Dependent Choice (DC) is one the most useful choice principles with many equivalents (including the Downward Löwenheim–Skolem and the Baire Category Theorem). When we violate the Axiom of Choice via symmetric extensions we often want to preserve at least that much. In this talk we will discuss a few older results about the preservation of DC in generic and symmetric extensions, and we will present a recent breakthrough from a work-in-progress with Jonathan Schilhan.
The fields of set theory and homological algebra are both centrally concerned with
questions of compactness, regarding the extent to which a structure's global properties are
determined by its local properties. It is thus no surprise that there has been considerable interplay
between these two fields. In this talk we will discuss some recent applications of set-theoretic
techniques to the study of the derived functor of the inverse limit, with further applications to the study
of strong homology and to the developing field of condensed mathematics. We then relate these
applications back to one of the oldest questions in set theory, that of the cardinality of the continuum.
At their core, these applications reduce to simple, purely combinatorial problems that are of interest in
their own right. No prior knowledge of either set theory or homological algebra will be assumed.
Counting the number of subgroups in a finite group has numerous applications, ranging from enumerating certain classes of finite graphs (up to isomorphism), to counting how many isomorphism classes of finite groups there are of a given order. In this talk, I will discuss the history behind the question; why it is important; and what we currently know.
This talk will not assume any knowledge of derived categories.
Two rings are "Morita equivalent" if they have equivalent module categories, and
if a property of rings depends only on the module category, then it is called
"Morita invariant". More generally, two rings are "derived equivalent" if they
have equivalent derived categories, and if a property of rings depends only on
the derived category, then it is called "derived invariant".
Fairly recently, Manuel Saorin asked me if I knew whether right coherence was a
derived invariant property. I didn't, but when my long term memory kicked in, I
realised that an example hidden in my 36 year old PhD thesis could be used to
give a counterexample.
Most of the talk will be an introduction to the background to this question and
variants, but it will finish with some fun examples of rings with strange
properties.
NOTES: unusual hour.
Abstract: Many key properties and applications of magnetic materials are strongly intertwined with the spatial distribution of magnetic moments inside the corresponding specimens. In addition to classical magnetic structures, magnetic skyrmions have raised interest in spintronics as carriers of information for future storage devices. In this talk, we present an advance in the mathematical modeling of magnetic skyrmions by analyzing the interplay of stochastic microstructures and chirality. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional, including the Dzyaloshinkskii-Moriya contribution. Eventually, we present an explicit characterization of minimizers of the effective model in the case of magnetic multilayers. This talk is based on a joint work with E. Davoli and J. Ingmanns.
We study how to construct a stochastic process on a finite interval with given `roughness'. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization, we provide a better (pathwise) estimator of Hölder exponent. Furthermore, we study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.
The Ramsey theorem was the first example of a natural result escaping the Big Five phenomenon, which had so far identified every result to have equivalent strength to one of five classical axiomatic bases. It has since then been thoroughly studied, and we present a form of generalization of said theorem, as "Ramsey like" theorems. A Ramsey like theorem is of the form: "For every k-coloring of n-tuples of integers, there exists an infinite set avoiding a set of finite patterns P". Depending on the properties of the patterns in P, these results have vastly different strengths. In particular, we will use the notions of preservation of hyperimmunity (or hyperimmunities) and 2-dimensional hyperimmunity as strength quantifiers, and find necessary and sufficient conditions for the patterns in P for each to hold. We also study Ramsey-like theorems that don't ask to avoid every pattern in P, but at least one. We also see colorings as graphs and prove the existence of a computable infinite graph of which every computable subgraph contains every possible finite subgraph.
