L4

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given 
instance. This lecture illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a gem from
the 19th century, namely the 3264 conics that are tangent to five given conics in the plane. Thereafter we turn to current problems in statistics, with focus on 
maximum likelihood estimation for linear Gaussian covariance models.
 

I will review the fishnet model, which is an integrable scalar QFT, obtained by an extreme gamma deformation of N=4 super Yang-Mills. The theory has a peculiar perturbative expansion in which many quantities at a fixed loop order are given by a single Feynman diagram. This feature allows the reinterpretation of Feynman loop integrals as integrable systems.

In this talk I will illustrate how solutions of nonlinear nonlocal Fokker-Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined outside such domain. Two different approaches are analysed, making crucial use of uniform estimates for energy and entropy functionals respectively. In both cases we prove that the problem in a bounded domain can be seen as a limit problem in the whole space involving a suitably chosen sequence of large confining potentials.
This is joint work with Maria Bruna and José Antonio Carrillo.
 

Let k be a characteristic $p>0$ perfect field, V be a complete DVR whose residue field is $k$ and fraction field $K$ is of characteristic $0$. We denote by $\mathcal{E}  _K$ the Amice ring with coefficients in $K$, and by $\mathcal{E} ^\dagger _K$ the bounded Robba ring with coefficients in $K$. Berthelot's classical theory of Rigid Cohomology over varieties $X/k((t))$ gives $\mathcal{E}  _K$-valued objects.  Recently, Lazda and Pal developed a refinement of rigid cohomology,
i.e. a theory of $\mathcal{E} ^\dagger _K$-valued Rigid Cohomology over varieties $X/k((t))$. Using this refinement, they proved a semistable version of the variational Tate conjecture. 

The purpose of this talk is to introduce to a theory of arithmetic D-modules with $\mathcal{E} ^\dagger _K$-valued cohomology which satisfies a formalism of Grothendieck’s six operations. 
 

Landmark-based human action recognition in videos is a challenging task in computer vision. One crucial step is to design discriminative features for spatial structure and temporal dynamics. To this end, we use and refine the path signature as an expressive, robust, nonlinear, and interpretable representation for landmark-based streamed data. Instead of extracting signature features from raw sequences, we propose path disintegrations and transformations as preprocessing to improve the efficiency and effectiveness of signature features. The path disintegrations spatially localize a pose into a collection of m-node paths from which the signatures encode non-local and non-linear geometrical dependencies, while temporally transform the evolutions of spatial features into hierarchical spatio-temporal paths from which the signatures encode long short-term dynamical dependencies. The path transformations allow the signatures to further explore correlations among different informative clues. Finally, all features are concatenated to constitute the input vector of a linear fully-connected network for action recognition. Experimental results on four benchmark datasets demonstrated that the proposed feature sets with only linear network achieves comparable state-of-the-art result to the cutting-edge deep learning methods. 

In the past decade, deep learning methods have achieved unprecedented performance on a broad range of problems in various fields from computer vision to speech recognition. So far research has mainly focused on developing deep learning methods for Euclidean-structured data. However, many important applications have to deal with non-Euclidean structured data, such as graphs and manifolds. Such data are becoming increasingly important in computer graphics and 3D vision, sensor networks, drug design, biomedicine, high energy physics, recommendation systems, and social media analysis. The adoption of deep learning in these fields has been lagging behind until recently, primarily since the non-Euclidean nature of objects dealt with makes the very definition of basic operations used in deep networks rather elusive. In this talk, I will introduce the emerging field of geometric deep learning on graphs and manifolds, overview existing solutions and outline the key difficulties and future research directions. As examples of applications, I will show problems from the domains of computer vision, graphics, high-energy physics, and fake news detection. 

The Ring Learning With Errors problem (RLWE) comes in various forms. Vanilla RLWE is the decision dual-RLWE variant, consisting in distinguishing from uniform a distribution depending on a secret belonging to the dual OK^vee of the ring of integers OK of a specified number field K. In primal-RLWE, the secret instead belongs to OK. Both decision dual-RLWE and primal-RLWE enjoy search counterparts. Also widely used is (search/decision) Polynomial Learning With Errors (PLWE), which is not defined using a ring of integers OK of a number field K but a polynomial ring Z[x]/f for a monic irreducible f in Z[x]. We show that there exist reductions between all of these six problems that incur limited parameter losses. More precisely: we prove that the (decision/search) dual to primal reduction from Lyubashevsky et al. [EUROCRYPT 2010] and Peikert [SCN 2016] can be implemented with a small error rate growth for all rings (the resulting reduction is nonuniform polynomial time); we extend it to polynomial-time reductions between (decision/search) primal RLWE and PLWE that work for a family of polynomials f that is exponentially large as a function of deg f (the resulting reduction is also non-uniform polynomial time); and we exploit the recent technique from Peikert et al. [STOC 2017] to obtain a search to decision reduction for RLWE. The reductions incur error rate increases that depend on intrinsic quantities related to K and f.

Based on joint work with Miruna Roșca and Damien Stehlé.

We consider a well-studied optimal execution problem under little assumptions on the underlying midprice process. We do so by using signatures from rough path theory, that allows converting the original problem into a more computationally tractable problem. We include a few numerical experiments where we show that our methodology is able to retrieve the theoretical optimal execution speed for several problems studied in the literature, as well as some cases not included in the literatture. We also study some estensions of our framework to other settings.