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I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis.

We show how traders use immediate execution limit orders (IELOs) to liquidate a position when the time between a trade attempt and the outcome of the attempt is random, i.e., there is latency in the marketplace and latency is random. We frame our model as a delayed impulse control problem in which the trader controls the times and the price limit of the IELOs she sends to the exchange. The contribution of the paper is twofold: (i) Our paper is the first to study an optimal liquidation problem that accounts for random delays, price impact, and transaction costs. (ii) We introduce a new type of impulse control problem with stochastic delay, not previously studied in the literature. We characterise the value functions as the solution to a coupled system of a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) and a partial differential equation. We use a Feynman-Kac type representation to reduce the system of coupled value functions to a non-standard HJBQVI, and we prove existence and uniqueness of this HJBQVI in a viscosity sense. Finally, we implement the latency-optimal strategy and compare it with three benchmarks:  (i)  optimal execution with deterministic latency, (ii) optimal execution with zero latency, (iii) time-weighted average price strategy. We show that when trading in the EUR/USD currency pair, the latency-optimal strategy outperforms the benchmarks between ten USD per million EUR traded and ninety USD per million EUR traded.

Smectic A liquid crystals are of great interest in physics for their striking defect structures, including curvature walls and focal conics. However, the mathematical modeling of smectic liquid crystals has not been extensively studied. This work takes a step forward in understanding these fascinating topological defects from both mathematical and numerical viewpoints. In this talk, we propose a new (two- and three-dimensional) mathematical continuum model for the transition between the smectic A and nematic phases, based on a real-valued smectic order parameter for the density perturbation and a tensor-valued nematic order parameter for the orientation. Our work expands on an idea mentioned by Ball & Bedford (2015). By doing so, physical head-to-tail symmetry in half charge defects is respected, which is not possible with vector-valued nematic order parameter.

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Motivated by the advent of machine learning, the last few years saw the return of hardware-supported low-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but can be extremely susceptible to rounding errors. An application that can largely benefit from the advantages of low-precision computing is the numerical solution of partial differential equations (PDEs), but a careful implementation and rounding error analysis are required to ensure that sensible results can still be obtained. In this talk we study the accumulation of rounding errors in the solution of the heat equation, a proxy for parabolic PDEs, via Runge-Kutta finite difference methods using round-to-nearest (RtN) and stochastic rounding (SR). We demonstrate how to implement the numerical scheme to reduce rounding errors and we present \emph{a priori} estimates for local and global rounding errors. Let $u$ be the roundoff unit. While the worst-case local errors are $O(u)$ with respect to the discretization parameters, the RtN and SR error behaviour is substantially different. We show that the RtN solution is discretization, initial condition and precision dependent, and always stagnates for small enough $\Delta t$. Until stagnation, the global error grows like $O(u\Delta t^{-1})$. In contrast, the leading order errors introduced by SR are zero-mean, independent in space and mean-independent in time, making SR resilient to stagnation and rounding error accumulation. In fact, we prove that for SR the global rounding errors are only $O(u\Delta t^{-1/4})$ in 1D and are essentially bounded (up to logarithmic factors) in higher dimensions.

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We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin–Smoluchowski type, the Fokker–Planck probability density flow maximizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in the backward direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker–Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. As a bonus of our trajectorial approach we derive the HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).

Research on robot manipulation has focused, in recent years, on grasping everyday objects, with target objects almost exclusively rigid items. Non–rigid objects, as textile ones, pose many additional challenges with respect to rigid object manipulation. In this seminar we will present how we can employ topology to study the ``state'' of a rectangular textile using the configuration space of $n$ points on the plane. Using a CW-decomposition of such space, we can define for any mesh associated with a rectangular textile a vector in an euclidean space with as many dimensions as the number of regions we have defined. This allows us to study the distribution of such points on the cloth and define meaningful states for detection and manipulation planning of textiles. We will explain how such regions can be defined and computationally how we can assign to any mesh the corresponding region. If time permits, we will also explain how the CW-structure allows us to define more than just euclidean distance between such mesh-distributions.

I will discuss various operator algebras in supersymmetric quantum field theories in various dimensions. The operator algebras are induced and classified by generalised topological twists. Omega deformation plays an important role in connecting different sectors. This talk is based on previous works and a work in progress with Junya Yagi.

In this talk I will present an optical theorem for perturbative CFTs, which directly computes the double discontinuity of CFT correlators in terms of the discontinuities of correlators at lower loops or lower points, in analogy to the optical theoreom for scattering amplitudes. I will then discuss the application of this theorem to high-energy scattering of type IIb strings in AdS at one loop and finite 't Hooft coupling. Tidal excitations are taken into account and shown to be efficiently described by an AdS vertex function. The result is related to the 1987 flat space result of Amati, Ciafaloni and Veneziano via the flat space limit in impact parameter space.