Geometry and Topology

Speaker: Sanjeevi Krishnan (Univ. of Pennsylvania)

"Directed Poincare Duality"

Time: 15:30

Room: MC 107

Sheaves of semimodules on locally ordered spaces model real-world local constraints on dynamical systems indescribable in the language of modules. This talk generalizes a homotopical construction of singular sheaf (co)homology for semimodule-valued sheaves over locally ordered spaces; terminal compactifications of ordered Euclidean space play the role of spheres. Examples of singular (co)homology semimodules include flows on a network subject to capacity constraints and causal singularities on spacetimes. Various calculational tools, incorporating previous cubical approximation theorems, will be presented. The main result presented is a Poincare Duality for sheaves over spacetimes and generalizations admitting top-dimensional topological singularities. Calculations on spacetime surfaces and directed graphs follow.

This talk assumes no familiarity with directed topology or semimodules.

Analysis Seminar

Speaker: Grigoris Fournodavlos (University of Toronto)

"On a characterization of Arakelian sets"

Time: 14:30

Room: MC 108

We are going to discuss some standard problems regarding uniform approximation in the complex domain. In particular, we are going to focus our attention on Arakelian's theorem, i.e., uniform holomorphic approximation on closed sets. These topics have been extensively studied in the past since the celebrated theorem by Mergelyan appeared in the early 50's. What we are going to address in this talk is perhaps a not so noted topological viewpoint of the subject. More precisely, we will motivate the following characterization of Arakelian sets and examine applications: "A closed set $F\subset \mathbb{C}$ is an Arakelian set in the plane, if and only if it has a base of simply connected open neighborhoods."

Homotopy Theory

Speaker: Mike Misamore (Western)

"Morava K-theories and the chromatic tower"

Time: 14:30

Room: MC 108

Comprehensive Exam Presentation

Speaker: Josue Rosario-Ortega (Western)

"Complex Monge-Ampère Equations, the Space of Kähler metrics and complexification of Hamiltonian flows"

Time: 13:00

Room: MC 108

In this talk I will explain some geometric aspects of Complex Monge-Ampère Equations and its relation with the space of Kähler metrics in a fixed cohomology class. I will also explain some recent applications to the complexification of Hamiltonian flows on a compact Kähler manifold.

Algebra Seminar

Speaker: Daniel Schaeppi (Western)

"Which tensor categories come from algebraic geometry?"

Time: 14:30

Room: MC 107

Tannakian duality as developed by Grothendieck, Saavedra, Deligne and Milne is a duality between geometric objects (affine group schemes, gerbes) and tensor categories (Tannakian categories). A Tannakian category is a tensor category equipped with additional structure which ensures that it is equivalent to the category of representations of an affine group scheme or gerbe. In characteristic zero Deligne has found a particularly simple description of Tannakian categories. This description is convenient since it only involves talking about properties of the tensor category. The properties in question are enough to construct the required additional structure.

Tannakian duality can be extended to a broader class of geometric objects including schemes and certain algebraic stacks. The corresponding tensor categories (that is, the categories of coherent sheaves on these objects) are the weakly Tannakian categories. I will review the notion of weakly Tannakian category, and I will talk about work in progress to generalize Deligne's description of Tannakian categories in characteristic zero to an intrinsic description of weakly Tannakian categories in characteristic zero.