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.
Infinite-time convergence of geometric flows, as even for finite-dimensional gradient flows, is a notoriously subtle problem. The best (or only) bet is to get a Łojasiewicz(-Simon) inequality stating that a power of the gradient dominates the distance to the critical energy value. I'll introduce a Łojasiewicz inequality between the tension field and Dirichlet energy of a map from the 2-sphere to itself, removing the technical restrictions from an estimate of Topping (Annals ‘04). The inequality guarantees convergence of weak solutions of harmonic map flow from $S^2$ to $S^2$ assuming that the body map is nonconstant.
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.
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.
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.
Skeins on tori. Monica Vazirani, University of California, Davis.
Abstract
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.
