We introduce the mapping space signature, a generalization of the path signature for maps from higher dimensional cubical domains, which is motivated by the topological perspective of iterated integrals by K. T. Chen. We show that the mapping space signature shares many of the analytic and algebraic properties of the path signature; in particular it is universal and characteristic with respect to Jacobian equivalence classes of cubical maps. This is joint work with Chad Giusti, Vidit Nanda, and Harald Oberhauser.

Only a decade ago the detection of gravitational waves seemed like a fantasy to most, and merely a handful of 
people in the world believed in the validity and even great potential of using the powerful framework of EFT, and 
more generally -- advances in QFT to study gravity theory for real-world gravitational waves. I will present the 
significant advancement accomplished uniquely via the tower of EFTs with the EFT of spinning gravitating objects, 
and the incorporation of QFT advances, which my work has pioneered since those days. Today, only 6 years after 
the official birth of precision gravity with a rapidly growing influx of gravitational-wave data, and a decade of great 
theoretical progress, the power and insight of using modern QFT for real-world gravity have become incontestable.

We present a computationally tractable method for simulating arbitrage free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.
We illustrate our method with two examples. First, we propose a dynamic factor model for the implied volatility surface, and show how our method may be used to remove static arbitrage from model scenarios. As a second example, we propose a nonparametric generative model for implied volatility surfaces based on a Generative Adversarial Network (GAN).

In this talk, we consider the sensitivity to the model uncertainty of an optimization problem. By introducing adapted Wasserstein perturbation, we extend the classical results in a static setting to the dynamic multi-period setting. Under mild conditions, we give an explicit formula for the first order approximation to the value function. An optimization problem with a cost of weak type will also be discussed.

Quantum field theories (QFTs) in d dimensions that posses a (d-1)-form symmetry are conjectured to decompose into disjoint “universes”, each of which is itself a (local and unitary) QFT. I will give an overview of our current understanding of decomposition, and then discuss how this phenomenon occurs in the fusion of condensation defects of certain 3d QFTs. This gives a “microscopic” explanation of why in these instances, the fusion coefficient can be taken as an integer rather than a general TQFT.

Donaldson-Thomas theory is classically defined for moduli spaces of sheaves over a Calabi-Yau threefold. Thanks to recent foundational work of Cao-Leung, Borisov-Joyce and Oh-Thomas, DT theory has been extended to Calabi-Yau 4-folds. We discuss how, in this context, one can define natural K-theoretic refinements of Donaldson-Thomas invariants (counting sheaves on Hilbert schemes) and Pandharipande-Thomas invariants (counting sheaves on moduli spaces of stable pairs) and how — conjecturally — they are related. Finally, we introduce an extension of DT invariants to Calabi-Yau 4-orbifolds, and propose a McKay-type correspondence, which we expect to be suitably interpreted as a wall-crossing phenomenon. Joint work (in progress) with Yalong Cao and Martijn Kool.

Over the last years, two different approaches to construct symmetry algebras acting on the cohomology of Nakajima quiver varieties have been developed. The first one, due to Maulik and Okounkov, exploits certain Lagrangian correspondences, called stable envelopes, to generate R-matrices for an arbitrary quiver and hence, via the RTT formalism, an algebra called Yangian. The second one realises the cohomology of Nakajima varieties as modules over the cohomological Hall algebra (CoHA) of the preprojective algebra of the quiver Q. It is widely expected that these two approaches are equivalent, and in particular that the Maulik-Okounkov Yangian coincides with the Drinfel’d double of the CoHA.

Motivated by this conjecture, in this talk I will show how to identify the stable envelopes themselves with the multiplication map of a subalgebra of the appropriate CoHA. 

As an application, I will introduce explicit inductive formulas for the stable envelopes and use them to produce integral solutions of the elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equation associated to arbitrary quiver (ongoing project with G. Felder and K. Wang). Time permitting, I will also discuss connections with Cherkis bow varieties in relation to 3d Mirror symmetry (ongoing project with R. Rimanyi).