I will talk about recent joint work with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold.

We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.

Most networks and graphs encountered in empirical studies, including internet and web graphs, social networks, and biological and ecological networks, are very sparse.  Standard spectral and linear algebra methods can fail badly when applied to such networks and a fundamentally different approach is needed.  Message passing methods, such as belief propagation, offer a promising solution for these problems.  In this talk I will introduce some simple models of sparse networks and illustrate how message passing can form the basis for a wide range of calculations of their structure.  I will also show how message passing can be applied to real-world data to calculate fundamental properties such as percolation thresholds, graph spectra, and community structure, and how the fixed-point structure of the message passing equations has a deep connection with structural phase transitions in networks.

Unordered configuration spaces on (connected) manifolds are basic objects
that appear in connection with many different areas of topology. When the
manifold M is non-compact, a theorem of McDuff and Segal states that these
spaces satisfy a phenomenon known as homological stability: fixing q, the
homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)
denotes the space of k-point configurations and homology is taken with
coefficients in Z. However, this statement is in general false for closed
manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we
extend all the previously known results in this situation. One key idea is
to introduce so-called "replication maps" between configuration spaces,
which in a sense replace the "stabilisation maps" that exist only in the
case of non-compact manifolds. One corollary of our results is to recover a
"homological periodicity" theorem of Nagpal -- taking homology with field
coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is
eventually periodic in k -- and we obtain a much simpler estimate for the
period. Another result is that homological stability holds with Z[1/2]
coefficients whenever M is odd-dimensional, and in fact we improve this to
stability with Z coefficients for 3- and 7-dimensional manifolds.

We propose a non-perturbative formulation of structure constants of single trace operators in planar N=4 SYM.  We match our results with both weak and strong coupling data available in the literature. Based on work with Benjamin Basso and Pedro Vieira.

: Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale.

This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault.

We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2.

We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility.

Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it.

This sheds light on why long memory of volatility has been widely accepted as a stylized fact.

Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

This is joint work with Jim Gatheral and Thibault Jaisson.