Talks

(Abstract): How is Artificial Intelligence changing your life and the world? How do you expect your data to be kept secure and private in the future? Artificial intelligence (AI) refers to the science of utilizing data to formulate mathematical models that predict outcomes with high assurance. Such predictions can be used to make decisions automatically or give recommendations with high confidence. Cryptography is the science of protecting the privacy and security of data. This talk will explain the dynamic relationship between cryptography and AI and how AI can be used to attack post-quantum cryptosystems.

(Abstract): This talk in on the method of modified energy for the a priori control of the H2-norm for strong solutions for dispersive equations.

(Abstract): A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG. This is research in collaboration with J. Dávila, A. Fernández, M. Musso and J. Wei.

(Abstract): Model Theory has been effective in mathematics in part due to its ability to deal with definability issues and linking them to structural properties of model classes - the most extreme example of such structural properties being categoricity. On the other hand, many important classes of mathematical structures do not quite naturally fit the constraints given by logics where model theory has its sharpest tools. Toward the end of the last century, newer ways of doing model theory, relying less directly on definability and more on controlling different sorts of embeddings between many structures in a class started to become more central. In more recent years, new ways of capturing syntax for "abstract" classes of models (given semantically) have emerged and seem to be starting turning the tables toward new possibilities.

(Abstract): In the last several decades there has been a gradual shift from first order model theory to its non-elementary versions. This is driven by discoveries both of new model-theoretic notions and constructions and of new applications in number theory and algebraic geometry. I will discuss one such development: categoricity and stability of abstract elementary classes and their connection to analytic aspects of algebraic/arithmetic geometry.

(Abstract): The characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms will be given. In particular, necessary and sufficient conditions of Titchmarsh type will be presented. The method is based on two-sided estimate for the rate of approximation of a β-admissible family of multipliers operators in terms of decay properties of Fourier transforms ([1]).

References:

[1] Jordão, T. Decay of Fourier transforms and generalized Besov spaces, Constructive Mathematical Analysis 3 (2020), 20–35

(Abstract): The interest on quaternionic linear operators stems from early works of 1930s (see, e.g. G. Birkhoff and J. von Neumann’s theory on quaternionic quantum mechanics). However, the lack of commutativity of the underlying algebra generates several problems. For once, it implies a distinction between left- or right-eigenvalues. The classic Grothendieck-Lidskii formula connects the spectrum of an operator with the Fredholm determinant and the trace of the operator. But while the concept of Fredholm determinant and trace for complex linear operators is well understood one lacks an appropriated analogue for linear operators with entries on higher dimensional algebras.

(Abstract): We study the exact boundary controllability of the nonlinear Benney-Luke equation at the right endpoint of a bounded domain, applying the control first to the first spatial derivative and to the second spatial derivative. We use the Hilbert Uniqueness Method to get the exact controllability of the linearized problem. From this fact and the Banach contraction fixed point principle, we establish the exact boundary controllability for the nonlinear Benney-Luke equation in bounded domains, for sufficiently small initial and final states.

(Abstract): Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. One can tropicalize algebraic varieties in this way to get polyhedral complexes that retain some information about the original varieties, and which can be combinatorially studied.

In this talk I will give a brief introduction to tropical geometry and some of its applications to enumerative algebraic geometry. In addition, I will talk about recent efforts to develop algebraic foundations for the field. In particular, I will present the notion of tropical ideals and discuss ongoing work studying some of their algebraic and geometric properties.

(Abstract): (based on joint projects with Frederick R. Cohen, Michael Crabb, Wolfgang Lück & Günter M. Ziegler) A great amount of fundamental leaps in mathematics come from the transfer of ideas and methods across fields. At first we will illustrate a part of a thick network connecting problems from various areas, like the Birkhoff (1927) problem on the existence of many closed orbits in smooth billiards, the Borsuk (1957) problem about the existence of regular embeddings, the Grünabaum–Hadwiger–Ramos (1960) hyperplane mass partition problem, the topological Tverberg problems (1981) of Bárány, Shlosman and Szücs, and the Smale (1987) problem of topological complexity of computational algorithms.

For this, we build an hourglass, decorating both bases with various problems and making connections between them through the network placed in the neck of the hourglass. The upper base of the hourglass contains embeddability problems like, the (non-)existence of k-fold immersions, k-regular- and k-skew-embeddings. The lower base contains more versatile questions, like counting of periodic billiard trajectories, the Nandakumar & Ramana-Rao partition problem, the Bárány–Larman conjecture, and the Farber’s topological complexity.

To illustrate the power of the hourglass we will start with problems from both sides of hourglass. Slowly, while turning the hourglass we will come to the neck and unravel new open questions about the topology of unordered configuration spaces. Answering these questions, from the neck of the hourglass, we will push out and solve both problems at the same time.