Abstracts

CALCULATIONS IN THE MOTIVIC STABLE HOMOTOPY CATEGORY
Hana Kong
Institute for Advanced Study

Abstract: Calculations in the motivic category are interesting and have close connections to number theory as well as classical stable homotopy theory. One example of motivic applications in the classical theory is the Adams spectral sequence computations by Isaksen—Wang—Xu. They base their approach on a theoretical result by Gheorghe—Wang—Xu about a t-structure on the p-complete cellular C-motivic category. I will first talk about a generalization of this result, joint with Tom Bachmann, Guozhen Wang, and Zhouli Xu. The generalization leads to computational applications in the classical and motivic Adams spectral sequences.

Another important computational tool in the classical theory is the Adams--Novikov spectral sequence. I will talk about its motivic analog, the motivic slice spectral sequence, and how it computes the motivic 'image-of-j' spectrum defined by Bachmann--Hopkins. The slice computation is joint with Eva Belmont and Dan Isaksen.

The comparison between the motivic slice spectral sequence and the motivic Adams spectral sequence parallels the classical Adams and Adams--Novikov spectral sequences.

ORIENTED AND UNITARY EQUIVARIANT BORDISM OF SURFACES
Bernardo Uribe
Universidad del Norte

Abstract: Together with collaborators Angel, Segovia and Samperton we have been able to determine explicitly the torsion part of both the equivariant oriented and unitary bordism groups of surfaces. The key result is the calculation of the obstruction class for a surface with free action to bound equivariantly. I will present this obstruction class as well as an explicit group where the invariant is non trivial.

EQUIVARIANT HOCHSCHILD THEORIES FROM A SHADOW PERSPECTIVE
Inbar Klang
Columbia University

Abstract: Bicategorical shadows, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. I'll begin the talk by reviewing this framework and introducing the equivariant Hochschild invariants we will discuss, twisted THH and Hochschild homology for Green functors. I will then talk about joint work with Adamyk, Gerhardt, Hess, and Kong, in which we prove that these equivariant Hochschild invariants are bicategorical shadows, from which we deduce that they satisfy Morita invariance. We also show that they receive trace maps from an equivariant version of algebraic K-theory.

FREE LOOP SPACES AND TOPOLOGICAL CO-HOCHSCHILD HOMOLOGY
Anna Marie Bohmann
Vanderbilt University

Abstract: Free loop spaces arise in many areas of geometry and topology. Simply put, the free loops on a space X is the space of maps from the circle into X. This is a main object of study in string topology and has important connections to geodesics on manifolds. In this talk, we discuss a new approach to computing the homology of free loop spaces via topological coHochschild homology, which is an invariant of coalgebras arising from homotopy theory techniques. This approach produces a spectral sequence for the homology of free loop spaces with algebraic structure that allows us to make new computations. We will explain the connection between free loop spaces and coalgebras in homotopy theory and give an example of the new computations. This is joint work with Teena Gerhardt and Brooke Shipley.

EMBEDDING CALCULUS IN CODIMENSION ZERO
Alexander Kupers
University of Toronto

Abstract: In this talk I will give an introduction to a recent approach to the classification of smooth manifolds and the homotopy theory of their diffeomorphism groups. This approach is based on embedding calculus, which is a tool that provides approximations to spaces of embeddings by restricting them to open balls. Classically it is used in codimension at least 3, where it is known the approximation map is an equivalence, but one can apply it in lower codimensions as well. Of particular interest is codimension 0, when embeddings between closed manifolds are diffeomorphisms. I will explain some results on the strengths and weaknesses of the resulting approach to the study of smooth manifolds, as well as connections to operads and some open problems. This is joint work with Ben Knudsen and with Manuel Krannich.

LARGE N PHENOMENA IN PROBABILITY AND HOMOLOGICAL ALGEBRA
Mahmoud Zeinalian
Lehmann College, CUNY

Abstract: I will discuss how the Loday-Quillen-Tsygan (LQT) Theorem admits a BV quantization. I will then discuss an application in calculating matrix integral and their multi-point correlation functions. One byproduct is the Harer-Zagier relations in terms of this noncommutative geometry. One central goal is to offer a new view on a class of large N phenomena by relating a homological result, such as the LQT theorem, to quantum and probabilistic systems, such as Gaussian random matrices. Time permitting, I will discuss LQT theorem’s the relation to Wilson loop observables in gauge theory and string topology. These are recent joint works with Alastair Hamilton, Owen Gwilliam, and Gregory Ginot.

