(d_1+\cdots +d_r)$ and he gave some lower bound estimates for $N(f,\mathbb{F}_q)$, which improve the classical estimate of $q^n-(d_1+\cdots +d_r)$. In this talk I will present theorems that improve the estimates of Heath-Brown.
JOINT WORK WITH RACHEL PETRIK.

17h30 - 18h10
Jean Carlos de Aguiar Lelis (UFPA - Brazil) Title: ON THE EXCEPTIONAL SET OF TRANSCENDENTAL ENTIRE FUNCTIONS IN SEVERAL VARIABLES.
PDF | Abstract

In this work, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ under complex conjugation and which contains the origin) is the exceptional set of un- countable many transcendental entire functions over $\mathbb{C}^m$with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.
JOINT WORK WITH ALVES, MARQUES, AND TROJOVISKY.

Thursday, February 08, 2024.

14h - 14h40
Victor Gonzalo Lopes Neumann (Universidade Federal de Uberlândia, Minas Gerais, Brazil) Title: INTEGRAL VALUES OF GENERATING FUNCTIONS OF RECURRENCE SEQUENCES.
PDF | Abstract

Suppose that $a_0,a_1, \cdots $ is an integer sequence which satisfies a recurrence relation with constant coefficients, and let $T(x) = f(x)/g(x)$ be its generating function, where $f(x)$ and $g(x)$ have no common factors in $\mathbb{Z}[x]$. In this talk, we study the problem of finding the rational values of $x$ such that $T(x)$ is an integer. We say that such a number is {\emph{good}} for the sequence. Our first main result is that if $g(x)$ has at least two different irreducible factors, or if $g(x)$ has a single irreducible factor of degree at least $3$, then the sequence has only finitely many good values. We also study sequences of the form $0, 1,\cdots$ for which the recurrence relation has order $2$. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them.
JOINT WORK WITH MICHAEL KNAPP AND ABÍLIO LEMOS.
Keywords: Generating function, Recurrence sequences, Diophantine equations.

14h50 - 15h30
Abílio Lemos (Universidade Federal de Viçosa, Minas Gerais, Brazil) Title: ON FUNCTIONAL GRAPHS OVER FINITE FIELDS.
PDF | Abstract

The dynamic of iterations of polynomials over finite fields have attracted much attention in recent years in part due to their applications in cryptography and integer factorization methods like {\emph Pollard rho algorithm}. We define the functional graph $f : \mathbb{F}_{q^2} \rightarrow \mathbb{F}_{q^2}$ as the directed graph ${\cal G}(f) = ({\cal V}, {\cal E})$ where ${\cal V} = \mathbb{F}_{q^2}$ and ${\cal E} = \{ \langle x,f(x)\rangle \, |\, x \in \mathbb{F}_{q^2} \}$. We define the iterations of $f$ as $f^{(0)}(x) = x$ and $f^{(n+1)}(n+1)(x) = f(f^{(n)}(x))$. Since $f$ is defined over finite field, fixing $\alpha \in\mathbb{F}_q$, there are integers $0 \leq i < j$, minimal, such that $f^{(i)}(\alpha) = f^{(j)}(\alpha)$. In the case when $i > 0$, we call the list $\alpha,f(\alpha),f^{(2)}(\alpha), \cdots ,f^{(i-1)}(\alpha)$ the pre-cycle and $f^{(i)}(\alpha),f^{(i+1)}(\alpha),\cdots ,f^{(j-1)}(\alpha)$ the cycle of length $(j-i)$ or the $(j-i)$-cycle. If $\alpha$ is an element of a cycle, we call it a periodic element and, if $f(\alpha) = \alpha$ we say it is a fixed point. In this talk, we present some results in this topic and some results of the functional graph ${\cal G}(f)$ of the map $a\mapsto f(a)$, where $f(X) = X(X^{q-1} - c)^{q+1}$, for $c \in \mathbb{F}_q^{\ast}$.
JOINT WORK WITH JOSIMAR J.R. AGUIRRE AND VICTOR G.L. NEUMANN.
Keywords: Dynamics of polynomials, finite fields.

