Weekly Bulletin

Motivated by the comparison of sets of holonomies of saddle connections on translation surfaces to Poisson point processes, we show that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic quadratic growth. The proof techniques combine ergodic methods for counting saddle connections with the fact that the Siegel--Veech transform is in L2. The talk will not assume prior knowledge of translation surfaces, and we will attempt to keep it broadly accessible. This is joint work with Sam Fairchild and Howard Masur.

一个不欢而散本征函数在有限域是harmonic function whose normal derivative at the boundary is proportional to the function itself. I will tell you most of what I know about the zero sets of such functions. A nice fact is that there are many zeros near the boundary: I will give a gentle proof of this in the first part of the talk. In the second, perhaps a little less gentle, part I will discuss some lower and upper bounds for the Hausdorff measure of the zero set; several questions remain unanswered. Comparisons with the (slightly) better understood case of eigenfunctions of the Laplace-Beltrami operator will also be provided.

A Diophantine equation is a polynomial equation to be solved in integers with the most famous example being X^n + Y^n = Z^n to be solved in integers X, Y and Z. A popular topic in number theory is to consider a family of Diophantine equations and ask how often it is soluble. We will give an introductory overview of this area.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/Peter-Koymans.htmlcall_made

In field theory, from a local Lagrangian it is possible to derive the phase space of the system. Roughly speaking, it represents the space of Cauchy data of the theory. Such space is equipped with a closed 2-form, whose kernel is often given by Hamiltonian vector fields generating (gauge) symmetries. The reduced phase space (RPS) is obtained as a symplectic manifold via quotienting the original phase space. In this talk, after a brief introduction and an example of RPS in a simple case, I will present the derivation (using a construction due to Kijowski and Tulcjiew) of the RPS of N=1 supergravity in dimension 4. If time allows, the corresponding BFV description is then carried out.

在本课程中,我们将检查转化m behaviors of the two- imensional incompressible Boussinesq equations without thermal diffusion. While the Boussinesq equations exhibit energy conservation, the flow's small-scale creation may induce the growth of finer norms in the solutions over time. During this talk, we will construct smooth initial data to demonstrate such norm growth phenomena, both with and without considering kinematic viscosity. Additionally, we will extend our results to provide an example of temperature patch solutions, the curvature and perimeter of which increase as time progresses. This work is a joint work with A. Kiselev and Y. Yao.

In 2000, Shiu proved that there are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. In particular, we'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3. We'll show further that all patterns of length 3 occur infinitely often: for any modulus q, every sequence (a mod q, b mod q, c mod q) appears infinitely often among consecutive sums of two squares. We'll discuss some of the proof techniques, and explain why they fail for primes. Joint work with Noam Kimmel.

现在需要可再生的贝叶斯模型, generative, predictive, robust, computationally scalable, and able to provide sound inferential conclusions. In this wide framework, Bayes factors still represent one of the most well-known and commonly adopted tools to perform model selection and hypothesis testing; however, they are usually criticized due to their intrinsic lack of calibration, and they are rarely used to measure the predictive accuracy arising from competing models. We propose two distinct approaches relying on BFs from our most recent research. With regard to prediction, we propose a new algorithmic protocol to transform Bayes factors into measures that evaluate the pure and intrinsic predictive capabilities of models in terms of posterior predictive distributions, by assessing some preliminary theoretical properties (joint work with Ioannis Ntzoufras). Then, regarding the analysis of replication studies (Held, 2020), we follow the stream outlined by Pawel and Held (2022) and propose a skeptical mixture prior which represents the prior of an investigator who is unconvinced by the original findings. Its novelty lies in the fact that it incorporates skepticism while controlling for prior-data conflict (Egidi et al., 2021). Consistency properties of the resulting skeptical BF are provided together with a thorough analysis of the main features of our proposal (joint work with Guido Consonni). Short Bibliography: Egidi, L., Pauli, F., & Torelli, N. (2022). Avoiding prior–data conflict in regression models via mixture priors. Canadian Journal of Statistics, 50(2), 491-510. Held, L. (2020). A new standard for the analysis and design of replication studies. Journal of the Royal Statistical Society Series A: Statistics in Society, 183(2), 431-448. Pawel, S., & Held, L. (2022). The sceptical Bayes factor for the assessment of replication success. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(3), 879-911.