Title. Erdos Covering Systems

Abstract. Since their introduction by Erdos in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In 1950, Erdos asked if there exist covering systems with distinct arbitrary large moduli. In 1965, Erdos and Selfridge asked if there exist covering systems with distinct odd moduli. In 1967, Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. In 2015, Hough resolved Erdos' problem showing that a finite collection of arithmetic progressions with distinct sufficiently large moduli does not cover the integers. We established a quantitative version of Hough's theorem estimating the density of the uncovered set, thus answering a question posed by Filaseta, Ford, Konyagin, Pomerance and Yu from 2007. Additionally, we resolved the Erdos-Selfridge problem in the square free case as well as Schinzel's conjecture in full generality. In this talk, we discuss these results and present a gentle exposition of the methods used. This talk is based on joint work with Paul Balister, Bela Bollobas, Rob Morris and Julian Sahasrabudhe.

The existence of spectral asymptotics of Laplace or Schrödinger operators acting on Riemannian manifolds is a classical problem known for more than 100 years. It has been known for a long time that obstacles to the existence of spectral asymptotic expansions are periodic and looping trajectories of the geodesic flow. A conjecture formulated in 2016 stated that these trajectories are the only such obstacles. I will discuss the history of this problem and describe the resent progress: proving this conjecture in special cases, as well as constructing some counterexamples.

I explore decoupling limits that lead to matrix theories on D-branes, focusing on their BPS nature and the emergence of non-Lorentzian target space geometries. In these limits, D-branes experience instantaneous gravitational forces, and when applied to curved geometries, it is shown that a single decoupling limit leads to the AdS/CFT correspondence. By applying two such limits, we generate new holographic examples, including those with non-Lorentzian bulk geometries. I also discuss the relationship between matrix theories and non-relativistic string theory, and their uplift to M-theory. Finally, we demonstrate that reversing these decoupling limits connects to the TTbar deformation in two dimensions. This provides a new perspective on the near-horizon brane geometry and leads to TTbar-like flow equations for the Dp-brane DBI action.

Matrix denoising is central to signal processing and machine learning. Its statistical analysis when the matrix to infer has a factorised structure with a rank growing proportionally to its dimension remains a challenge, except when it is rotationally invariant. The reason is that the model is not a usual spin system because of the growing rank dimension, nor a matrix model due to the lack of rotation symmetry, but rather a hybrid between the two. I will discuss recent findings on Bayesian matrix denoising when the hidden signal XX^t is not rotationally invariant. I will discuss the existence of a « universality breaking » phase transition separating a regime akin to random matrix theory with strong universality properties, from one of the mean-field type as in spin models, treatable by spin glass techniques.
In the second part, I will connect this model and phenomenology to learning in neural networks. We will see how these findings allow to analyse neural networks with an extensively large hidden layer trained near their interpolation threshold, a model that has been resisting for a long time. I will show that the phase transition in matrix denoising translates in this context into a sharp learning transition. The related papers are: https://arxiv.org/pdf/2411.01974 \DSEMIC https://arxiv.org/pdf/2501.18530

One half of mirror symmetry for Fano varieties is typically stated as a relation between the symplectic geometry of a Fano variety Y and the complex geometry of a Landau-Ginzburg model, realized as a pair (X,W) with X a quasi-projective variety and W a regular function on X. The pair (X,W) itself is expected to reflect a pair on the Fano side, namely a decomposition of Y into a disjoint union of an affine log Calabi-Yau and an anticanonical divisor D, thought of as mirror to W. We will discuss recent work which shows how the intrinsic mirror construction of Gross and Siebert naturally produce potential LG models assuming milder conditions on the singularities of D than typically required for the intrinsic mirror construction. In particular, we show that classical periods of this LG model recover the quantum periods of Y. In the setting when Y\D is an affine cluster variety, we will describe how these LG models naturally give rise to Laurent polynomial mirrors and encode certain toric degenerations of Y. As an example, we consider Y = Gr(k,n), D a particular choice of anticanonical divisor with affine cluster variety complement and give an explicit description of the intrinsic LG model in terms of Plücker coordinates on Gr(n-k,n), recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams. 

Title: Why Arithmetic Jet Spaces?

Abstract: The theory of arithmetic jet spaces is rooted in δ-geometry, which has emerged as an elegant and powerful framework in recent advances in p-adic geometry. In this talk, I will provide an overview of arithmetic jet spaces and explore their applications in Diophantine geometry and p-adic Hodge theory. Along the way, I will also present a brief survey of key developments in this area.

SOFR derivatives market is still illiquid and incomplete so it is not amenable to classical risk-neutral term structure models which are based on the assumption of perfect liquidity and completeness. We develop a statistical SOFR term structure model that is well-suited for risk management and derivatives pricing within the incomplete markets paradigm. The model incorporates relevant macroeconomic factors that drive central bank policy rates which, in turn, cause random jumps often observed in the SOFR rates. The model is easy to calibrate to historical data, current market quotes, and the user’s views concerning the future development of the relevant macroeconomic factors. The model is illustrated by indifference pricing of SOFR derivatives. This is joint work with Teemu Pennanen.