I will discuss some recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge, and their connection to combinatorics (of coloured partitions) and representation theory (of affine Lie algebras and related algebras such as their W-algebra). 

Given a prime p and an elliptic curve E (say over Q), one can associate a "mod p Galois representation" of the absolute Galois group of Q by considering the natural action on p-torsion points of E.

In 1972, Serre showed that if the endomorphism ring of E is "minimal", then there exists a prime P(E) such that for all p>P(E), the mod p Galois representation is surjective. This raised an immediate question (now known as Serre's uniformity conjecture) on whether P(E) can be bounded as E ranges over elliptic curves over Q with minimal endomorphism rings.

I'll sketch a proof of this result, the current status of the conjecture, and (time permitting) some extensions of this result (e.g. to abelian varieties with appropriately analogous endomorphism rings).

We show the existence of an infinite family of complex saddle-points at large N, for the matrix model of the superconformal index of SU(N) N = 4 super Yang-Mills theory on S3 × S1 with one chemical potential τ. The saddle-point configurations are labelled by points (m,n) on the lattice Λτ = Z τ + Z with gcd(m, n) = 1. The eigenvalues at a given saddle are uniformly distributed along a string winding (m, n) times along the (A, B) cycles of the torus C/Λτ . The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and its values at (m,n) saddles are determined by Fourier averages of the latter along directions of the torus. The actions of (0,1) and (1,0) agree with that of pure AdS5 and the Gutowski-Reall AdS5 black hole, respectively. The actions of the other saddles take a surprisingly simple form. Generically, they carry non vanishing entropy. The Gutowski-Reall black hole saddle dominates the canonical ensemble when τ is close to the origin, and other saddles dominate when τ approaches rational points. 

This talk will be a survey on the applications of random matrix theory in neuroscience. We will explain why and how we use random matrices to model networks of neurons in the brain. We are mainly interested in the study of neuronal dynamics, and we will present results that cover two parallel directions taken by the field of theoretical neuroscience. First, we will talk about the critical point of transitioning to chaos in cases of random matrices that aim to be more "biologically plausible". And secondly, we will see how a deterministic and a random matrix (corresponding to learned structure and noise in a neuronal network) can interact in a dynamical system.

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Arakelov theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

Given a morphism of schemes of characteristic p, we can construct a morphism from the exterior algebra of Kahler differentials to the cohomology of De Rham complex, which is an isomorphism when the original morphism is smooth.

We present some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient a = a(·, ·). The model case is driven by the following nonlocal double phase operator,

$$\int \frac{|u(x) − u(y)|^{p−2} (u(x) − u(y))} {|x − y|^{n+sp}} dy+ \int a(x, y) \frac{|u(x) − u(y)|^{ q−2} (u(x) − u(y))} {|x − y|^{n+tq}} dy$$

where $q ≥ p$ and $a(·, ·) = 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

We will present three problems that we are interested in:

Forecast of volatility both at the instrument and portfolio level by combining a model based approach with data driven research
We will deal with additional complications that arise in case of instruments that are highly correlated and/or with low volumes and open interest.
Test if volatility forecast improves metrics or can be used to derive alpha in our trading book.

Price predication using physical oil grades data
Hypothesis:
Physical markets are most reflective of true fundamentals. Derivative markets can deviate from fundamentals (and hence physical markets) over short term time horizons but eventually converge back. These dislocations would represent potential trading opportunities.
The problem:
Can we use the rich data from the physical market prices to predict price changes in the derivative markets?
Solution would explore lead/lag relationships amongst a dataset of highly correlated features. Also explore feature interdependencies and non-linearities.
The prediction could be in the form of a price target for the derivative (‘fair value’), a simple direction without magnitude, or a probabilistic range of outcomes.

Modelling oil balances by satellite data
The flow of oil around the world from being extracted, refined, transported and consumed, forms a very large dynamic network. At both regular and irregular intervals, we can make noisy measurements of the amount of oil at certain points in the network.
In addition, we have general macro-economic information about the supply and demand of oil in certain regions.
Based on that information, with general information about the connections between nodes in the network i.e. the typical rate of transfer, one can build a general model for how oil flows through the network.
We would like to build a probabilistic model on the network, representing our belief about the amount of oil stored at each of our nodes, which we refer to as balances.
We want to focus on particular parts of the network where our beliefs can be augmented by satellite data, which can be done by focusing on a sub network containing nodes that satellite measurements can be applied to.