15h50 - 16h30
Guilherme Tizziotti (UFU - Brazil) Title: THE CORNER ELEMENT OF GENERALIZED NUMERICAL SEMIGROUPS.
PDF | Abstract

In this talk we present the concept of corner element of a generalized numerical semigroup (GNS), which extends in a sense the idea of conductor of a numerical semigroup to GNS. Also, we present properties of this new notion and its relations with existing invariants in the literature, we provide an algorithm to compute all the GNSs with fixed corner, and lower and upper bounds on the number of GNSs having a fixed corner element.
JOINT WORK WITH M. BERNARDINI AND W. TENÓRIO.
Keywords: generalized numerical semigroup, genus, corner.

16h40 - 17h20
David Leep (University of Kentucky - USA) Title: PAIRS OF QUADRATIC FORMS OVER P-ADIC FIELDS.
PDF | Abstract

ILet $K$ be a $p$-adic field and let ${\cal Q}_1, {\cal Q}_2 \in K[x_1,\cdots , x_n]$ be quadratic forms in $n$ variables with coefficients in $K$.
Heath-Brown proved that if $n = 8$, ${\cal Q}_1, {\cal Q}_2 $ have a nontrivial common zero defined over $K$, and the variety defined by $\{{\cal Q}_1, {\cal Q}_2 \}$ is nonsingular, then there exist $a,b \in K$, not both zero, such that $ a{\cal Q}_1 + b{\cal Q}_2 $ splits off at least $3$ hyperbolic planes.
This rather technical theorem, whose proof is very long, was an important ingredient to Heath-Brown’s proof that nonsingular pairs of quadratic forms in $8$ variables defined over a number field satisfy the Hasse Principle. More concretely: Suppose that $\{{\cal Q}_1, {\cal Q}_2 \}$ is a pair of quadratic forms in $8$ variables defined over a number field $F$ and assume that the variety defined by $\{{\cal Q}_1, {\cal Q}_2 \}$ is nonsingular. If ${\cal Q}_1, {\cal Q}_2$ have a nontrivial zero defined over each nonarchimedean completion of $F$ and also over each real completion of $F$ (if any), then ${\cal Q}_1, {\cal Q}_2$ have a nontrivial zero defined over $F$.
Heath-Brown’s theorem on pairs of quadratic forms over $p$-adic fields raised many questions in my mind, including why his proof was so difficult. Also, it seemed that this result for $n = 8$ should be part of a bigger theorem for general values of $n$.
This talk will explore the situation for general values of $n$.
JOINT WORK WITH JOHN HALL.
Keywords: Pairs of Quadratic Forms, $p$-adic fields.

Talks

All activities will be hybrid
Join Zoom Meeting https://us02web.zoom.us/j/88233010577
Meeting ID: 882 3301 0577 - BRT time zone (GMT-3)

Wednesday, February 7, 2024.

14h - 15h
Christophe Ringeissen (Inria, Nancy) Title.Building and Combining Unification and Matching Procedures.
PDF | Abstract

The concept of unification is ubiquitous in logic programming and in automated reasoning, for instance to perform deduction in theorem proving. Equational unification consists in finding instances of terms so that these instances are equivalent with respect to an equational theory given by some set of axioms. Equational unification being undecidable in general, it is important to identify equational theories and simple cases where it is possible to obtain sound, complete and terminating unification procedures. Matching is a particular (simple) case of unification of greatest interest to execute rule-based programs and to perform simplification in theorem proving. We show a systematic approach to build and to combine unification and matching algorithms for a large class of equational theories. We illustrate the approach on equational theories of practical interest in the analysis of security protocols.

15h - 16h
Edward Hermann Haeusler (PUC-Rio - Brazil) Title.Propositional Proofs Compression.
PDF | Abstract

This talk presents a novel compressing propositional proofs algorithm and discusses its relevance in solving some computational complexity well-known conjectures.

