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Discrete microlocal Morse theory
No full textWe establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities
Stuck in uncertainty: A predictive processing/ active inference account of procrastination-like behaviour in autism
No full textUn fenómeno a menudo asociado con el autismo es un modo atípico de función ejecutiva, cuyas manifestaciones incluyen dificultad para iniciar tareas. En algunos casos, esto va acompañado de sentimientos de inercia y sensaciones que pueden describirse como inquietud y parálisis simultáneas. En consecuencia, la dificultad para iniciar las tareas puede dar lugar a la procrastinación, ya sea simplemente posponiendo el trabajo en la tarea objetivo o realizando otras tareas no relacionadas antes de dedicarse a la tarea objetivo. Curiosamente, sin embargo, también está documentado que, una vez iniciada una tarea, los autistas pueden centrarse en ella intensamente y durante periodos prolongados de tiempo, especialmente cuando les resulta interesante.
Este trabajo utiliza el procesamiento predictivo y la inferencia activa para modelar la relación entre la función ejecutiva, la procrastinación y la hiperfocalización en el autismo. Este modelo integra las causas conocidas y propuestas de los déficits en la función ejecutiva y el papel que desempeña el interés en la regulación de la atención y la motivación. El modelo propone que la procrastinación es el resultado de procesos diferenciales de minimización de errores de predicción, como la ponderación de estímulos sensoriales. Se discuten los vínculos con modelos propuestos previamente, como la coherencia central débil (CCC), y la teoría de los priores altos e inflexibles de los errores de predicción en el autismo (HIPPEA).A phenomenon often associated with autism is an atypical mode of executive function, manifestations of which include difficulty in initiating tasks. In some cases, this is accompanied by feelings of inertia and sensations that can be described as simultaneous restlessness and paralysis. Consequently, difficulty in getting started on tasks can result in procrastination, either by simply postponing working on the target task or by performing other unrelated tasks before engaging in the target task. Interestingly, however, it is also documented that once a task has been started, autistic persons may focus on it intensely and for prolonged periods of time, especially when it is interesting to them.This paper uses predictive processing and active inference to model therelationship between executive function, procrastination, and hyperfocus in autism. This model integrates the known and proposed causes of deficits in executive function and the role played by interest in attention regulation and motivation. The model proposes that procrastination is the outcome of differential prediction-error minimizing processes, such as weighting of sensory stimuli. Links to previously proposed models such as weak central coherence (WCC), and the theory of high, inflexible priors of prediction errors in autism (HIPPEA) are discusse
PyDaddy: A Python Package for Discovering SDEs from Time Series Data
No full textPyDaddy is an open source package which is a key contribution of the manuscript Nabeel et al, arXiv:2205.02645. The basic scientific premise for this package is to discover the nature of stochasticity in ecological time series datasets. It is well known that the stochasticity can affect the dynamics of ecological systems in counter-intuitive ways. Without understanding the equations (typically, in the form of stochastic differential equations or SDEs, in short) that govern the dynamics of populations or ecosystems, it's challenging to determine the impact of randomness on real datasets. In this manuscript and accompanying package, we introduce a methodology for discovering equations (SDEs) that transforms time series data of state variables into stochastic differential equations. This approach merges traditional stochastic calculus with modern equation-discovery techniques. We showcase the generality of our method through various applications and discuss its limitations and potential pitfalls, offering diagnostic measures to address these challenges
Response theory for locally gapped systems
No full textWe introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings.
On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system.
More precisely, we say that an equilibrium state ρ0 of a Hamiltonian H0 is locally gapped in Λgap⊂Λ, whenever the Liouvillian −i[H0,⋅] is almost invertible on local observables supported in Λgap when tested in ρ0.
To put this into context, we provide other alternative notions of a local gap and discuss their relations.
The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs).
By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation acts in a region which is further than |logϵ| away from the non-gapped region Λ∖Λgap
ISTA Thesis
No full textInstant messaging applications like Whatsapp, Signal or Telegram have become ubiquitous in today's society.
Many of them provide not only end-to-end encryption, but also security guarantees even when the key material gets compromised.
These are achieved through frequent key update performed by users.
In particular, the compromise of a group key should preserve confidentiality of previously exchanged messages (forward secrecy), and a subsequent key update will ensure security for future ones (post-compromise security).
Though great protocols for one-on-one communication have been known for some time, the design of ones that scale efficiently for larger groups while achieving akin security guarantees is a hard problem.
A great deal of research has been aimed at this topic, much of it under the umbrella of the Messaging Layer Security (MLS) working group at the IETF.
Started in 2018, this joint effort by academics and industry culminated in 2023 with the publication of the first standard for secure group messaging [IETF, RFC9420].
At the core of secure group messaging is a cryptographic primitive termed Continuous Group Key Agreement, or CGKA [Alwen et al. 2021], that essentially allows a changing group of users to agree on a common key with the added functionality security against compromises is achieved by users asynchronously issuing a key update. In this thesis we contribute to the understanding of CGKA across different angles.
First, we present a new technique to effect dynamic operations in groups, i.e., add or remove members, that can be more efficient that the one employed by MLS in certain settings.
Considering the setting of users belonging to multiple overlapping groups, we then show lowerbounds on the communication cost of constructions that leverage said overlap, at the same time showing protocols that are asymptotically optimal and efficient for practical settings, respectively. Along the way, we show that the communication cost of key updates in MLS is average-cost optimal.
An important feature in CGKA protocols, particularly for big groups, is the possibility of executing several group operations concurrently. While later versions of MLS support this, they do at the cost of worsening the communication efficiency of future group operations.
