Orthogonal calculus is a version of functor calculus that sits at the interface between geometry and homotopy theory; the calculus takes as input functors defined on Euclidean spaces and outputs a Taylor tower of functors reminiscent of a Taylor series of functions from differential calculus. The interplay between the geometric nature of the functors and the homotopical constructions produces a calculus in which computations are incredibly complex. These complexities ultimately result in orthogonal calculus being an under-explored variant of functor calculus.

The Nielsen-Thurston classification theorem in the theory of mapping class groups and Thurston's theorem in complex dynamics are cornerstone theorems in their respective fields. They give normal forms for homeomorphisms and branched covers of surfaces, respectively. In joint work with Belk and Winarski, we state an ubertheorem that contains both theorems as special cases. We prove the ubertheorem as a consequence of Teichmuller's theorem, just as Bers proved the Nielsen-Thurston classification. Our work resolves several open questions. I will aim for the talk to be accessible to a wide audience of topologists.

Twists of supersymmetric QFTs give rich examples of topological field theories connected to mathematics including the 2d A/B models of mirror symmetry and the theory of Donaldson—Thomas invariants. In this talk, I will show how twists of supersymmetric quantum mechanics with a Kahler target exhibits a model for BV quantization. Using this, I will give a proposal for the Hilbert space of states in terms of the cohomology of a certain perverse sheaf. I’ll mention a few examples including the ordinary A-model and the categorified DT invariants of a CY 3-fold. This is based on joint work with Pavel Safronov.

I will talk about pseudo-isotopy, a notion important for understanding self-diffeomorphisms of manifolds up to isotopy. Pseudo-isotopies of manifolds in dimensions 5 and up were understood in the 70s by work of Cerf for simply connected manifolds, and by Hatcher and Wagoner in the non-simply connected case, using invariants from algebraic K-theory. Quinn later proved Cerf’s result topologically in dimension 4, leading to a classification of self-homeomorphisms of simply connected 4-manifolds up to isotopy. I will talk about my work on what Hatcher and Wagoner’s K-theoretic invariants can say about pseudo-isotopies of non-simply connected 4-manifolds, and how they can be used to construct diffeomorphisms of certain 4-manifolds which are pseudo-isotopic but not isotopic to the identity.

We ask to what extend the SL(2,C)-character variety of the fundamental group of the complement of a knot in S^3 determines the knot. Our methods use results from group theory, classical 3-manifold topology, but also geometric input in two ways: The geometrisation theorem for 3-manifolds, and instanton gauge theory. In particular this is connected to SU(2)-character varieties of two-component links, a topic where much less is known than in the case of knots. This is joint work with Michel Boileau, Teruaki Kitano and Steven Sivek.

The mapping class group of a compact simply-connected 4-manifold is the set of self-diffeomorphisms (or self-homeomorphisms,in the topological category), up to isotopy. For a manifold with nonempty boundary, one assumes the self-automorphisms fix the boundary pointwise. In both the smooth and topological categories, I will describe sufficient conditions for two automorphisms to be pseudoisotopic. Pseudoisotopy is weaker than isotopy, but in the topological category we are able to use this theorem to compute the mapping class group in many situations, extending results of Quinn from the closed case. We use our theorem to prove new topological unknotting results for embedded 2-spheres in 4-manifolds homotopic to the 2-sphere. This is joint work with Mark Powell.

The symplectic (A-infinity,2)-category Symp, which is currently under construction by myself and my collaborators, is a 2-category-like structure whose objects are symplectic manifolds and where hom(M,N) := Fuk(M^- x N). Symp is a coherent algebraic structure which encodes the functoriality properties of the Fukaya category. This talk will begin with the following question: what can we say about the part of Symp that knows only about a single symplectic manifold M, and the diagonal Lagrangian correspondence from M to itself? We expect that the answer to this question should be a chain-level algebraic structure on symplectic cohomology, and in this talk I will present progress toward confirming this. Specifically, I will present a "simplicial version" of the 2-dimensional Fulton-MacPherson operad, which may be of independent topological interest. I will emphasize the topological aspects of this project; in particular, I will assume no symplectic background.