16h30 - 17h10
Daniele Nantes-Sobrinho (UNB/Imperial College London) Title.Matching Plans for Frame Inference in Compositional Reasoning.
PDF | Abstract

The use of function specifications and manipulation of user- defined predicates are two essential ingredients of modern compositional verification tools. To execute these operations successfully, these tools must be able to solve the frame inference problem, that is, understand the parts of the state relevant for the operation at hand. We introduce matching plans, which facilitate frame inference and advance the state-of- the-art in that they allow tool users to write intuitive and understandable specifications and the tools to efficiently infer the information required to perform frame inference.
(Joint work with Andreas Lööw, Petar Maksimovic, Philippa Gardner, Sacha Ayoun)

17h10 - 17h40
Gabriela de Souza Ferreira (UNB -Brazil) Title.Anti-Unification on Terms With Different Types.
PDF | Abstract

Anti-unification is the problem of expressing the most common structure between two given expressions. The different problems of anti-unification depend on how such commonalities are established and structured. In this talk the focus is in the universe of Typed Lambda Terms, more precisely, in how to identify the common structures of expressions that may have different types. First, a fast presentation of the anti- unification problem for inputs with the same type will be given, with the propose of highlighting the difference between the standard problem and well-known more general problems. Secondly, some preliminar ideas about how to solve anti-unification for terms with different types will be discussed. The presentation will use several examples and illustrations.

17h40 - 18h10
Andrés Felipe González Barragán (UNB -Brazil) Title.Anti-Unification on Absorption Theories.
PDF | Abstract

Anti-unification or generalization is the process of finding commonalities between expressions. There are many applications of generalization as clone and plagiarism detection. While syntactic forms of anti-unification are enough for many applications, some aspects of software analysis methods are more appropriately modeled by reasoning modulo equational theories. Some important theories include the absorption property; i.e., operators with axioms f(x, εf ) ≈ f(εf , x) ≈ εf. This work presents a sound and complete anti- unification algorithm for such theories. Additionally, it shows that anti- unification of absorption theories is of type infinitary, and provides a finitary representation of the minimal complete set of generalizations.

18h10 - 18h40
Guilherme Borges Brandão (UNB -Brazil) Title.Bounded ACh Unification.
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E-Unification is a procedure to find solutions to a set of equations between firstorder terms with respect to an equational theory E. ACh theory takes a function symbol h that is an homomorphism over an associative-commutative function symbol. Previous work proved that ACh Unification is an undecidable problem [1]. Recently, however, it was defined an approximation of the problem called Bounded ACh Unification [2], that consists in bounding the number of times that h is applied recursively. In this talk, we will discuss the inference rules for solving this problem and verify the proofs of termination, soundness and completeness.

Thusrday, February 8, 2024 .

14h - 15h
Chelsea Edmonds (University of Sheffield) Title: Formalising Combinatorial Mathematics: A Modular Approach.
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Formalised mathematics has historically focussed on the goal of formalising specific well-known theorems, which can lead to issues around reusability and accessibility, especially as formal libraries continue to grow at a rapid pace. My work on formalising combinatorics in Isabelle/HOL presents an approach which places the primary focus on the formalisation of proof techniques and underlying structures, making use of locales, Isabelle's module system. In this talk, I'll give an introduction to the locale-centric methodology for formalising mathematical structures to create flexible, modular and extensible libraries. Additionally, I'll demonstrate how the same locale-centric methods can be applied to set up proof contexts to support general proof techniques. I'll conclude by demonstrating an example probabilistic technique to formalise some results on hypergraphs.

15h - 16h
María Inés de Frutos-Fernández (University of Sheffield) Title: Formalising local fields in the Lean theorem prover.
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Local fields, and more generally fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued fields, as well as the abstract definition and some fundamental properties of local fields.
This is joint work with Filippo A. E. Nuccio.

16h30 - 17h10
Maurício Ayala-Rincón (UNB - Brazil) Title: Exemplifying Contemporary Formal Mathematics.
PDF | Abstract

Computer-verified theories are the class of "complete" formalizations mathematicians have aimed for centuries. Theorem's proofs formalized in "proof assistants" provide certificates of their "correctness" and the required grade of granularity to be considered "complete." Such proof's mechanizations also come with tools to be ported by mathematicians and other professionals into robust technological tools independently of mathematical intuition, skills, hierarchy, or folklore. Computational formalizations provide absolutely correct proofs except for remote possibilities of computer failures. In this talk, we will select simple mathematical formalizations developed by the local group members to illustrate how such formalizations are developed in the proof assistant PVS.

