Wednesday, 5 March 2025, 17:15-19:00 (UTC+1)
A HIGH-TEMPERATURE PHASE TRANSITION FROM NUMBER THEORY Let S be the semidirect product of the multiplicative positive integers acting on the integers, with the operation (a,m)(b,n) = (ab,bm+n), where a and b are positive. In previous joint work with Astrid an Huef and Iain Raeburn, we studied the Toeplitz C*-algebra generated by the left regular representation of S on l^2(S), and showed that the extremal KMS equilibrium states with respect to the natural dynamics, for inverse temperatures above the critical value 1, are parametrized by the point masses on the unit circle. I will talk about what happens for inverse temperatures between 0 and 1. Surprisingly, the system has an unprecedented high-temperature phase transition with extremal KMS states parametrized by averages of point masses at roots of unity of the same primitive order together with Lebesgue measure. The quotients associated to these extremal states embed in the Bost-Connes algebra, and establish a link to the Bost-Connes phase transition with spontaneous symmetry breaking. This is current joint work with Tyler Schulz.https://uw-edu-pl.zoom.us/j/95105055663?pwd=TTIvVkxmMndhaHpqMFUrdm8xbzlHdz09Meeting ID: 951 0505 5663 Passcode: 924338
A HIGH-TEMPERATURE PHASE TRANSITION FROM NUMBER THEORY
Let S be the semidirect product of the multiplicative positive integers acting on the integers, with the operation (a,m)(b,n) = (ab,bm+n), where a and b are positive. In previous joint work with Astrid an Huef and Iain Raeburn, we studied the Toeplitz C*-algebra generated by the left regular representation of S on l^2(S), and showed that the extremal KMS equilibrium states with respect to the natural dynamics, for inverse temperatures above the critical value 1, are parametrized by the point masses on the unit circle. I will talk about what happens for inverse temperatures between 0 and 1. Surprisingly, the system has an unprecedented high-temperature phase transition with extremal KMS states parametrized by averages of point masses at roots of unity of the same primitive order together with Lebesgue measure. The quotients associated to these extremal states embed in the Bost-Connes algebra, and establish a link to the Bost-Connes phase transition with spontaneous symmetry breaking. This is current joint work with Tyler Schulz.https://uw-edu-pl.zoom.us/j/95105055663?pwd=TTIvVkxmMndhaHpqMFUrdm8xbzlHdz09Meeting ID: 951 0505 5663 Passcode: 924338
The Organizers:
Paul F. Baum, Francesco D'Andrea, Ludwik D?browski, Søren Eilers, Piotr M. Hajac, Frédéric Latrémolière, Tomasz Maszczyk, Ryszard Nest, Marc A. Rieffel, Andrzej Sitarz, Wojciech Szyma?ski, Adam Wegert
Using objective priors in Bayesian applications has become a common practice to analyze data without subjective information. Formal rules usually obtain these prior distributions, and the data provide the dominant information in the posterior distribution. However, these priors are typically improper and may lead to improper posterior. Here, for a general family of distributions, we show that the obtained objective priors for the parameters either follow a power-law distribution or have an asymptotic power-law behavior. As a result, we observed that the exponents of the model are between 0.5 and 1. Understanding these behaviors allows us to easily verify if such priors lead to proper or improper posteriors directly from the exponent of the power-law. The general family considered in our study includes essential models such as Exponential, Gamma, Weibull, Nakagamim, Half-Normal, Rayleigh, Erlang, and Maxwell Boltzmann distributions, to list a few. In summary, we show that comprehending the mechanisms describing the shapes of the priors provides essential information that can be used to understand the properties of the posterior distributions.