Seminars from until

O desenvolvimento de Modelos Estatísticos para Detecção de Fraudes em Testes tem ganhado relevância nos últimos, particularmente aqueles baseados na Teoria da Resposta ao Item (TRI). Exames e avaliações podem ter suspeitas de fraude associadas se os resultados estiverem vinculados a vantagens financeiras ou vagas em instituições de ensino. Serão apresentados os principais modelos, comportamentos estatísticos associados, desempenho computacional para execução dos mesmos e uma aplicação a dados reais. Foi construído um pacote computacional no R que será apresentado e disponibilizado ao público.

In this work we consider the initial value problem for the cubic Schrödinger equation, posed on cylinder $\mathbb{R}× \mathbb{T}$, with fractional derivatives in the periodic direction. The spatial operator includes elliptic and hyperbolic regimes. We prove $L^2$ global well-posedness results in the case of higher order derivatives in the periodic direction by proving a $L^4× L^2$ Strichartz inequality.

Further, these results remain valid on the euclidean environment $\mathbb{R}^2$, so well-posedness in $L^2$ are also achieved in this case.

Our proof in the elliptic/hyperbolic case does not work for small order derivatives in the periodic direction.

Warm-up question: Given two binary strings $s\in \{0,1\}^n$ and $t\in \{0,1\}^{n+k}$, we say $s$ is the result of a k-deletion on $t$ if $s$ is a (not necessarily contiguous) substring of $t$, which we denote by $s\subseteq t$. Given $s\in \{0,1\}^n$ and $k\in \mathbb{N}$, consider
\begin{equation*}
A_{s,k} =\{ t\in \{0,1\}^{n+k} \colon s\subseteq t \}.
\end{equation*}
Determine the size of $A_{s,k}$ in terms of $s$ and $k$.

Bootstrap percolation is a classical statistical physics model displaying metastable behaviour. Let each site of the square lattice be infected independently with a fixed probability. At each round, infect each site with at least two infected neighbours and do not remove any infections. How long does it take before the origin is infected? We start by reviewing the rich history of this problem and some of the classical arguments used to tackle it. We then give a very precise answer to the above question in the relevant regime of sparse infection. The key to the proof is a new locality approach to bootstrap percolation, which also resolves the bootstrap percolation paradox concerning the failure of numerical predictions in the field. The talk is based on joint work with Augusto Teixeira available at https://arxiv.org/abs/2404.07903.

Nonstandard analysis (NSA), founded by Abraham Robinson in the 1960s, was to a great extent inspired by Leibniz’s ideas and intuitions towards the use of infinitesimal and infinitely large quantities. One of the greatest features of NSA is that, by allowing a correct formulation of infinitesimals, one is now able to reason using orders of magnitude. This means that one can give precise meaning, and reason formally, about otherwise vague terms such as "small" or "large". Recently, accounts of vagueness relying on NSA were introduced [2, 8, 4]. In particular, and unlike other accounts of vagueness, the so-called nonstandard primitivist account [4, 5] embraces transitivity for marginal differences (i.e. "small" differences), but not for large differences in a soritical series. Nonstandard primitivism also seems to be particularly adequate to deal with the ship of Theseus paradox [3, 6] and may also shed some light in doxastic reasoning by considering infinitesimal probabilities and associating them to infinitesimal credences [1, 7]. We aim at assessing the relative merits of nonstandard primitivism and to show some lines of future research regarding the connections between NSA and philosophy.

(This is joint work with Bruno Jacinto)

1. Benci, Vieri and Horsten, Leon and Wenmackers, Sylvia. Infinitesimal Probabilities. The British Journal for the Philosophy of Science, 69 (2): 509–552, 2018.
2. Walter Dean. Strict finitism, feasibility, and the sorites. Review of Symbolic Logic, 11 (2):295–346, 2018.
3. Bruno Dinis. Equality and near-equality in a nonstandard world. Log. Log. Philos., 32 (1):105–118, 2023. ISSN 1425-3305,2300-9802.
4. Bruno Dinis and Bruno Jacinto. A theory of marginal and large difference. Erkenntnis, 2023.
5. Bruno Dinis and Bruno Jacinto. Marginality scales for gradable objects. (preprint), 2023.
6. Bruno Dinis and Bruno Jacinto. Counterparts as Near-equals. (preprint), 2024.
7. Kenny Easwaran. Regularity and Hyperreal Credences. Philosophical Review, 123 (1):1- 41, 2014.
8. Yair Itzhaki. Qualitative versus quantitative representation: a non-standard analysis of the sorites paradox. Linguistics and Philosophy, 44:1013–1044, 2021.

