More Like this

1. A note on limiting Calderon–Zygmund theory for transformed n$$n$$‐Laplace systems in divergence form

   https://doi.org/10.1112/blms.13147  

   Martino, Dorian; Schikorra, Armin (November 2024, Bulletin of the London Mathematical Society)

   Abstract We consider rotated ‐Laplace systems on the unit ball of the formwhere , , and for some . We prove that with estimates. As a corollary, we obtain that solutions to , where is the Hardy space, have a higher integrability, namely, . 

   more » « less
2. Integrability of SLE via conformal welding of random surfaces

   https://doi.org/10.1002/cpa.22180  

   Ang, Morris; Holden, Nina; Sun, Xin (May 2024, Communications on Pure and Applied Mathematics)

   Abstract We demonstrate how to obtain integrability results for the Schramm‐Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating‐of‐trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called . Our proof is built on two connections between SLE, LCFT, and mating‐of‐trees. Firstly, LCFT and mating‐of‐trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call theuniform embeddingof an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper Ang, Holden and Sun (2023). Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating‐of‐trees based on these two connections. 

   more » « less
3. Chaotic roots of the modular multiplication dynamical system in Shor's algorithm

   https://doi.org/10.1103/PhysRevResearch.6.L032046  

   Patoary, Abu Musa; Vikram, Amit; Shou, Laura; Galitski, Victor (August 2024, Physical Review Research)

   Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the “maximally random” Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an “integrable” quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized $A$-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other “chaotic” quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity, and chaos in and via quantum algorithms. Published by the American Physical Society2024 

   more » « less
4. Maximally supersymmetric RG flows in 4D and integrability

   https://doi.org/10.1007/JHEP12(2021)119  

   Caetano, João; Peelaers, Wolfger; Rastelli, Leonardo (December 2021, Journal of High Energy Physics)

   A bstract We revisit the leading irrelevant deformation of $$ \mathcal{N} $$ N = 4 Super Yang-Mills theory that preserves sixteen supercharges. We consider the deformed theory on S 3 × ℝ . We are able to write a closed form expression of the classical action thanks to a formalism that realizes eight supercharges off shell. We then investigate integrability of the spectral problem, by studying the spin-chain Hamiltonian in planar perturbation theory. While there are some structural indications that a suitably defined deformation might preserve integrability, we are unable to settle this question by our two-loop calculation; indeed up to this order we recover the integrable Hamiltonian of undeformed $$ \mathcal{N} $$ N = 4 SYM due to accidental symmetry enhancement. We also comment on the holographic interpretation of the theory. 

   more » « less
5. Limits of semistatic trading strategies

   https://doi.org/10.1111/mafi.12366  

   Nutz, Marcel; Wiesel, Johannes; Zhao, Long (December 2022, Mathematical Finance)

   Abstract We show that pointwise limits of semistatic trading strategies in discrete time are again semistatic strategies. The analysis is carried out in full generality for a two‐period model, and under a probabilistic condition for multiperiod, multistock models. Our result contrasts with a counterexample of Acciaio, Larsson, and Schachermayer, and shows that their observation is due to a failure of integrability rather than instability of the semistatic form. Mathematically, our results relate to the decomposability of functions as studied in the context of Schrödinger bridges. 

   more » « less