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In this talk, we study a self-similar fragmentation process with index a>0, modeling the evolution of particles that break into smaller fragments over time. In this setting, the fragmentation rate of a particle of size u is proportional to u^a. We present asymptotic results describing the size of the largest fragment in the system. In particular, we establish a precise connection between the asymptotic behavior of the largest fragment and that of the so-called dislocation measure, which governs the underlying fragmentation mechanism. This is based on joint work with Samuel G. G. Johnston, Sandra Palau, and Joscha Prochno: https://arxiv.org/pdf/2409.11795

We study a natural generalization of the Unsplittable Flow Problem (UFP) on a path. We generalize the problem by allowing multiple parallel paths (machines). The objective is to find a profit-maximizing feasible assignment of n given tasks to m machines. Each task has a given machine-independent start and finish time, as well as a profit and a resource demand. Each machine is characterized by a given time-varying capacity. At any point in time, the total demand of tasks assigned to a machine must not exceed its available capacity. To the best of our knowledge, this is the first work to study the multiple path generalization of the UFP on a path. Assuming that all machines have uniform capacities and the number of machines is constant, we present a polynomial-time (2 + ϵ)-approximation algorithm for this NP-hard problem, derived by combining randomized rounding techniques with dynamic programming. Furthermore, under the no-bottleneck assumption (NBA) and assuming that the tasks are small, i.e., their demand-to-capacity ratio is less than a fixed value δ between their start and finish times for all the machines, we present a randomized-rounding-based polynomial-time constant factor approximation algorithm. Additionally, for the case of uniform capacities where demands are at most half of the capacity, we present a generalization of the "List Algorithm" given in [2]. We prove that this LP-rounding-based method achieves a polynomial-time 3-approximation. This result extends the 2-approximation from the single-machine setting, with the increased ratio reflecting the additional disjointness constraints required when assigning tasks across multiple parallel machines. Finally, we exhibit instances with unit demands that possess an integrality gap greater than 1, a phenomenon that does not occur in the standard UFP on a path (i.e., the single machine case).

Calderbank-Shor-Steane (CSS) codes, such as the toric code, are a widely studied class of quantum error-correcting codes. Understanding the thermalization time of these systems is important not only for error correction but also for applications like Gibbs sampling. We show that CSS codes on a lattice satisfy a modified logarithmic Sobolev inequality and thus thermalize rapidly in any dimension at sufficiently high temperatures. For a special subclass, including the toric code, this rapid thermalization even holds at all positive temperatures. The central idea underlying our approach is to exploit the structure of CSS codes to decompose a quantity into two simpler, (almost) classical components, allowing us to apply tools from classical statistical mechanics to analyze the thermalization. In the last part I will show how this method generalizes to 2D Abelian quantum double models. This is joint work with Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, Antonio Pérez-Hernández, Cambyse Rouzé and Simone Warzel.

Divergent asymptotic expansions are ubiquitous in mathematical physics, yet they often encode far more information than their formal nature suggests. In this talk, I will present ideas from resurgence theory, which provide a systematic way to reconstruct analytic functions from such expansions. As a an example, I will consider the exact WKB method, where asymptotic series arise as formal solutions to Schrödinger operators. Resurgence reveals how different analytic realizations of these series are related through Stokes phenomena—discrete jumps that encode nonperturbative effects, and geometrically encode changes of triangulations of an underlying Riemann Surface.