In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this talk, I will discuss some aspects of this modeling problem. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum, Marvin Qi and Daniel Spiegel, David Stephen, Xueda Wen.

Severals SPDEs arise from SDE dynamics under partial conditioning of the noise. My talk will circulate on three concrete examples, the Zakai equation from non-linear filtering, the pathwise control problem suggested by Lions-Sougandis, and last not least a rough PDE approach to pricing in non-Markovian stochastic volatility models. Underlying all these examples is the notion of rough stochastic differential equations, recently introduced by K. Lê, A. Hocquet and the speaker.

Many natural systems can be maintained in a stationary state through the exchange of matter, energy or information with their surroundings. These various currents break time-reversal invariance, generating a continuous increase of entropy in the universe. Such systems are out of equilibrium and can not be described by the Laws of Thermodynamics, or by using the classical principles of statistical physics, à la Gibbs-Boltzmann. In the last decades, however, important advances in our understanding of non-equilibrium processes have been achieved. Concepts of rares events, large deviations, fluctuations relations and macroscopic fluctuations provide a unified framework with the emergence of some universal features. The objective of this talk is to review these new ideas in non-equilibrium statistical physics and to illustrate them by examples inspired from soft-condensed matter.

The asymmetric simple exclusion process is a model of interacting particles that appears in many realistic descriptions of low-dimensional transport with constraints and plays the role of a paradigm to understand the behaviour of non-equilibrium systems. The aim of this talk is to review some representative exact results about this model, to describe the methods involved and present some recent developments. In particular, by using the mathematical arsenal of integrable probabilities developed to solve the one-dimensional Kardar-Parisi-Zhang equation, we shall derive the exact finite-time distribution of a tagged particle in the symmetric simple exclusion process.

The main aim of this talk is to discuss the existence of nontrivial (and non semi-trivial) least energy solutions for a Neumann elliptic system with a critical nonlinearity, characterized by a cooperative-competitive behaviour, namely
\[ \begin{cases} -\Delta u+\lambda_1 u=u^3+β uv^2 & \text{ in } \Omega\\ -\Delta v+\lambda_2 v=v^3+\beta u^2v & \text{ in } \Omega\\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial \nu}=0 &\text{ on } \partial\Omega,\\ \end{cases} \label{eq:pbeta} \tag{$\mathcal P_\beta$} \] where $\Omega\subset \mathbb{R}^4$ is a $C^2$ bounded domain, and $\lambda_1,\lambda_2>0$ and the parameter $\beta\in\mathbb{R}$ captures the essence of cooperation-competition, assuming positive or negative values respectively.

The approach is variational and the idea is to minimize the energy functional on a suitable manifold of the Nehari type. In addition, to deal with the critical power, we estimate the energy level, using the solutions of $-\Delta w=w^3$ in $ \mathbb{R} ^4$ and the solution for the scalar equations $-\Delta u_i+\lambda_iu_i=u_i^3$ in $\Omega$, to establish a compactness condition based on the classical Cherrier's inequality: if $\partial\Omega\in C^1$ then for each $\varepsilon\gt 0$ there exists $M_{\varepsilon}>0$ such that $$\|u\|_{2^*}\le\left(\frac{2^{2/N}}{S}+\varepsilon\right)^{1/2}\|\nabla u\|_2+M_{\varepsilon}\|u\|_2,\ \ ∀u\in H^1(\Omega),$$ where $S$ is the best Sobolev constant. Additionally, I will discuss the more difficult cases in which $\lambda_1,\lambda_2\le0$, that I have started to study recently.

Gianni Jona-Lasinio and his collaborators have proposed in the early 2000’s a non-linear action functional that encodes the macroscopic fluctuations and the large deviations for a wide class of diffusive systems out of equilibrium, by generalizing a variational principle due to Kipnis, Olla and Varadhan. This theory, called the Macroscopic Fluctuation Theory (MFT) shows that large deviations far from equilibrium can be found by solving two coupled non-linear hydrodynamic equations. In this talk, we shall show that the MFT equations for the symmetric exclusion process are classically integrable and can be solved with the help of the inverse scattering method, originally developed to study solitons in dispersive non-linear wave equations (such as KdV or NLS).

