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Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. For example, Hochschild-Pirashvili homology for a wedge of circles or Jacobi diagrams in handlebodies give rise to interesting functors on gr. Some of these natural examples satisfy further properties: they are analytic and/or exponential. Pirashvili proves that the category of exponential contravariant functors from gr to the category k-Mod of k-modules is equivalent to the category of cocommutative Hopf algebras over k. Powell proves an equivalence between the category of analytic contravariant functors from gr to k-Mod, and the category of linear functors on the linear PROP associated to the Lie operad when k is a field of characteristic 0. In this talk, after explaining these two equivalences of categories, I will explain how they interact with each other. (This is a joint work with Minkyu Kim).[-]

Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. For example, Hochschild-Pirashvili homology for a wedge of circles or Jacobi diagrams in handlebodies give rise to interesting functors on gr. Some of these natural examples satisfy further properties: they are analytic and/or exponential. Pirashvili proves that the category of exponential contravariant ...[+]

18A25 ; 16T05 ; 18M70

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We will explain a new setting of genuine equivariant homotopy theory for infinite groups (like the integers) inspired by work of Kaledin. This is different from the usual approach and allows to prove a comparison to a new relative version of cyclotomic spectra that we introduce (called polygonic spectra). We shall also explain the relation to TR and to Witt vectors with coefficients. This is joint work with A. Krause and J. McCandless.

13D03 ; 13F35 ; 19D55

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In this work, we start with a new method to adjoin roots to ring spectra. Upon making a new definition of logarithmic THH, we prove that root adjunction is logarithmic THH etale and that it results in interesting splittings in algebraic K-theory. Using this, we obtain new algebraic K-theory computations. This is a joint work with Christian Ausoni and Tasos Moulinos.

55P99 ; 19D99 ; 55N15

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Hilbert's Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry. In this talk, I will discuss a version of Hilbert's Nullstellensatz in chromatic homotopy theory, where Lubin-Tate theories play the role of algebraically closed fields. Time permitting, I will then indicate some of the applications of the chromatic nullstellensatz including to redshift for the algebraic K-theory of commutative algebras. This is joint work with Tomer Schlank and Allen Yuan.[-]

Hilbert's Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry. In this talk, I will discuss a version of Hilbert's Nullstellensatz in chromatic homotopy theory, where Lubin-Tate theories play the role of algebraically closed fields. Time permitting, I will then indicate some of the applications of the chromatic nullstellensatz including to redshift for the algebraic K-theory ...[+]

55P43 ; 19D99

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Inspired by the triad of rational, trigonometric and elliptic functions appearing in representation theory, Grojnowski defined in 1995 a higher analogue of equivariant ordinary cohomology and equivariant K-theory: equivariant elliptic cohomology. However, his approach only works over the complex numbers. Based on ideas of Lurie, David Gepner and I have recently defined equivariant elliptic cohomology without these restrictions. This allows in particular to refine topological modular forms to a genuine equivariant theory.[-]

Inspired by the triad of rational, trigonometric and elliptic functions appearing in representation theory, Grojnowski defined in 1995 a higher analogue of equivariant ordinary cohomology and equivariant K-theory: equivariant elliptic cohomology. However, his approach only works over the complex numbers. Based on ideas of Lurie, David Gepner and I have recently defined equivariant elliptic cohomology without these restrictions. This allows in ...[+]

55N34 ; 55P91 ; 14A30

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The classical Barratt-Priddy-Quillen theorem states that the $K$-theory spectrum of the category of finite sets and isomorphisms is equivalent to the sphere spectrum. A more general statement is that for an unbased space $X$, the suspension spectrum $\Sigma_{+}^{\infty} X$ is equivalent to the spectrum associated to the free $E_{\infty}$ space on $X$. In this talk we will present a categorical construction of the latter that is lax monoidal. This compatibility with multiplicative structures is necessary when using this functor to change enrichments, as in the work of Guillou-May.This is joint work with Bert Guillou, Peter May and Mona Merling.[-]

