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We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications.

In this talk we will discuss a collection of convolution inequalities for real valued functions on the hypercube, motivated by combinatorial applications.

In this talk, we propose a new approach to solving the Buckley-Leverett System, which is to consider a compressible approximation model characterized by a stiff pressure law. Passing to the incompressible limit, the compressible model gives rise to a Hele-Shaw type free boundary limit of Buckley-Leverett System, and it is shown the existence of a weak solution of it.

We will present a MCMC quantum algorithm to sample ground states of Hamiltonian systems.

We study transitions from chaotic to integrable Hamiltonians in the double scaled SYK and $p$-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bi-local Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition.

In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this talk, I will discuss some aspects of this modeling problem. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum, Marvin Qi and Daniel Spiegel, David Stephen, Xueda Wen.

Nonstandard analysis (NSA), founded by Abraham Robinson in the 1960’s, was to a great extent inspired by Leibniz’s ideas and intuitions towards the use of infinitesimal and infinitely large quantities. One of the greatest features of NSA is that, by allowing a correct formulation of infinitesimals, one is now able to reason using orders of magnitude. This means that one can give precise meaning, and reason formally, about otherwise vague terms such as "small" or "large". Recently, accounts of vagueness relying on NSA were introduced [2, 8, 4]. In particular, and unlike other accounts of vagueness, the so-called nonstandard primitivist account [4, 5] embraces transitivity for marginal differences (i.e. "small" differences), but not for large differences in a soritical series. Nonstandard primitivism also seems to be particularly adequate to deal with the ship of Theseus paradox [3, 6] and may also shed some light in doxastic reasoning by considering infinitesimal probabilities and associating them to infinitesimal credences [1, 7]. We aim at assessing the relative merits of nonstandard primitivism and to show some lines of future research regarding the connections between NSA and philosophy. (This is joint work with Bruno Jacinto)
[1] Benci, Vieri and Horsten, Leon and Wenmackers, Sylvia. Infinitesimal Probabili- ties. The British Journal for the Philosophy of Science, 69 (2): 509–552, 2018. doi: 10.1093/bjps/axw013
[2] Walter Dean. Strict finitism, feasibility, and the sorites. Review of Symbolic Logic, 11 (2):295–346, 2018. doi: 10.1017/S1755020318000163.
[3] Bruno Dinis. Equality and near-equality in a nonstandard world. Log. Log. Philos., 32 (1):105–118, 2023. ISSN 1425-3305,2300-9802.
[4] Bruno Dinis and Bruno Jacinto. A theory of marginal and large difference. Erkenntnis, 2023.
[5] Bruno Dinis and Bruno Jacinto. Marginality scales for gradable objects. (preprint), 2023.
[6] Bruno Dinis and Bruno Jacinto. Counterparts as Near-equals. (preprint), 2024.
[7] Kenny Easwaran. Regularity and Hyperreal Credences. Philosophical Review, 123 (1):1- 41, 2014.
[8] Yair Itzhaki. Qualitative versus quantitative representation: a non-standard analysis of the sorites paradox. Linguistics and Philosophy, 44:1013–1044, 2021. URL https: //doi.org/10.1007/s10988-020-09306-7.

I will present some models in ecology and epidemiology using a transport equation approach, so called structured models. The first models are of predator-prey type and include a variable hunger structure. They take the form of nonlocal transport equations coupled to ODEs. Then, we use a similar approach in an epidemiological model including disease awareness and variable susceptibility. We show well-posedness results, asymptotic behavior, and numerical simulations. This is joint work with C. Rebelo, A. Margheri, and P. Lafargeas.