Convex projective geometry is a flexible generalization of hyperbolic geometry that retains many of the nice features of its hyperbolic counterpart. It comes in two distinct flavors: properly convex geometry and strictly convex geometry. In the strictly convex setting work of Benoist and Cooper, Long, and Tillmann have shown that strictly convex manifolds are geometrically and structurally similar to hyperbolic manifolds. On the other hand, general properly convex manifolds can exhibit extremely non-hyperbolic behavior. However in recent work with Darren Long we show that if the properly convex manifold is "almost" strictly convex then it will remain structurally similar to a hyperbolic manifold. I will discuss this last phenomenon by way of a concrete example.

Given two smooth maps of manifolds $f:M \to L$ and $g:N \to L,$ if they are not transverse, the fibered product $M \times_L N$ may not exist, or may not have the expected dimension. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In fact, every (quasi-smooth) derived manifold is locally of this form. In this talk, we briefly explain what derived manifolds ought to be, why one should care about them, and how one can describe them. We end by explaining a bit of our joint work with Dmitry Roytenberg in which we make rigorous some ideas of Kontsevich to give a simple model for derived intersections as certain differential graded manifolds.

A Margulis spacetime is the quotient of three-dimensional space by a free group of affine transformations acting properly discontinuously. Each of these manifolds is equipped with a flat Lorentzian metric compatible with the affine structure. I will survey some recent results, joint with Francois Gueritaud and Fanny Kassel, about the geometry, topology, and deformation theory of these flat spacetimes. In particular, we give a parameterization of the moduli space in the same spirit as Penner's cell decomposition of the decorated Teichmuller space of a punctured surface. I will also discuss connections with the negative curvature (anti de Sitter) setting.

Given a pair of collections of disjoint simple closed curves on a closed, oriented surface, it is straightforward to check when they have the same orbits under the mapping class group using the topology of the cut open surface and the dual graphs to the systems of curves. When the curve systems have intersections, the dual graph may not be well-defined. We will explore a useful fix for this problem, Sageev's dual cube complex, which we show characterizes mapping class group orbits for filling systems of curves. We will use this invariant to give new bounds for the number of mapping class group orbits of maximal complete 1-systems of curves on a surface. This is joint work with Tarik Aougab.

Going back to Zwiebach, there is a connection between the geometry of moduli spaces and master equations. We discuss that this setup generalizes to what we call an odd version. This means that for any type of operadic structure there is a certain twist on the chain level which produces the correct operations (BV and Gerstenhaber brackets) for the master equation. The equation itself then appears as the obstruction for algebras over a so--called Feynman transform to be compatible with a natural differential. We will discuss the geometry involving moduli spaces as well as the algebraic aspects in the most common examples. We will finish with a novel categorical setup, that of Feynman categories, which collects all these under a common roof.

I will study a stabilization of the symplectic category of Weinstein. This category can be described as a suitable category of motives and serves as a domain for geometric quantization. We will also show that monoidal symmetries of the stable symplectic category can be identified with a quotient of the Grothendieck-Teichmuller group. This provides evidence to a conjecture of Kontsevich that claims that the Grothendieck-Teichmuller group acts on the moduli space of certain field theories.

The systole of a Riemannian manifold M is its shortest closed geodesic. If M is a hyperbolic 3-manifold of finite volume there is an upper bound for the length of the systole which is logarithmic in its volume. Though systole length is purely geometric information, Mostow rigidity means it is also a topological invariant of M, and thus we may study how systole length varies under Dehn surgery. A result of Adams and Reid provides a universal bound on the systole length of any hyperbolic link complement M-L, provided M is non-hyperbolic. In this talk I'll discuss the relationship between systole length and volume, with the goal of generalizing this theorem to an arbitrary 3-manifold M. This is joint work with Chris Leininger.

Let S_g denote the closed, genus g surface. In this talk, we'll discuss the space of flat circle bundles over S_g, also known as the "representation space" Hom(pi_1(S_g), Homeo+(S^1)). The Milnor-Wood inequality gives a lower bound on the number of components of this space (4g-3), but until very recently it was not known whether this bound was sharp. In fact, we still don't know whether the space has finitely many or infinitely many components (!)

I'll report on recent work and new tools to understand Hom(\pi_1(S_g), Homeo+(S^1)). In particular, I use dynamical methods to give a new lower bound on the number of components, draw analogies with more familiar character varieties and representation spaces, and find surprising rigidity phenomena for geometric representations.

KR-theory is the K-theory of vector bundles with a conjugate-linear involution. It arises in physics in the classification of D-brane charges in what are called orientifold string theories, where space-time is equipped with an involution and the fields satisfy an equivariance condition. (This kind of theory is quite useful for getting rid of physically unrealistic states, such as tachyons.) We describe how fuller analysis forces one to consider twists of KR-theory, and we explain how this works for orientifold string theories on elliptic curves. This is joint work with Chuck Doran and Stefan Mendez-Diez.

Topological phases, aka. Berry phases, appear in systems with non-trivial topology. They are related to Chern classes of line bundles on the complement of a set of "degenerate" points. Motivated by a certain material geometry we discuss this type of setup in what is called momentum space. We will touch upon the singularities of the degenerate locus as well as local and global structure of the charges.

Atiyah-Segal and others define a twisted form of K-theory associated to classes in H^3(X). Their method is geometric, using the Fredholm operator model for the spaces which define K-theory. Homotopically, this amounts to a multiplicative map from K(Z, 2) to the space of units of K-theory, GL_1(K). In joint work with Hisham Sati, we extended this construction to higher-chromatic versions of K-theory, Morava's E-theories, E_n. We computed the space of E-infinity maps from K(Z, n+1) to GL_1(E_n), thereby introducing a natural form of twisted E-theory. I will talk about these constructions and subsequent work which applies them to the study of the stable homotopy groups of the (K(n)-local) sphere.