EMV is the international protocol standard for smartcard payments and is used in billions of payment cards worldwide. Despite the standard’s advertised security, various issues have been previously uncovered, deriving from logical flaws that are hard to spot in EMV’s lengthy and complex specification, running over 2,000 pages.

We have formalized various models of EMV in Tamarin, a symbolic model checker for cryptographic protocols. Tamarin was extremely effective in finding critical flaws, both known and new. For example, we discovered multiple ways that an attacker can use a victim's EMV card (e.g., Mastercard or Visa Card) for high-valued purchases without the victim's supposedly required PIN. Said more simply, the PIN on your EMV card is useless! We report on this, as well as followup work with an EMV consortium member on verifying the latest, improved version of the protocol, the EMV Kernel C-8. Overall our work provides evidence that security protocol model checkers like Tamarin have an essential role to play in developing real-world payment protocols and that they are up to this challenge.

Phishing remains one of the most effective cyber threats, affecting millions of organizations. Phishing education, training, and awareness programs are used to address employees’ lack of knowledge about phishing attacks. However, despite being very expensive, these interventions are not always effective, mainly due to the lack of customization of training materials based on the employees’ needs and profiles. In fact, creating customized training content for each employee and each context would require a huge effort from security practitioners and educators thus increasing costs even more. The proposal we present in this talk is to use Large Language Models to automate some steps in the design process of training content, which is tailored to the specific user profile. Joint work with Giuseppe Desolda and Francesco Greco of the University of Bari.

In this talk, we will discuss a recent result concerning variational problems that encompass both bulk and interface energies, which are used to describe a broad spectrum of phenomena in applied sciences. A key feature of these models is the presence of a free boundary whose regularity is intricate to establish due to the interaction between the bulk term and the perimeter term. We will present an $\varepsilon$-regularity result for almost-minimizers of a class of problems with bulk energy of Dirichlet type and surface energy exhibiting anisotropic Finsler behavior, defined by an ellipsoidal density that is Hölder continuous with respect to the position variable. This presentation is based on joint work with L. Esposito and L. Lamberti.

Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.

A topological field theory (TFT) with particles exhibits distinguished state spaces, where the incoming and outgoing particles match. These "endo-state spaces" occur naturally in physical applications and possess interesting mathematical structures: There is a natural gauge action by conjugation and a natural stabilization map. We will show that the gauge action has a non-trivial orbit structure, leading to quiver moduli spaces, and the stabilization map leads to a treatment of infinite particle content and AF-algebras.

The talk will be rather introductory and assumes no knowledge of quivers or AF-algebras.

I will present some models in ecology and epidemiology using a transport equation approach, so called structured models. The first models are of predator-prey type and include a variable hunger structure. They take the form of nonlocal transport equations coupled to ODEs. Then, we use a similar approach in an epidemiological model including disease awareness and variable susceptibility. We show well-posedness results, asymptotic behavior, and numerical simulations. This is joint work with C. Rebelo, A. Margheri, and P. Lafargeas.

Topological quantum field theories (TQFTs) have attracted much attention from the physics and mathematical communities over the last thirty years, and for good reason: in low dimensions they let simple topology inform less-understood algebraic constructions. In the first half of this talk we will introduce factorization homology, a powerful procedure for constructing TQFTs out of homotopical gadgets, called $\mathsf{E}_n$-algebras. We explore this in dimension $n=1$, by using string-nets. In the second half of the talk, we introduce TQFTs with defects and factorization homology for stratified spaces and for an appropriate notion of stratified $\mathsf{E}_n$-algebra. Once again, we focus on dimension $n=1$, and will end by offering a conjectural connection between string-nets on stratified cylinders, Drinfel'd centres for bimodule categories and quasiparticles in a topological quantum computer.

A what?!

We will give a gentle introduction to gerbes and other assorted "higher structures" from topology and mathematical physics. Gerbes are a generalization of line bundles. For a line bundle, the space of sections forms a vector space, and a little extra geometric structure can make it into a Hilbert space, beloved by quantum physicists everywhere. After introducing gerbes, we will ponder the analogous construction: how do we define a "Hilbert space of sections" for a gerbe?

This expository talk is based on the work of Bunk and Szabo.

A mathematical model for the pitch of a plate is given by the first eigenvalue of the bilaplacian over the domain representing the plate at rest. Therefore, the question in title amounts to finding the domain for which the first eigenvalue of the bilaplacian is minimal. To address this shape optimization problem, we will introduce classic tools such as symmetrization techniques and comparison principles.

