I will describe a new functor calculus, reminiscent of orthogonal calculus, for functors from FI, the category of finite sets and injections, to a stable presentable infinity-category. As in other functor calculi, homogeneous FI-objects are classified by "Taylor coefficients," and FI-calculus has the agreeable property that a Taylor tower can always be recovered from its coefficients along with canonical morphisms between those coefficients. I will also discuss how the phenomenon of representation stability emerges as a facet of FI-calculus, so that FI-calculus can be understood as the elaboration of representation stability into a functor calculus in the stable infinity-categorical setting.

The homology of the unordered configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. We show that the degree of this polynomial is captured by a connectivity invariant of the graph, and we compute its leading coefficient explicitly in terms of cut counts and vertex valences. This "stable" (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin. We also discuss conjectural analogous phenomena in the ordered setting. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole.

The Loday-Quillen-Tsygan theorem identifies the cyclic cohomology of an algebra A as the large N limit of Lie algebra cohomology of gl_N(A), i.e., matrices with coefficients in A. When A has a nondegenerate pairing, one can ask about deformations of these constructions ("quantizations") and compatibility with this LQT map. Remarkably, there is a special case that encodes the Gaussian unitary ensemble, and it offers a homological view on the emergence of ribbon graphs in this domain of probability theory. More broadly, this quantum LQT construction offers a model of gauge/string dualities in the topological setting, with potential applications to mirror symmetry. This work is joint with Greg Ginot, Alastair Hamilton, and Mahmoud Zeinalian.

We present a homotopy theoretic method for calculating Ext groups between polynomial functors from the category of (finitely generated, free) groups to abelian groups. It enables us to extend the range of what can be calculated. In particular, we can calculate torsion in the ext groups, about which very little seems to have been known. We will discuss some applications to the stable cohomology of Aut(F_n), based on a theorem of Djament. Joint with Marco Nervo.

Given a nonunital commutative S-algebra I, T(I), the Andre-Quillen homology of I, is a derived version of I/I^2. 

We describe how I is built from T(I), and how T(I) is built from I. 

On one hand the natural map from I to T(I) is the beginning of a tower under I, with nth fiber built from Com(n) and the n-fold smash product of T(I) with itself.  This can be identified with the Goodwillie tower of the identity in commutative S-algebras.

On the other hand this same natural map from I to T(I) is the beginning of an increasing filtration on T(I) with nth cofiber built from Lie(n) and the n-fold smash product of I with itself.  This is the filtration used by Behrens and Rezk in their work on chromatic unstable homotopy, and we give and outline of the construction that allows one to check that there is agreement between their construction of this filtration and my earlier one.

As an application, if X is a connective spectrum, with 0th space X(0), the above theory gives interesting spectral sequences for computing the homology of X from the homology of X(0) and vice-versa: one lets I(X) be (roughly) the suspension spectrum of X(0) and computes that T(I(X)) = X.

The result of Watanabe(’18) showed that Kontsevich’s invariants can distinguish smooth fiber bundles that are isomorphic as topological fiber bundles. Given a framed smooth fiber bundle E over M (with homology sphere fibers), Kontsevich’s invariants are defined by considering the (Fulton-MacPherson compactified) configuration space bundle of E and doing some intersections in its total space to get an intersection number. Since the Fulton-MacPherson compactification is obtained by doing a sequence of real blow-ups, which depends on the smooth structure in an essential way, yet intersection theoretical invariants usually do not depend on the smooth structure, it is plausible that different smooth structures on E yield different topological structures on the configuration space bundles of E, and Kontsevich’s invariants only depend on the topological structure of the configuration space bundles. We verify that this is indeed the case. 

For a finitely dominated map of spaces f:A-->B, the Becker-Gottlieb transfer is a certain "wrong-way'' map f^! from the suspension spectrum of B to that of A. Roughly speaking, this transfer encodes the Euler characteristics of the connected components of the fibers of f. For a composable pair f:A-->B and g:B-->C of finitely dominated maps, the composition gf:A-->C is again finitely dominated, and it is natural to ask whether (gf)^! = f^!g^!. That is, if the transfer is functorial with respect to composition of finitely dominated maps. Several attempts to prove this functoriality appeared in the literature, but it is still open.  

In my talk, I will explain the subtlety of this functoriality from several directions. First, I will discuss a wider context in which an analogous transfer can be defined and demonstrate that in this wider generality functoriality fails. I will then give a general formula from a joint work with Cnossen, Ramzi, and Yanovski, for the transfer along the composition gf. Besides f^!g^!, this formula involves the ``traces of monodromy'' along various free loops in B. Based on this formula, I will present several cases in which the transfers do compose and a result of Klein, Malkiewich, and Ramzi on the functoriality of transfers for arbitrary finitely dominated maps on the level of \pi_0.

Many of the functors typically studied using Orthogonal Calculus, such as BO or BTOP, admit a lax symmetric monoidal structure, yet such structures have not played a role in Orthogonal Calculus so far.

I will explain a proof of the fact that the Taylor approximations of a lax symmetric monoidal functor are themselves lax symmetric monoidal, using methods of infinity-category theory such as Day convolution. I will also explain what kind of maps are induced on the derivative spectra of a lax symmetric monoidal functor.

I will talk about joint work with Hebestreit, Weiss, and Winges. First, I will give a brief introduction to homology manifolds from a modern perspective and indicate that the underlying homotopy type of a compact homology manifold is a Poincare duality complex, and hence is equipped with a Spivak normal fibration. It is an old theorem of Ferry and Pedersen that this Spivak normal fibration admits a (canonical) reduction to a stable euclidean bundle, or equivalently that a compact homology manifold admits a (canonical) degree one normal map from a closed topological manifold. Much of the literature on homology manifolds rests on the existence of such a degree one normal map. I will then show that this fact and the seminal results of Bryant-Ferry-Mio-Weinberger on the surgery theory of homology manifolds contradict each other. 

In the final part, I will summarize which parts of BFMW should be considered open, and under what assumptions their main result on the surgery classification should hold true, based on communications with Weinberger. Taking this for granted, our approach in fact yields an example of a homology manifold whose Spivak fibration does not admit a lift to a stable euclidean bundle.

It was conjectured by Milnor that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated.  The main result of this paper (arXiv:2303.15347) is a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated. 

After a brief crash course in the background material we will discuss the topological and geometric construction of the space.  In particular we will see how the mapping class group of S^3xS^3 can be used to build a complete space with twisted actions that are geometrically compatible with Ricci curvature.