A closed surface can be endowed with a certain locally Euclidean metric structure called a translation surface. Moduli spaces that parametrize such structures are called strata, and there is still much to discover of their global topology. These strata admit a decomposition into finitely many polytopal regions parametrized by certain triangulations of translation surfaces (L-infinity Delaunay triangulations). These regions are adjacent to each other in pathological ways, but it was conjectured by Frankel that these pathologies can be nicely classified. We affirm this conjecture of Frankel, and use the resulting classification to give an explicit presentation of strata as quotients of locally finite simplicial complexes via the action of the mapping class group.

I will describe some approaches to getting at chromatic patterns in equivariant stable stems, mostly focusing on the group C_2. At height 1, these patterns are first seen in equivariant K-theory, and we will see how one may use equivariant K-theory to define and descend to the equivariant K(1)-local sphere. At height 2, equivariant elliptic cohomology plays the role of equivariant K-theory, and I will describe some work concerning C_2-equivariant TMF_0(3).

General covariance is a crucial notion in the study of field theories on curved spacetimes. Roughly, a generally covariant field theory is one that is defined with respect to a background semi-Riemannian metric such that it is only sensitive to the diffeomorphism classes of that metric. In other words, the bundle of theories over the space of semi-Riemannian metrics is equivariant with respect to the diffeomorphism group of the underlying spacetime. In this talk, we will make the preceding ideas precise using stacks and introduce examples, using the Batalin-Vilkovisky formalism to discuss classical field theories.

General covariance is a crucial notion in the study of field theories on curved spacetimes. Roughly, a generally covariant field theory is one that is defined with respect to a background semi-Riemannian metric such that it is only sensitive to the diffeomorphism classes of that metric. In other words, the bundle of theories over the space of semi-Riemannian metrics is equivariant with respect to the diffeomorphism group of the underlying spacetime. In this talk, we will make the preceding ideas precise using stacks and introduce examples, using the Batalin-Vilkovisky formalism to discuss classical field theories.

We will review a geometric approach to the construction of Steenrod operations and generalize it to construct RO(G)-graded Steenrod operations for all finite discrete group G. Some properties of the RO(G)-graded Steenrod operations will be discussed.

Factorization algebras are cosheaf-like objects which mix algebra and geometry. Costello and Gwilliam have constructed, for any quantum field theory on a manifold (without) boundary, a factorization algebra of observables for the field theory. In my dissertation, I extended the result of Costello and Gwilliam to a class of field theories on manifolds with boundary. In this talk, I will survey the results of my dissertation, beginning with a brief introduction to factorization algebras. Then, I will focus on two examples of bulk-boundary systems: topological mechanics and BF theory, both on the non-negative real line.

The cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces. The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic. In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum.The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

The cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces. The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic. In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum.The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

We consider versions of the Lusternik-Schnirelmann category with $\pi_1$-constraints of geometric origin, such as amenability or polynomial growth. I will survey the connection with topological volumes such as simplicial volume.

The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite abelian groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples.

Murray, Roberts and Wockel showed that there is no strict model of the string 2-group using the free loop group. Instead, they construct the next best thing, a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. Based on their expectations, we build a category of 2-representations for coherent Lie 2-groups and some interesting examples. We also discuss the relation between this category of 2-representations and the category of representations. Afterwards, we build the associated 2-vector bundles of the string group.

In further projects, we will construct the tensor product of 2-representations and compare it with the Connes fusion product of bimodules over von Neumann algebras. And I will explore the relation between elliptic cohomology and 2-groups by first studying Quasi-elliptic cohomology of coherent 2-groups.

This is joint work with Chenchang Zhu.

Any homology class of degree 2 in a simply connected 4-manifold can be represented by an oriented embedded surface as well as a normally immersed sphere. In order to measure the complexity of one such homology class, one can look for the minimum genus among representatives given by embedded surfaces. Similarly, the minimum number of double points among representatives given by immersed spheres provides another measure for complexity of the homology class. It is natural to ask how these two quantities are related to each other. In my talk, I'll discuss some tools which could be useful to study the difference between these two measures on the complexity of homology classes. In particular, I'll explain that how they can be used to show that positive clasp number of a knot can be arbitrarily larger than its slice genus, answering a question raised by Kronheimer and Mrowka. If time permits, I'll also talk about some evidence towards an extension of the slice-ribbon conjecture to torus knots. This talk is based on a joint work with Chris Scaduto.