17h10 - 17h40
Daniella Santaguida Magalhães de Souza (UNB -Brazil) Title: Closed Rewriting - Checking overlaps of Nominal Rewriting Rules.
PDF | Abstract

Nominal rewriting is not complete for equational reasoning in general; however, closed nominal rewriting is complete for equational reasoning with closed axioms [1]. Intuitively, no free atom occurs in a closed term, and, as a natural assumption, closed axioms do not allow abstracted atoms to become free. In this talk, we will analyze the confluence of nominal rewriting systems (NRS), we will check whether closedness can be useful/essencial to guarantee the confluence of a NRS, and finally we will make some observations on possible extensions for proving confluence modulo equational theories.

17h40 - 18h10
Nikson Bernardes Fernandes Ferreira (UNB -Brazil) Title: How do PVS strategies help to prove correct float-point implementations?
PDF | Abstract

Numerical programs are usually taught using real arithmetic but implemented in floating-point arithmetic. The Floats domain does not inherit known Real properties such as distributivity and carries propagating representational errors that can lead to unexpected behaviors. It is hazardous for safety-critical applications such as air traffic management. Following a methodology introduced by M. Moscato, L. Titolo et al., given a logically verified PVS Real specification, it is possible to get a provably correct floating-point implementation, for which it is also possible to obtain mathematical certificates (proofs) ensuring its correct behavior. The proof structure of such certificates is suitable for being fully proved using PVS strategies. This talk will show how we develop PVS strategies to improve proofs' automation degree.

Friday February 9, 2024 .

14h - 15h
Daniel Lima Ventura (UFG -Brazil) Title: Quantitative Weak Linearisation.
PDF | Abstract

Weak-linear terms are such that a (syntactic) linearity condition is demanded only on functions that may be applied, i.e consumed during some reduction path, while non- appliable functions, i.e that persists in any reduction path, can be non-linear. Weak linearisation was originally defined through a static characterisation of virtual redices, based on (legal) paths computed from the (syntactical) term tree. We revisit this notion through a quantitative type system, in which minimal typings are characterised through a notion of tightness, where term constructors are classified as either consumable or persistent, thus allowing us to define an expansion relation, between general $\lambda$-terms and weak-linear $\lambda$-terms, whilst preserving normal forms by reduction.
(Joint work with Sandra Alves, presented at ICTAC2022)

15h - 16h
Deivid Vale (Radboud University, Nijmegen, NL) Title: A New Characterisation of BFFs via Higher-Order Rewriting.
PDF | Abstract

The class of basic feasible functionals (BFF) is the analog of PF (polynomial time functions) for type-two functionals, that is, functionals that can take (first-order) functions as arguments. BFF can be defined by means of oracle Turing machines of time bounded by a second-order polynomial. On the other hand, higher-order term rewriting provides an elegant formalism for expressing higher-order computation. In this talk, address the problem of characterizing the class BFF by higher-order term rewriting. Various kinds of interpretations for first-order term rewriting have been introduced in the literature for proving termination and characterizing (first-order) complexity classes. In this recent work, we consider a notion of cost-size interpretations for higher-order term rewriting and see definitions as ways of computing functionals. We then discuss a novel characterization of BFFs by means of higher-order rewriting and the higher-order interpretation method. Finally, we will shed light on the soundness and completeness of such rewriting-based characterizations of higher-order complexity classes.
This is a joint work with Patrick Baillot, Ugo Dal Lago, and Cynthia Kop.

15h - 16h
Ali Khăn Caires Ribeiro Santos (UNB - Brazil) Title: On the Nominal Semantics of Nominal Algebra with Fixed-Point Constraints.
PDF | Abstract

Nominal algebra is a framework for interpreting and reasoning about systems with binding, using nominal terms as its term language. Its derivation rules are subject to freshness conditions and extend first-order terms with object-level variables, meta-level variables, and constructs for binding, alphaequivalence, and capture avoidance. The semantics of nominal algebra are in the class of nominal sets, which have a rich structure for involving names, permutation, and name binding. Previous work has proven the soundness and completeness of nominal algebra, with applications including interpreting solutions to nominal (dis)unification problems. However, recent investigations show that the intentional semantics of nominal algebra with fixed-point constraints is not sound. This talk presents a counterexample and proposes solutions by exploring a subclass of nominal sets called strong nominal sets and necessary modifications in derivation rules.