Abstract: In this talk, I will discuss a construction of TR with coefficients. For every ring R, we can form the topological Hochschild homology THH(R) of R and the cyclotomic structure on THH(R) allows us to extract the invariants given by TC(R) and TR(R). These invariants are important in the study of algebraic K-theory trough trace methods. More generally, if M is an R-bimodule, then we can form the topological Hochschild homology THH(R, M) with coefficients. This does not carry an action of circle group, so in particular it does not carry the structure of a cyclotomic spectrum. Instead, we introduce the notion of a polygonic spectrum which is designed to capture the structure present on THH(R, M). The main idea is to consider THH as a trace theory defined on cyclic graphs labelled by rings and bimodules. This will allow us to define a version of TR with coefficients by a corepresentability formula. This is joint work with Achim Krause and Thomas Nikolaus.
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Abstract: Stable homotopy theory is intimately related to the geometry of formal groups. Franke took a step towards making this precise by proposing a certain derived category of sheaves on the moduli stack of formal groups as an algebraic analog for the infinity category of spectra of chromatic height â¤n. He conjectured that at sufficiently large primes the homotopy truncation of the two categories coincide. Barthel-Schlank-Stapleton proved an asymptotic variant of Frankeâs conjecture using categorical ultraproducts. Later, Pstragowski used Goerss-Hopkins theory to prove a non-monoidal formulation of the conjecture. We discuss recent work, building on these results, in which we prove the symmetric monoidal formulation of Frankeâs conjecture.
Abstract: We recount Grothendieck's definition of a \lambda-ring and explain why it globalises the notion of p-typical \delta-rings as appearing e.g. in the context of prismatic cohomology. We then give an intrinsic definition of animated \lambda-rings in terms of animated rings equipped with Frobenius lifts and show that the \infty-category obtained this way is equivalent to the non-abelian derived category of the 1-category of \lambda-rings in the sense of Quillen and Lurie.
Abstract: In a recent paper, Carmeli, Schlank, and Yanovski built and classified the higher cyclotomic extensions of the T(n)-local sphere using higher semi-additive methods, and developed a generalization of Kummer theory, relating the Picard and Galois groups. In particular, they find a torsion subgroup of size p-1 in the Picard group of the category of T(n)-local spectra. In our work, we show the existence of a torsion subgroup of size 2p^n-2 in the Picard group of the T(n)-local category giving rise to more Galois extensions. We then use the cyclic Galois extensions to build non-split Azumaya algebras, giving us an infinite family of elements in the Brauer group.
Abstract: In recent work with Ben Antieau and Thomas Nikolaus, we develop new methods to compute K-theory of Z/p^n and related rings, based on prismatic cohomology. This approach can be turned into an algorithm, which we implement. The same methods also allow us to prove that K-theory of Z/p^n vanishes in large enough even degrees, and to give an explicit formula for the orders in large odd degrees. In this talk, I want to give an overview over the ingredients of these computations.
Abstract: It was an observation of Barry Mazur that led to the âprimes as knotsâ philosophy. The idea here is that prime numbers can be viewed as knots in RR^3, and squarefree integers as links with connected components corresponding to prime divisors. It is reasonable to ask what properties of links, and what statements of knot theory can be meaningfully interpreted in number theory with this philosophy in mind. In knot theory, one way to produce invariants of links is by âcoloringâ. Coloring methods are defined using keis, a type of algebraic structure. Each kei produces a different invariant - col(L), the number of colorings of a link L. This notion of colorings has a number-theoretical counterpart, col(N), which we define. We also make a statement about the growth rate of this function on average. In simplistic terms, using braid groups as a model for random links, we conjecture that the average order of col(N) is determined by the expected number of colorings of random links. This is verified for 8 of the 10 keis with at most 4 elements.
Abstract: Over characteristic 0 we give a full description of the periodic cyclic homology HP of animated commutative algebras. In particular we can deduce an explicit condition on the Hodge completed derived de Rham complex of an animated commutative algebra, that makes the HKR filtration on HP as introduced by Antieau and again by Bhatt-Lurie exhaustive.
Abstract: The redshift conjecture of Ausoni-Rognes says that there is a strong interaction between the algebraic K-theory and the chromatic filtration on spectra. Namely, if a ring spectrum R is of chromatic height n, then K(R) is of chromatic height n+1. Hopkins-Lurie, followed by Carmeli-Schlank-Yanovski, showed that the category of spectra of height n is higher semiadditive, that is, colimits and limits indexed by pi-finite spaces are canonically equivalent. In this talk, we will describe higher semiadditive K-theory, a variant of algebraic K-theory that takes higher semiadditivity into account. We will explain how semiadditive methods allow us to show that it satisfies a form of the redshift conjecture. Relevant background on algebraic K-theory, chromatic homotopy, and semiadditivity will be explained.
Abstract: In order to establish a Brown -Representability theorem in the context of genuine G-spectra and global spectra we propose a new equivalent set of axioms for cohomology theories on anima. Our formulation of the E.-S. axioms is applicable verbatim for G-anima and global anima and we expect these to model G-spectra and global spectra respectively.
Abstract: The spectrum of topological modular forms provides valuable insight into phenomena occurring in chromatic height up to 2. Lurie demonstrated that it can be realized as the global sections of a non-connective DM stack. While there does not exist a perfect analog for higher heights, Behrens and Lawson proposed a partial analog. I will explain the intended goals of their construction â proven, conjectured and speculative- and how to make precise their idea using the language of nonconnective spectral algebraic geometry. If time permits, I shall explain a new result about an algebro-geometric feature of this construction â one which is alien to the connective setting â and speculate about its significance.
Abstract: : I explain a generalization of the theory of recollements that enables one to reconstruct sheaves on an \infty-topos stratified by a finite poset in terms of their stratumwise pullbacks and gluing data thereamong. In terms of the \infty-categorical Hochster duality theorem of Barwick-Glasman-Haine applied to the Zariski stratification of the etale topos, this gives another description of etale sheaves on a (qcqs) scheme that amounts to a vast generalization of the recollement one has associated to any closed-open decomposition of the underlying spectral space.
Abstract: In higher category theory, the notion of a presentable (infinity-)category plays a fundamental role. These are categories which are not small but are governed by a small, or âset worthâ, amount of data. This enables one to study and manipulate these large objects without falling into various set-theoretical paradoxes. A fundamental result in the theory of presentable categories is that they can all be obtained as accessible localizations of categories of presheaves over small categories. It follows that every presentable category is a full subcategory of a category of presheaves closed under limits and sufficiently filtered colimits. The following question then arises naturally: is every full subcategory of a presentable category closed under limits and sufficiently filtered colimits presentable? In the classical case of presentable 1-categories, this question was given a positive answer by Adamek and Rosicky in 1989, in what came to be known as the reflection theorem. Their proof, however, does not directly apply to the world of infinity categories. We give a positive answer in the general infinity categorical case. In the lecture, we will outline our proof and discuss some applications.