Embedding calculus assigns to a manifold M the data of the collection of configuration spaces of various numbers of open balls in M and the natural maps between these. The Goodwillie-Klein-Weiss convergence results tell you that you can recover embeddings from their action on this data, as long as the codimension is at least 3. I will explain how this may be used to obtain information about moduli spaces of manifolds, and present some novel convergence results in low dimensions. This includes joint work with Manual Krannich and Mauricio Bustamante.

In this talk I will discuss the proof of a recent result: every action of a finite group G on a finite and contractible 2-complex X has a fixed point. This was conjectured by Carles Casacuberta and Warren Dicks and was also posed as a question by Michael Aschbacher and Yoav Segev. We build on the classification, due to Bob Oliver and Yoav Segev, of the groups G which act without fixed points on an acyclic 2-complex. Part of this work is joint with Kevin Piterman.

All 3-manifolds bound 4-manifolds, and many constructions of 3-manifolds automatically come with a 4-manifold bounding it. Often times these 4-manifolds have definite intersection forms. Using Heegaard Floer correction terms and an analysis of short characteristic covectors in bimodular lattices, we give an obstruction for a 3-manifold to bound a definite 4-manifold, and produce some concrete examples. This is joint work with Kyle Larson.

I will discuss work, joint with Dylan Wilson, in which we construct an E_3 algebra form of BP and study its algebraic K-theory. We prove various redshift statements conjectured by Ausoni and Rognes.

The universal enveloping algebra of a Lie algebra is one in a family of higher enveloping algebras: each dg Lie algebra g has an enveloping E_n algebra U_n(g). This construction admits a nice presentation via factorization algebras, by work of Knudsen, and we will discuss a factorization model for the center of U_n(g). This setting makes computing factorization homology tractable. We will discuss various consequences of this convenient model, in representation theory, topology, and physics. (This is joint work with Greg Ginot, Brian Williams, and Mahmoud Zeinalian.)

For random groups in the Gromov density model at $d<3/14$, we construct walls in the Cayley complex X which give rise to a nontrivial action by isometries on a CAT(0) cube complex, This extends results of Ollivier-Wise and Mackay-Przytycki at densities $d<1/5$ and $d<5/24$, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density $d<1/4$.

The present work describes a topological quantum field theory of Wilson surfaces. We start with a definition of a Wilson line observable in gauge theories, from which we construct first a Wilson surface observable as a 2-dimensional $\sigma$-model. Then a Wilson surface theory is formulated as a separate 2-dimensioanl topological quantum field theory with a 1-dimensional Hilbert space. It has a Lagrangian of a BF theory with a constraint on the B-field. On a closed surface, the Wilson surface theory defines a topological invariant of the principal G-bundle. The Wilson surface theory can interact with some backgroun gauge theory through the topolology of principal G-bundles. We compute explicitly the partition functions of teh 2-dimensioanl Yang-Mills theory interacting with a Wilson surfac for the cases $G=SO(3)$, $G=SU(N)/\mathbb{Z}_m$,$G=Spin(4N)/(\mathbb{Z}_2\oplus \mathbb{Z}_2)$ and obtain a general formula for $G$ any compact connected Lie group.

There is a rich interplay between the fields of knot theory and 3- and 4-manifold topology. In this talk, I will describe a weak notion of equivalence for knots called concordance, and highlight some historical and recent connections between knot concordance and the study of 4-manifolds, with a particular emphasis on applications of knot concordance to the construction and detection of small 4-manifolds which admit multiple smooth structures.

I will discuss a relatively short new proof of the classical fact that Poincare duality groups in dimension 2 are surface groups.

I will discuss the context and solution of the rectangular peg problem: for every smooth Jordan curve and rectangle in the Euclidean plane, one can place four points on the curve at the vertices of a rectangle similar to the one given. The solution involves symplectic geometry in a surprising way. Joint work with Andrew Lobb.

A major open problem in quantum topology is the construction of an oriented 4- dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4-manifolds. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of vector-space-valued oriented 4-dimensional TQFTs, including `unitary field theories' and `once-extended field theories' which assign algebras or linear categories to 2- manifolds. If time permits, I will give a concrete expression for the value of a semisimple TQFT on a simply connected 4-manifold and explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a non-simply connected 4-manifold. This is based on arXiv:2001.02288.