Irina Bobkova, IAS Spanier-Whitehead dual of TMF at p=2. I am a Tamarkin Assistant Professor of Mathematics and NSF Postdoctoral Fellow at Brown University. The philosophy of chromatic homotopy theory is that the homotopy groups of spheres (specifically, the p -local stable groups for a prime number p) are divided into families called the chromatic layers. Jeremy Hahn, MIT Toward the C_p-fixed points of Morava E-theory. Denition 1.1. We'llworkwiththecategory of nite polyhedra (or nite CW complexes) and homotopy classes of continuous maps between them. I wrote up a list elsewhere (see Sanath Devalapurkar's answer to What is covered in algebraic topology? Let me divide to this purpose chromatic homotopy theory pseudo-historically in different phases. Homotopy Theory August 4-10, 2019 Website. In spring 2021, I am running a learning seminar on stable homotopy theory and spectra.. Lecture 4. Department of Mathematics, University of California San Diego ***** Math 292 - Topology Seminar (Chromatic Homotopy Theory Student Seminar) This includes articles concerning both computations and the formal . What references and resources (e.g. This chapter explains how the solution of the Ravenel Conjectures by Ethan S. Devinatz, Michael J. Hopkins, D. C. Ravenel, and Jeffrey H. Smith leads to a canon In general our construction exhibits a kind of redshift, whereby BP<n-1> is used to produce a height n theory. REZK, C., Notes on the Hopkins-Miller theorem, in Homotopy Theory via Algebraic Geometry and Group Representations (Evanston, IL, 1997), pp. On Beyond Hatcher!. CHROMATIC HOMOTOPY THEORY D. CULVER CONTENTS 1. The articles cover a variety of topics spanning the current research frontier of homotopy theory. 2010 to 2012 NSF grant DMS-0805833, "Formal group laws in homotopy theory and K-theory." 2008 to 2012 16 Related Articles 3. At height 1 our construction is due to Snaith, who built complex K-theory from CP. chromatic homotopy theorists to study the homology theories v 1 n BP as n varies, in an effort to understand these patterns one at a time.1 Each of these homology theories v 1 n BP comes with its own Adams spectral sequence, and Bouseld's theory of localization shows that each of these spectral sequences converges to the homotopy of some . 4. Chromatic homotopy theory is asymptotically algebraic, with Tobias Barthel and Tomer M. Schlank, Invent. European Women in Mathematics/European Mathematical Society Mittag-Leffler Summer School Chromatic Homotopy Theory and Friends is scheduled to take place June 7 to 10, 2022 at the Mittag-Leffler Institute in Djursholm, Sweden. Topics include any part of homotopy theory that has a computational flavor, including but not limited to stable homotopy theory, unstable homotopy theory, chromatic homotopy theory, equivariant homotopy theory, motivic homotopy theory, and K-theory. The mapping path space P p of a map p:EB is the pullback of along p.If p is fibration, then the natural map EP p is a fiber-homotopy equivalence; thus, roughly speaking, one . Chromatic homotopy theory gives us a way to see through this veil by using the algebraic geometry of formal groups to organize theory and computations. Chromatic Homotopy Theory This is a joint seminar of the Center for Advanced Studies at Skoltechand International laboratory for Mirror Symmetry and Automorphic Formsoffered by myselfand Vladimir Shaydurov in the 2018-2019 academic year. Let be the category whose objects are nite nonempty totally ordered sets and maps are . 1. Let K(n) be the n-th Morava K-theory spectrum K(n) = Z=p[v1 n], K(0) = Q Let L K(n)be Bous eld localization with respect to K(n) . We'll start by saying what it means for a cohomology theory to be complex orientable. Chromatic homotopy theory gives us a way to see through this veil by using the algebraic geometry of formal groups to organize theory and computations. 313-366. Mini-conference on quantum field theory, December 3rd, 2012. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. Talbot workshop on chromatic homotopy theory, April 25th, 2013. Introduces chromatic homotopy theory, algebraic K-theory and higher semiadditivity, and describes the construction of higher semiadditive K-theory and certain redshift results for it. