Title: Directed univalence and the Yoneda embedding for synthetic (∞,1)-categoriesAbstract: I'll present recent advances in synthetic (∞,1)-category theory, more specifically a modal extension of Riehl--Shulman's simplicial homotopy type theory. This includes the construction of the univeral left fibration, the Yoneda embedding and Yoneda lemma, a study of cofinal functors, Quillen's Theorem A, and first steps in […]
Title: Towards a birational geometric version of the monodromy conjecture.Abstract: The monodromy conjecture of Denef—Loeser is a conjecture in singularity theory that predicts that given a complex polynomial f, and any pole s of its motivic zeta function, exp(2πis) is a "monodromy eigenvalue" associated to f. I will formulate a "birational geometric" version of the conjecture, and […]
Title: Surfaces and Differential GeometryAbstract: Informally, differential geometry is using calculus to study smooth objects. More formally, it is the study of smooth objects that locally look like R^n. Classically, differential geometry is the study of smooth curves (informally bent lines) and surfaces (informally bent paper) inside of three-dimensional space.In this talk, we will define and […]
Title: Poisson geometry and Azumaya loci of cluster algebrasAbstract: Roughly speaking, cluster algebra is a commutative algebra obtained from taking the intersection of Laurent polynomial rings associated a "seed". When a cluster algebra satisfies some compatibility condition, M. Gekhtman, M. Shapiro, and A. Vainshtein showed that one can connect a Poisson bracket to the cluster algebra. In […]
Title - Stability Theorems for the Width Abstract - In this talk, I'll discuss some recent and ongoing work about the stability of min-max widths of spheres under various lower curvature bounds. Some of this is joint with Davi Maximo and Paul Sweeney Jr.
Title: Shtukas and their cohomology. Abstract: I will introduce the moduli space of shtukas and explain a spectral description of their cohomology using categorical traces along a Frobenius endomorphism. Reference: https://arxiv.org/abs/1606.09608
We will be having a a mixer with the undergrad physics club SPS on Friday in Bloomberg 462 at 430pm. We will have jeopardy, lots of nice free food, an integration bee, and other fun activities planned! Come by for a fun time!
Title: Triple junction solution for the Allen-Cahn systemAbstract: In this talk, I will discuss recent results on the vector-valued Allen-Cahn system with a triple-well potential. We establish the existence of an entire minimizing solution that asymptotically converges to a triple junction, corresponding to a planar minimal cone, at infinity. Furthermore, we prove the uniqueness of […]
Title: Hodge symmetries of singular varieties.Abstract: The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge–Du Bois diamond. For example, the failure […]
Title: An Introduction to the Koszul ComplexAbstract: In this talk we will explore the motivation, definition, and usage of the Koszul complex. We will specifically pay attention to examples of Koszul complexes in geometry, representation theory, and analysis. Ideally the audience knows some linear algebra and abstract algebra.
Title: Higher Modularity of Elliptic Curves Abstract: Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), […]