Talks on Tuesdays, 3-4 pm in University 213.
(Talks listed with first-name only are tentative, and maybe move / be canceled depending on other speakers' schedules.)
October 5. Andrew Hanlon, "Categorical dimensions in (symplectic) topology"
Abstract: In geometry and topology, there are many ways of measuring the complexity of a manifold. Some of these invariants, such as the minimum number of critical values of a Morse function, describe how easy it is to build your space of interest from smaller pieces. For a triangulated category, one can similarly study the smallest number of cones needed to build every object from generators which is known as the Rouquier dimension. I will discuss how these notions are related and some of the many open questions on this topic. This is joint work with Jeff Hicks and Oleg Lazarev.
October 13. Elden Elmanto, "Splitting vector bundles and Suslin's isomorphism"
Abstract: I want to explain some of the details in my colloquium and introduce two surprisingly useful definitions of dimension of a variety: 1) its motivic dimension (obtained in work with Morrow) and 2) its Lichtenbaum--Quillen dimension (obtained with Addington) and explain how these interact with extensions of Bloch--Kato conjectures in singular settings following old theorems of Suslin.
October 20. Alexandru Suciu, "Non-formal Milnor fibers, from hyperplane arrangements to polyhedral products"
Abstract: The Milnor fiber F of a weighted-homogeneous polynomial f is a smooth affine variety whose topology encodes subtle information about the singularity of f at the origin. A basic question is whether F is formal, in the sense of rational homotopy theory. I will address this question in two complementary settings.
The first is that of complex hyperplane arrangements, where 1-formality of the Milnor fiber $F(\mathcal{A})$ obeys a combinatorial dichotomy. Arrangements that are decomposable over $\mathbb{Q}$ have 1-formal Milnor fibers, under a mild separation hypothesis; on the other hand, the presence of two or more reduced 3-multinets obstructs 1-formality. This yields a combinatorial criterion for non-1-formality, together with an explicit infinite family of examples, the monomial arrangements $\mathcal{A}(3k,3k,3)$.
The second setting is the highly connected range, where a realization theorem of Fernández de Bobadilla identifies every moment-angle complex $\mathcal{Z}_K$ with the Milnor fiber of an explicit weighted-homogeneous polynomial with trivial monodromy. Combining this with Massey product constructions in moment-angle complexes produces Milnor fibers that are non-formal and arbitrarily highly connected. For an explicit family of such examples, I will also determine the formality degree of the Milnor fiber and show that the Kato–Matsumoto connectivity bound is sharp.
November 3. Lenny Ng, TBD. In the Oregon Contact TOPology Unleashed Seminar (joint with Oregon State)
Abstract: TBD