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via . Chro-matic homotopy theory is an organizing principle which is highly devel-oped in the stable situation. Chro-matic homotopy theory is an organizing principle which is highly devel-oped in the stable situation. For every The goal of this summer school is to increase the number of women mathematicians working in chromatic homotopy theory and adjacent areas. There are 12 chromatic homotopy theory-related words in total (not very many, I know), with the top 5 most semantically related being stable homotopy theory, complex-oriented cohomology theory, daniel quillen, formal group and landweber exact functor theorem. Latest Revisions Discuss this page ContextElliptic cohomologyelliptic cohomology, tmf, string theorycomplex orientedcohomology chromatic level 2elliptic curvesupersingular elliptic curvederived elliptic curvemoduli stack elliptic curvesmodular form, Jacobi formEisenstein series, invariant, Weierstrass sigma function, Dedekind eta functionelliptic genus, Witten. The E2-term 7 1.3. Chromatic homotopy theory gives us a way to see through this veil by using the algebraic geometry of formal groups to organize theory and computations. It is called -nite if it is m-nite for some m.1 Theorem 1.1.3 (Hopkins-Lurie, [20]) Let A be a -nite space. In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. Inspired by the Ax-Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. Next you should get some familiarity with equivariant homotopy theory. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. . a bit more of a road map. The stable homotopy groups of any finite complex admits a filtration, called the chromatic filtration, where the height n stratum consists of periodic families of elements. Math., 2020. We use the theory of derivators as our model for homotopy theories, but no foreknowledge of the subject is assumed. The chromatic filtration stratifies the p-local stable homotopy category into layers, the K (n)-local categories, for each n 0.The process of moving from local to global involves patching together these K (n)-localizations.. Chromatic assembly We employ this theory to give an asymp- totic solution to the approximation problem in chromatic homotopy theory. The Classical Adams spectral sequence 2 1.1. Below is a list of chromatic homotopy theory words - that is, words related to chromatic homotopy theory. Using the v, self maps provided by the Hopkins-Smith periodicity theorem . Delta Power Operations , Talbot 2021: Ambidexterity in Chromatic Homotopy Theory , 26/10/2021 If there is time, describe the chromatic fracture square Ln(X) 'L n 1(X)h Ln 1(L K( )(X)) L K( )(X). ), which (essentially) gives one path to understanding chromotopy. Hi, my name is Yajit Jain. The image of J in 4 k 1 s ( S 0) is a cyclic group whose order is equal to the denominator of ( 1 2 k) / 2 (up to a factor of 2 ). The zen of -categories. In this approach, a spectrum E is approximated at a prime p by a tower {LfE})no of localizations away from the finite p-local spectra of type n + 1. Elliptic curves and chromatic stable homotopy theory Elliptic curves enter algebraic topology through "Elliptic cohomology"-really a family of cohomology theories-and their associated "elliptic genera". The simplest case is via K-theory: Bott periodicity gives rise to a repeating family of elements in the stable homotopy groups of spheres via the image of the J homomorphism. Let L The central theorem ( Mandell 01) says that . The chromatic picture is best described in terms of localization at a chosen prime p. After one localizes at a prime p, the moduli of formal groups admits a descending ltration, called the height ltration. Workshop at the Mathematisches Forschungsinstitut Oberwolfach on homotopy theory, organized with Jesper Grodal and Birgit Richter. In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. Much of this chapter is modeled on Kan's original papers [Kan58] and [Kan57]. Lecture 1. Title: An introduction to chromatic homotopy theory. This project studies two of the most important structural conjectures in this field, the telescope and chromatic splitting conjectures. Lecture 6. For any topological space X, one can attempt to compute the E-cohomology groups E (X) by means of the Atiyah-Hirzebruch spectral . Nat Stapleton, Kentucky Chromatic homotopy theory is asymptotically algebraic. In p -adic homotopy theory one studies, for any prime number p, simply connected homotopy types (of topological spaces, hence -groupoids) all of whose homotopy groups have the structure of (finitely generated) modules