We will discuss some recent results concerning weighted eigenvalue problems in bounded Lipschitz domains, under Neumann boundary conditions.
The optimization of the distribution of resources leads to minimize a principal eigenvalue with respect to the sign-changing weight. Important qualitative properties of the positivity set of the optimal weight, such as being connected, as well as its location, are still not known in general.

We will present some new achievements in the asymptotical study regarding these properties.

Joint works with Dario Mazzoleni (Università di Pavia), Lorenzo Ferreri (Scuola Normale Superiore di Pisa) and Gianmaria Verzini (Politecnico di Milano).

The intermediate long wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, providing a natural connection between two famous water-wave equations: the Benjamin-Ono equation (= the deep-water regime) and the Korteweg-de Vries equation (= the shallow-water regime).

I will discuss how to exploit the completely integrable nature of the equation to establish the statistical convergence of ILW to both BO and KdV, namely, the convergence of the higher order conservation laws for ILW and of their associated invariant measures. Here, we observe a stark contrast between the two regimes, with distinct modes of convergence as well as a novel 2-to-1 collapse of the invariant ILW dynamics to the corresponding invariant KdV dynamics.

Feynman graph integrals of topological field theories have been proved to be ultraviolet finite by Axelrod and Singer, and Kontsevich independently. This result leads to many applications including universal finite type knot invariants and the formality of $E_n$ operads. In this talk, I will extend the finiteness results (and some anomaly cancellation results) to Feynman graph integrals of topological-holomorphic theories on flat spaces. The main technique for the proof is compactification of the moduli space of metric graphs. As a result, we can construct many factorization algebras from quantum topological-holomorphic theories. In the special case of 4d Chern–Simons theory, the factorization algebra structure encodes the Yang–Baxter equation. If time permits, I will sketch how to extend these results to Feynman graph integrals on Kähler manifolds. Part of this work is joint with Brian Williams.

Reference: https://arxiv.org/abs/2401.08113

Sharp restriction theory and the finite field extension problem have both received much attention in the last two decades, but so far they have not intersected. In this talk, we discuss our first results on sharp restriction theory on finite fields. Even though our methods for dealing with paraboloids and cones borrow some inspiration from their euclidean counterparts, new phenomena arise which are related to the underlying arithmetic and discrete structures. The talk is based on recent joint work with Cristian González-Riquelme.

We study skeins on the 2-torus and 3-torus via the representation theory of the double affine Hecke algebra of type A and its connection to quantum D-modules. As an application we can compute the dimension of the generic $SL_N$- and $GL_N$-skein module of the 3-torus for arbitrary N. This is joint work with Sam Gunningham and David Jordan.

For measuring possible concentrations of the eigenfunctions of the Laplace operator on a manifold, Burq-Gerard-Tzvetkov studied $L^p$ norm of the restrictions of the eigenfunctions to submanifolds. They proved sharp $L^p$ estimates restricted to the geodesic or a curve having nonvanishing geodesic curvature. I will talk about $L^p$ estimates restricted to a curve which is not geodesic and has vanishing geodesic curvature. The proof involves semiclassical analysis.

We define the atomic Hardy space $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$, $0\lt p≤ 1$, for the twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy space defined using the maximal function corresponding to the heat semigroup $e^{-\mathcal{L}t}$, $t\gt 0$. We also prove sharp $L^p$, $0\lt p≤ 1$, estimates for $\mathcal{L}^{β/2}e^{i\sqrt{\mathcal{L}}}$. More precisely, we prove that it is a bounded operator on $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$ when $β≥(2n-1)(1/p-1/2)$.

We consider a torus embedded in the 3-dimensional Euclidean space. It has a natural two-parameter scaling structure. Under this structure, we can consider a two parameter maximal average over the tori. We study the sharp boundedness of this maximal function, its Sobolev regularity, and local smoothing properties. We compare this result with the one parameter maximal function.

We will discuss a weak universality phenomenon in the context of two-dimensional fractional nonlinear wave equations. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $Φ_2^4$, we will present a sufficient and almost necessary criteria for the convergence of invariant measures to the fractional $Φ_2^4$. Then we will discuss the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. This extends a result of Gubinelli-Koch-Oh to a situation where we do not have any local Cauchy theory with highly supercritical nonlinearities. This is a joint work with Chenmin Sun and Weijun Xu.