Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. However, little is known about the collective dynamical phenomena involving topological signals. Typically, topological signals of a given dimension are investigated and filtered using the corresponding Hodge Laplacians. In this talk, I will introduce the topological Dirac operator that can be used to process simultaneously topological signals of different dimensions.  I will discuss the main spectral properties of the Dirac operator defined on networks, simplicial complexes and multiplex networks, and their relation to Hodge Laplacians.   I will show that topological signals treated with the Hodge Laplacians or with the Dirac operator can undergo collective synchronization phenomena displaying different types of critical phenomena. Finally, I will show how the Dirac operator allows to couple the dynamics of topological signals of different dimension leading to the Dirac signal processing of signals defined on nodes, links and triangles of simplicial complexes. 

Irreversible processes are accompanied by an increase in the internal entropy of a continuum, and as such the entropy production function is fundamental in determining the overall state of the system. In this talk, it will be shown that the entropy production function can be utilized for a variational analysis of certain dissipative continua in two different ways. Firstly, a novel unified Lagrangian-Hamiltonian formalism is constructed giving phase space extra structure, and applied to the study of fluid flow and brittle fracture.  Secondly, a maximum entropy production principle is presented for simple bodies and its implications to the study of fluid flow discussed. 

Second-order regularity results are established for solutions to elliptic equations and systems with the principal part having a Uhlenbeck structure and square-integrable right-hand sides. Both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, if the domain is convex, no regularity of its boundary is needed. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on joint investigations with A.Kh.Balci, L.Diening, and V.Maz'ya.

We introduce the mapping space signature, a generalization of the path signature for maps from higher dimensional cubical domains, which is motivated by the topological perspective of iterated integrals by K. T. Chen. We show that the mapping space signature shares many of the analytic and algebraic properties of the path signature; in particular it is universal and characteristic with respect to Jacobian equivalence classes of cubical maps. This is joint work with Chad Giusti, Vidit Nanda, and Harald Oberhauser.

Only a decade ago the detection of gravitational waves seemed like a fantasy to most, and merely a handful of 
people in the world believed in the validity and even great potential of using the powerful framework of EFT, and 
more generally -- advances in QFT to study gravity theory for real-world gravitational waves. I will present the 
significant advancement accomplished uniquely via the tower of EFTs with the EFT of spinning gravitating objects, 
and the incorporation of QFT advances, which my work has pioneered since those days. Today, only 6 years after 
the official birth of precision gravity with a rapidly growing influx of gravitational-wave data, and a decade of great 
theoretical progress, the power and insight of using modern QFT for real-world gravity have become incontestable.

We present a computationally tractable method for simulating arbitrage free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.
We illustrate our method with two examples. First, we propose a dynamic factor model for the implied volatility surface, and show how our method may be used to remove static arbitrage from model scenarios. As a second example, we propose a nonparametric generative model for implied volatility surfaces based on a Generative Adversarial Network (GAN).

In this talk, we consider the sensitivity to the model uncertainty of an optimization problem. By introducing adapted Wasserstein perturbation, we extend the classical results in a static setting to the dynamic multi-period setting. Under mild conditions, we give an explicit formula for the first order approximation to the value function. An optimization problem with a cost of weak type will also be discussed.