over the p-adic integers \mathbb {Z}_p - the p -adic homotopy types. Tomer Schlank Anabelian Bousfield lattice and presentable modes Video not available This shows that chromatic homotopy theory at a xed height is Lecture 3. Abstract: In Chromatic homotopy theory, one tries to understand the homotopy groups of spheres using the height filtration on formal group laws. Lecture Notes in Math., 1051. . BRST quantization of a relativistic point-particle, with Kevin Wray. Chromatic homotopy theory is based on Quillen's and Landweber's work on complex oriented cohomology theories and formal group laws. I will also survey how the theory applies to computations. Abstract: At the center of homotopy theory is the classical problem of understanding the stable homotopy groups of spheres. In the 1960's, Adams computed the image of the J -homomorphism in the stable homotopy groups of spheres. Berkeley topology seminar, November, 2012. [] Model categories for algebraists, or: What's really going on with injective and projective . 2 Chromatic homotopy theory Chromatic homotopy theory is born from the observation, perhaps dating to the work of Miller-Ravenel-Wilson [12] and Devinatz-Hopkins-Smith [3], that p-local spectra tend to split into various 'layers,' each of which has a certain kind of 'periodicity.' Speci cally, for Ind The Spanier-Whitehead category. Stacks & Chromatic Homotopy Theory Christian Carrick January 12, 2020 The goal of this seminar is to study the chromatic picture of stable homotopy theory via the language of stacks. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via . (9) Nilpotence and Periodicity. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via . Convergence of the classical ASS 8 1.4. We want to generalize orientabil-ity of manifolds to other contexts. Part 1. approximation problem in chromatic homotopy theory. Chromatic Homotopy Theory (252x) Lectures: . The goal of this talk is to . More precisely, we show that the ultraproduct of the E(n;p)-local categories over any non-principal ultra lter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. Speaker Title (Click to view video) Comment. NSF grant DMS-1560699, "FRG: Collaborative Research: Floer homotopy theory." 2016 to 2019 NSF grant DMS-1206008, "Methods of algebraic geometry in algebraic topology." 2012 to 2016 Alfred P. Sloan Research Fellowship. 1. The articles cover a variety of topics spanning the current research frontier of homotopy theory. UIUC topology seminar, January 22nd, 2013. We'llworkwiththecategory of nite polyhedra (or nite CW complexes) and homotopy classes of continuous maps between them. topy theory and ho-motopy coherent dia-grams 1. Simplicial homotopy theory The standard reference for simplicial homotopy theory is the book by Goerss and Jardine [GJ09]. This filtration is intimately tied to the algebraic geometry of formal group laws, and via this connection computations in stable homotopy theory can be tied to certain . It hides beauty and pattern behind a veil of complexity. Chromatic Homotopy Theory, Journey to the Frontier May 16-20, 2018 Website. Lubin-Tate theory, character theory, and power operations, Handbook of Homotopy Theory, 2020. There are 12 chromatic homotopy theory-related words in total (not very many, I know), with the top 5 most semantically related being stable homotopy theory, complex-oriented cohomology theory, daniel quillen, formal group and landweber exact functor theorem. April 8th: Lyne Moser, Max Planck Institute. Before that I was an undergraduate student at MIT. 1148 S. Carmeli et al. This extends work of Hovey (for model categories) and Lurie (for infinity categories) and repairs an earlier attempt of Heller. Short talks by postdoctoral membersTopic: Chromatic homotopy theorySpeaker: Irina BobkovaAffiliation: Member, School of MathematicsDate: September 26, 2017 video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)? This is an expository essay extracted from the introductory chapter of my thesis. The electronic Computational Homotopy Theory Seminar is an online international research seminar. This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17-21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL. Homotopy theory deals with spaces of large but nite dimension. There is one family for each natural number n (called the height ) and it corresponds to collections of elements that repeat at a certain frequency.