I will begin with an introduction to algebraic K-theory, ring spectra and the chromatic redshift conjecture. After this, I will talk about our new proof of the redshift conjecture for Lubin-Tate spectra and our algebraic K-theory computations. This work is partially joint with Christian Ausoni and Tasos Moulinos.
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Some spaces seem to have different dimensions at different scales. A long thin strip might appear one-dimensional at a distance, then two-dimensional when zoomed in on, but when zoomed in on even closer it is seen to be made of a finite array of points, so at that scale it seems zero-dimensional. I will present a way of quantifying this phenomenon using a couple of measures of the size of metric spaces, namely magnitude and spread. I will show lots of examples for finite metric spaces.
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I explain some cancellation and non-cancellation phenomena in algebraic geometry and relate them to the structure of the Grothendieck ring of varieties and to the groups of birational self-maps of algebraic varieties, in particular the Cremona groups.
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In ancient Greece, geometry was about points, lines, circles, and communicated through pictures. The 17th Century marked a transformative shift, connecting geometry with algebra, and lead to working with equations over visual representations. Algebraic geometry emerged as a magical blend of geometric intuition and algebraic methods. Commutative algebra, mainly the study of polynomial rings and their ideals, dominated the field for an extensive period. Then with the emergence of noncommutative algebras, such as matrix algebras, our unstoppable geometric intuition hit an immovable wall. The solution? A return to pictures as representations. In this expository talk, I will introduce a visual perspective on algebras, exploring path algebras and their captivating connections to different fields.
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Aperiodic sequences and sequence spaces form prototypical mathematical models of quasicrystals. The most quintessential examples include subshifts of Sturmian words and substitutions, which are ubiquitous objects in ergodic theory and aperiodic order. Two of the most striking features these shift spaces have, are that they have zero topological entropy and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions, and in stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy and exhibit uncountably many ergodic measures. Moreover, they have been shown to provide mathematical models for physical quasicrystals with defects.
We will begin by talking about subshifts generated by Sturmian words and ways to measure their complexity beyond topological entropy, and show how this measure of complexity can be used to build a classification via Jarník sets. We will then build a bridge between these subshifts and subshifts of random substitutions. We will conclude with some recent dynamical results on subshifts of random substitutions and ways to visualise these subshifts. Namely, we will present a method to build a new class of Rauzy fractals.
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We use 3d defect TQFTs to give a gauge theoretical formulation of (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers, groupoids and their representations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms of spans of groupoids and representations of spans. This is work in progress with João Faría-Martins, University of Leeds.
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Over the past decade or so, homotopy type theory (HoTT) has emerged as a novel foundation for mathematics. Rather than taking sets as the basic entities of mathematics, HoTT provides us with a synthetic theory of structures, expressing naturally notions of structural equivalence. These structures are infinity-groupoids, or what Peter Scholze has called ‘anima’. Evidence that the underlying dependent type theory is well-suited to present mainstream mathematics comes from the success of Kevin Buzzard’s program to use Lean as an automated proof assistant to verify contemporary results.
In HoTT itself it is possible to develop what is called ‘synthetic homotopy theory’. But mathematicians also treat further varieties of structure, such as cohesion, smooth structure, equivariance and linear structure. It turns out that these may all be treated synthetically by the addition of ‘modalities’ to HoTT. With the close relationship between HoTT and computation, it appears that Linear HoTT has things to say about quantum computation.
In this talk I shall be giving a gentle introduction to these ideas.
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