The classical Barratt-Priddy-Quillen theorem states that the $K$-theory spectrum of the category of finite sets and isomorphisms is equivalent to the sphere spectrum. A more general statement is that for an unbased space $X$, the suspension spectrum $\Sigma_{+}^{\infty} X$ is equivalent to the spectrum associated to the free $E_{\infty}$ space on $X$. In this talk we will present a categorical construction of the latter that is lax monoidal. ...[+]

19D23 ; 19L47 ; 55P48

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We will start with recognition criteria for equivalences in (infty,n)-categories, introducing and discussing the latter along the way. From there, we will investigate how the notion of equivalence is encoded in strict higher categories, and time permitting discuss the relation of those notions. This is based on joint work (partially in progress) with Martina Rovelli.

18N65 ; 55U35 ; 18N10 ; 18N50

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In joint work with Gabriel Angelini-Knoll, Christian Ausoni, Dominic Leon Culver and Eva Höning, we calculate the $\bmod \left(p, v_1, v_2\right)$ homotopy $V(2)_* T C(B P\langle 2\rangle)$ of the topological cyclic homology of the truncated Brown-Peterson spectrum $B P\langle 2\rangle$, at all primes $p \geq 7$, and show that it is a finitely generated and free $\mathbb{F}_p\left[v_3\right]$-module on $12 p+4$ generators in explicit degrees within the range $-1 \leq * \leq 2 p^3+2 p^2+2 p-3$. Our computation is the first that exhibits chromatic redshift from pure $v_2$-periodicity to pure $v_3$-periodicity in a precise quantitative manner.[-]

In joint work with Gabriel Angelini-Knoll, Christian Ausoni, Dominic Leon Culver and Eva Höning, we calculate the $\bmod \left(p, v_1, v_2\right)$ homotopy $V(2)_* T C(B P\langle 2\rangle)$ of the topological cyclic homology of the truncated Brown-Peterson spectrum $B P\langle 2\rangle$, at all primes $p \geq 7$, and show that it is a finitely generated and free $\mathbb{F}_p\left[v_3\right]$-module on $12 p+4$ generators in explicit degrees ...[+]

19D50 ; 19D55 ; 55P43 ; 55Q51

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There are many different notions of 'being algebraic' used in stable homotopy theory. The relationships between those turn out to be unexpectedly subtle. We will explain the different ways in which a model category of interest can be algebraic, explore the different implications between them and illustrate those with plenty of examples. This is joint work with Jocelyne Ishak and Jordan Williamson.

18N40 ; 55P42

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I will discuss ways in which representation spheres of specific finite (or profinite) groups can describe elements in chromatically interesting Picard groups, including the quintessential Picard group of the K(n)-local category. This is based on work joint with Beaudry, Bobkova, Goerss, Henn, Hill, and Pham.

55Q50 ; 55Q51

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Documents Vespa, Christine 16 résultats

On analytic exponential functors on free groups - Vespa, Christine (Auteur de la Conférence) | CIRM H Nouveau

Equivariant homotopy theory for infinite groups and THH with coefficients - Nikolaus, Thomas (Auteur de la Conférence) | CIRM H Nouveau

Adjoining roots to ring spectra and algebraic K-theory - Bayindir, Haldun Özgür (Auteur de la Conférence) | CIRM H Nouveau

The chromatic Nullstellensatz - Burklund, Robert (Auteur de la Conférence) | CIRM H Nouveau

Equivariant elliptic cohomology - Meier, Lennart (Auteur de la Conférence) | CIRM H Nouveau

(2)-categorical constructions and the multiplicative equivariant Barratt- Quillen-Priddy theorem - Osorno, Angélica (Auteur de la Conférence) | CIRM H Nouveau

Equivalences in higher categories - Ozornova, Viktoriya (Auteur de la Conférence) | CIRM H Nouveau

Topological cyclic homology of the second truncated Brown–Peterson spectrum - Rognes, John (Auteur de la Conférence) | CIRM H Nouveau

How algebraic is a stable model category? - Roitzheim, Constanze (Auteur de la Conférence) | CIRM H Nouveau

Representation spheres and chromatic Picard groups - Stojanoska, Vesna (Auteur de la Conférence) | CIRM H Nouveau