OPERATIONS ON HOCHSCHILD COMPLEXES
Ralph Kaufmann
Purdue University

Abstract: The fact that Hochschild cochain complexes are not commutative, but only commutative up to a controlled homotopy was a fundamental insight of Gerstenhaber. There is moreover a series of higher operations that can be neatly organized into several types of operadic structures of which Deligne's conjecture is a part. In the case of a Frobenius algebra, and some weaker versions, there is moreover a whole package of operations on the chain level based on surfaces or alternatively graphs with extra structures. In geometric situations this captures algebraic string topology operations, for instance an operation corresponding to the Goresky-Hingston coproduct. In newer work with Manuel Rivera and Zhengfang Wang, we extend these operations to Hochschild chain complexes and the Tate Hochschild complex, which connects the chain and cochain complexes and plays a role in singularity theory. There is a particular dualization which dualizes a higher multiplication found in this complex to a double Poisson bracket which also appears as a natural homotopy.

POINT COUNTING TO DETECT NON-PERMUTATIVE ELEMENTS OF K_1(\Var)
Inna Zakharevich
Cornell University

Abstract: The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field k) modulo the relation that for a closed immersion Y --> X there is a relation that [X] = [Y] + [X \ Y]. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field k is finite for a variety X we can consider the "almost-finite" set X(\bar k), which represents the local zeta function of X. In this talk we will discuss how to detect interesting elements in K_1(\Var) (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on X.

STRING TOPOLOGY AND THE CYCLIC DELIGNE CONJECTURE

Nick Rozenblyum
University of Chicago

Abstract: I will describe a new approach to genus zero operations in string topology via noncommuative geometry. Specifically, I will explain the cyclic version of the Deligne conjecture which gives an action of the framed E_2 operad on the Hochschild chains of a Calabi-Yau category. In the setting of relative Calabi-Yau structures, this gives a generalization of the string topology operations to manifolds with boundary. Moreover, this structure has a natural interpretation in terms of deformation theory which gives a vast generalization of Turaev's theorem relating the Lie algebra of loops on a Riemann surface to the Poisson algebra of functions on the character variety. This is joint work with Christopher Brav.

EQUIVARIANT ALGEBRAIC K-THEORY OF G-MANIFOLDS
Mona Merling
University of Pennsylvania

Abstract: Quillen’s algebraic K-theory of rings has deep connections to problems in number theory, and Waldhausen's A-theory, an extension of algebraic K-theory to spaces, is central to the classification of diffeomorphisms of manifolds. The first two talks will assume very minimal background in algebraic topology and will survey algebraic K-theory and equivariant stable homotopy theory. In the last talk, building on the previous two talks, I will discuss recent advances on the equivariant A-theory of G-manifolds.

OPERATIONS ON HOHCSCHILD AND CYCLIC COMPLEXES: AN OVERVIEW
Boris Tsygan
Northwestern University

Abstract: Hochschild and cyclic complexes are noncommutative analogues of forms and multivectors. The latter carry a rich algebraic structure; generalizing this structure to the noncommutative case had been much studied in the last sixty years. The subject started in the early sixties by Gerstenhaber, Hochschild-Kostant-Rosenberg, and Rinehart. It further developed in the early eighties with the cyclic homology theory of Connes and myself. In the nineties, powerful results on the algebraic structure of Hochschild cochains were proved by Getzler-Jones, McLure-Smith, Kontsevich, Tamarkin, and others. Since the beginning of this century much work had been done to advance the topics in the case of Frobenius algebras or/and (pre) Calabi-Yau categories (Tradler-Zeinallian, Kaufmann, Ward, Rivera-Wang, Kontsevich-Takeda-Vlassopoulos, Iudiu-Kontsevich, Wahl-Westerland, Waikit Yeung, Brav-Rozenblyum, and others). Another body of work studies noncommutative generalizations of operations arising on forms in positive or mixed characteristic, such as Frobenius maps (Kaledin). It had been understood at least for the last twenty years that the correct question to ask (following Drinfeld) is: what do DG categories form? Results in this direction were obtained by Batanin, Tamarkin, and others. A general structure that unites the results above is not yet clear. I will give an overview and formulate some conjectures and questions.