Title: Pyknotic setsAbstract: In this talk, I will introduce the idea of Pyknotic/Condensed sets and how they can potentially be used to solve the problem of the category of topological spaces not being abelian. I will also go over the prerequisite concepts in category theory such as the notion of Grothendieck topology, pro-object in a […]
Title: On the cone conjecture for log Calabi-Yau threefolds.Abstract: Let $Y$ be a smooth projective threefold admitting a $K3$ fibration $f: Y rightarrow mathbb{P}^{1}$ with $-K_{Y} = f^{ast} mathcal{O}(1)$. Then the extremal rays of the cone of curves of $Y$ in the region $K_{Y} < 0$ are of two types: the blowup of a smooth curve (Type […]
Title: Modularity of trianguline Galois representations Abstract: The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation rho: Gal_Q -> GL2(Qp) is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an […]
Title: Speculations on Computing with the Relative Trace FormulaAbstract: I will discuss and motivate the problem of computing statistics of automorphic representations. Many statistics have been computed previously using "explicit" forms of Arthur's trace formula. I will present a very speculative plan to develop similar "explicit" versions of relative trace formulas and mention some new specific […]
Title: A Generalization of the Geroch Conjecture with Arbitrary EndsThe Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles. In particular, I will talk about why the […]
Title: Tannaka duality for algebraic 2-groupsAbstract: The Tannakian formalism is a web of statements which, in its most basic form, attempts to recover a group G from its symmetric monoidal category of representations. The goal of this talk is to explain a Tannaka duality result that applies to groups in Artin stacks. In this case, […]
Title: Pyknotic/condensed sets II Abstract: We will continue the discussion of pyknotic sets, and introduce the notion of condensed sets.
Title: Symmetries of Fano varieties.Abstract: A landmark result of Birkar, Prokhorov, and Shramov shows that automorphism groups of Fano (or more generally rationally connected) varieties over C of a fixed dimension are uniformly Jordan. This means in particular that there is some upper bound on the size of symmetric groups acting faithfully on rationally connected varieties of […]
Title: An introduction to Hopf Rings.Abstract: An old guy’s perspective on what homotopy theory used to be about and why complex cobordism has become central to the study.And, I’ll show one approach to getting more (and different) information about complex cobordism.If time permits (it won’t), I’ll show some of how chromatic homotopy theory comes out […]
Title: Dirac geometryAbstract: Whatever it is that animates anima and breathes life into higher algebra, this something gives the homotopy groups of a commutative algebra in spectra the structure of a commutative algebra in the symmetric monoidal category of graded abelian groups. Being commutative, these algebras form the affine building blocks of a geometry, which […]
Title: Elliptic curves and modular forms.Abstract: If you’ve heard of Fermat’s last theorem, you might have heard that its proof involves some sophisticated geometric objects that seemingly have nothing to do with whether a^n + b^n = c^n for n > 2. The objects in question are elliptic curves and modular forms. I will attempt […]