Abu Dhabi Stochastics Days

Abstract: We write G–> (H_1, H_2,…, H_k) if any colouring of the edges of a graph G with k colors results in a monochromatic copy of a graph  H_i for some i=1,2,…,k. In Ramsey Theory we want to find the smallest graph G with this property for a given k-tuple (H_1,H_2, …, H_k), where the size of G is measured by the number of its vertices or/and edges. In the talk we demonstrate how tools used in random graph theory, such as Sparse Regularity Lemma and Sandwich Theorem, can be applied to estimate Ramsey numbers for cycles and paths.

Abstract: Using tools from free probability, we describe an algorithm for finding the ground state energy in the Sherrington-Kirkpatrick model.  The SK model is an instance of a spin glass, where we consider a graph $(V,E)$ and (random) interaction weights for each pair of adjacent vertices, and study the possible configurations assigning $\pm 1$ to each vertex.  In the SK model, $G$ is a complete graph and the interactions are independent Gaussian, so the Hamiltonian is $H(\sigma) = \langle \sigma, A \sigma \rangle$, where $A$ is an $n \times n$ GOE (Gaussian orthogonal ensemble) random matrix and $\sigma \in \{-1,1\}^n$ is the spin configuration.  Thus, finding the ground state energy means maximizing $\langle \sigma, A \sigma \rangle$ over $\{-1,1\}^n$.  We want to find near-maximizers algorithmically in polynomial time in $n$ even though the size of the system is exponential in $n$.  We follow a similar approach to that which Subag used on the sphere; we extend the objective function $\langle \sigma, A \sigma \rangle$ to the convex hull $[-1,1]$ while adding a certain potential $\sum_j \Lambda(\sigma_j)$ on the interior.  Here $\Lambda$ is a function obtained from the Parisi formula that describes the large-$n$ behavior of the maximum.  The new objective function will have a path of near-maximizers from the origin to the somewhere near the corners of the cube.  We follow such an approximate path by a sequence of iterations which move in a random direction in the bottom part of the spectrum of the Hessian of the objective.  We are able to control the diagonal of the approximate spectral projection using a certain conditional expectation formula in free probability together with asymptotic expansions and concentration of measure.  Controlling the diagonal in turn allows us to show convergence of the empirical distribution of our vectors toward the Auffinger-Chen SDE and hence show near-optimality.

Abstract: Economic Fitness and Complexity (EFC) is the recent economic discipline and methodology we have developed in the past ten years. EFC makes use and develops the modern techniques of data analysis to build economic models based on a scientific methodology inspired by the science of Complex Systems with special attention to quantitative tests to provide a sound scientific framework. It consists of a data based and bottom up approach that considers specific and concrete problems without economic ideologies and it acquires information from the previous growth data of all countries with methods of Complex Networks, Algorithms and Machine Learning. Its main characteristics are the scientific rigor, the precision in the analysis and in the forecasting, transparency and adaptability. According to Bloomberg Views: “New research has demonstrated that the “fitness” technique systematically outperforms standard methods, despite requiring much less data”. Recently we have developed these methodologies to study also the impact of AI on the Job Market. Introducing the concept of Job Fitness, on average we observe an inverse proportionality between Job Fitness and AI impact. However, there are also important outliers which require additional considerations. Stimulated by these observations we have developed a radically different and more scientific approach to estimate these effects. This has led us to reconsider the standard information for the impact of AI on the skills. The common wisdom is based on subjective experts opinion while we have now introduced a new approach based on real Start Up investment, which is more objective and scientific and it leads to important differencies. In general, up to now we have considered mostly the analysis of countries. The present challenge is to extend these methodology also for firms. This requires new data and new concepts.