In this thesis we introduce two new protocols that permit concurrency without any negative effect on efficiency. Our protocols circumvent previously existing lower bounds by satisfying a new notion of post-compromise security that only asks for security to be re-established after a certain number of key updates have taken place. While this can be slower than MLS in terms of rounds of communication, we show that it leads to more efficient overall communication.
Additionally, we introduce a new technique that allows group members to decrease the information they need to store and download, which makes one of our protocols enjoy much lower download cost than any other existing CGKA constructions
Dynamics of morphogen source formation in a growing tissue
No full textA tight regulation of morphogen production is key for morphogen gradient formation and thereby for reproducible and organised organ development. Although many genetic interactions involved in the establishment of morphogen production domains are known, the biophysical mechanisms of morphogen source formation are poorly understood. Here we addressed this by focusing on the morphogen Sonic hedgehog (Shh) in the vertebrate neural tube. Shh is produced by the adjacently located notochord and by the floor plate of the neural tube. Using a data-constrained computational screen, we identified different possible mechanisms by which floor plate formation can occur, only one of which is consistent with experimental data. In this mechanism, the floor plate is established rapidly in response to Shh from the notochord and the dynamics of regulatory interactions within the neural tube. In this process, uniform activators and Shh-dependent repressors are key for establishing the floor plate size. Subsequently, the floor plate becomes insensitive to Shh and increases in size due to tissue growth, leading to scaling of the floor plate with neural tube size. In turn, this results in scaling of the Shh amplitude with tissue growth. Thus, this mechanism ensures a separation of time scales in floor plate formation, so that the floor plate domain becomes growth-dependent after an initial rapid establishment phase. Our study raises the possibility that the time scale separation between specification and growth might be a common strategy for scaling the morphogen gradient amplitude in growing organs. The model that we developed provides a new opportunity for quantitative studies of morphogen source formation in growing tissues
Birkhoff conjecture for nearly centrally symmetric domains
No full textIn this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 2018) and show that a domain close enough to a centrally symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) by proving that a notion of rational integrability is equivalent to the C0-integrability condition used in their paper
Reconciling theories of dominance with the relative rates of adaptive substitution on sex chromosomes and autosomes
No full textThe dominance of beneficial mutations is a key evolutionary parameter affecting the rate and genetic basis of adaptation, yet it is notoriously difficult to estimate. A leading method to infer it is to compare the relative rates of adaptive substitution for X-linked and autosomal genes, which—according to a classic model by Charlesworth et al. (1987)—is a simple function of the dominance of new beneficial mutations. Recent evidence that rates of adaptive substitution are faster for X-linked genes implies, accordingly, that beneficial mutations are usually recessive. However, this conclusion is incompatible with leading theories of dominance, which predict that beneficial mutations tend to be dominant or overdominant with respect to fitness. To address this incompatibility, we use Fisher’s geometric model to predict the distribution of fitness effects of new mutations and the relative rates of positively selected substitution on the X and autosomes. Previous predictions of faster-X theory emerge as a special case of our model in which the phenotypic effects of mutations are small relative to the distance to the phenotypic optimum. But as mutational effects become large relative to the optimum, we observe an elevated tempo of positively selected substitutions on the X relative to the autosomes across a broader range of dominance conditions, including those predicted by theories of dominance. Our results imply that, contrary to previous models, dominant and overdominant beneficial mutations can plausibly generate patterns of faster-X adaptation. We discuss resulting implications for genomic studies of adaptation and inferences of dominance
PMLR
No full textWe introduce a new framework for studying meta-learning methods using PAC-Bayesian theory. Its main advantage over previous work is that it allows for more flexibility in how the transfer of knowledge between tasks is realized. For previous approaches, this could only happen indirectly, by means of learning prior distributions over models. In contrast, the new generalization bounds that we prove express the process of meta-learning much more directly as learning the learning algorithm that should be used for future tasks. The flexibility of our framework makes it suitable to analyze a wide range of meta-learning mechanisms and even design new mechanisms. Other than our theoretical contributions we also show empirically that our framework improves the prediction quality in practical meta-learning mechanisms
ISTA Thesis
No full textState-of-the-art quantum computers, with roughly a thousand qubits, face a crucial technological challenge of scaling up. Spins confined in quantum dots (QDs) are a promising candidate
for qubits due to their long coherence, tunability, control, and readout. However, their natural
coupling is the short-ranged (∼ 100 nm) exchange interaction, limited to nearest neighbours.
Long-ranged (∼ 1 mm) qubit interactions mediated by a photon could be engineered through a
coherent spin-photon coupling. Achieving a strong coupling to a photon is inherently challenging in QDs due to the small dipole moment of the confined charge. However, the potential of
high-impedance resonators to compensate for this has gained significant attention in the past
decade. Nevertheless, previous QD circuit quantum electrodynamics implementations have not
exceeded the impedance of ∼ 3.8 kΩ, leaving opportunities for significant improvement. The
large kinetic inductance of granular aluminium (grAl) could provide an order-of-magnitude
enhancement. However, fully exploiting the potential of disordered or granular superconductors
is challenging as their impedances close to the superconductor-to-insulator transition are
difficult to control reproducibly. We report on the realization of a wireless ohmmeter which
allows in situ resistance measurements during film deposition and, therefore, indirect control
of the kinetic inductance of grAl films. This allows us to reproducibly fabricate resonators
with characteristic impedance exceeding the resistance quantum, even reaching 22.3 kW, due
to the large sheet kinetic inductance of up to 3 nH □−1
. By integrating an 8 kW resonator
with a germanium double QD, we demonstrate a strong charge-photon coupling with the
highest rate reported, 566 MHz. The demonstrated method and grAl properties make these
resonators suitable for boosting the spin-photon coupling strength, a crucial requirement for
fast, high-fidelity, long-distance two-qubit gates