For two thousand years geometry was synonymous with the perfectly flat expanse imagined by Euclid. But nineteenth‑century investigations into the parallel postulate lifted a veil from our eyes, revealing the richer realms charted by Gauss and Riemann. In this talk we’ll take an “insider’s tour” of those curved landscapes.

We begin by thinking carefully about what it means to see, and use this to step inside new geometries, by tracing light rays along their geodesics. Modern computing affords us the ability to make this thought experiment a reality, with interactive ray-traced demos and experiments.  Using these, we will explore curved spaces important to modern mathematics, and physics, including the curved spacetime we live in.

Just as a current impacts the effort a swimmer must make, so too does the research atmosphere in a community or conference affect research output.  In this talk, I will discuss various examples of this, both long-standing programs of others, and many examples that I have experienced or witnessed.  In particular, I will discuss different branches of my research program and how their development was impacted by the atmosphere in conferences, seminars, and research communities. I also discuss what I have learned from times when my actions have created counter currents for others.

Content note: This talk will include some descriptions of harassment.

Kontsevich’s formality theorem in deformation theory asserts that the Hochschild-Kostant-Rosenberg map lifts to a quasi-isomorphism of homotopy Gerstenhaber algebras between polyvector fields and polydifferential operators. By exhibiting the sheaf of polydifferential operators as a universal enveloping algebra, this can be interpreted as a type of Duflo isomorphism. In this talk, I will present my work towards understanding this geometric Duflo isomorphism in the context of ∞-operads and higher centers. In particular, I will explain how higher centers can be used to equip the Hochschild cochain complex with a universal property.

Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative (i.e. $\mathbb{E}_\infty$) ring spectra. In this talk, I will discuss an algebro-geometric analogue of this framework, called the theory of normed motivic ring spectra. As a particular example of interest, I'll show that (very effective) Hermitian K-theory can be equipped with a normed ring structure.

Definable coherent sheaves (with respect to an o-minimal structure) were introduced by Bakker, Brunebarbe and Tsimerman  (BBT) and used as an essential tool in their proof of Griffiths' conjecture that the image of the period map is algebraic.   The category of these definable sheaves on a complex algebraic variety X sits in between the category of algebraic and analytic sheaves.  More precisely, there is a definablization functor taking coherent algebraic sheaves to definable coherent sheaves and an analytification functor going from the category of definable coherent sheaves to the category of coherent analytic sheaves.  This makes them useful for answering questions about analytic maps involving algebraic varieties.  I'll explain these two functors and the concept of o-minimality necessary to define the BBT category of definable coherent sheaves.  Then I'll state a couple of results I obtained recently with Adam Melrod on the cohomology groups of definable coherent sheaves both in the case where X is projective (when, for reasonable o-minimal structures,  the groups are the same as the usual cohomology groups) and the general case (when they very much aren't).