The junior geometry seminar meets every Tuesday at 3:45-4:45pm central in PMA 12.166
What is the junior geometry seminar?
The organizational meeting was on January 10 at 3:45 pm central.
Email reminders will be sent to the mailing list. Click here to be added to the mailing list.
List of past organizers (aka JG ORGANIZER HALL OF FAME )
The titles and abstracts of the talks are also available here. at the official math department seminar calendar.
Title: Fans of Geometry Rejoice
Abstract: What does combinatorics have to do with schemes? In the spirit of adding more representation to the Bernd Siebert/Sam Payne brand of geometry, I’ll give an elementary answer to the question by introducing you to toric geometry. We’ll see how you can build a handy class of complex schemes using nothing more than polyhedral fans. What’s better, we’ll see how this allows one to give extremely concrete answers to sometimes nebulous questions such as “is this scheme smooth,” and “what is the divisor class group of this scheme?”, time allowing.
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Title: (Canceled due to weather)
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Title: The Brown Representability Theorem
Abstract: We provide an exposition of the Brown representability theorem. The theorem gives sufficient conditions for a contravariant functor from the homotopy category of based CW-complexes to the category of pointed sets to be representable. In particular, it proves the representability of cohomology theories satisfying the Eilenbeerg-Steenrod axioms.
Title: From Mechanics to Geometry
Abstract: In this talk we see how the progression of analytical mechanics, from Newtonian to Lagrangian to Hamiltonian, and careful examination of the equations describing Hamiltonian mechanics lead to the definition of a symplectic manifold and simple proofs of seemingly non-trivial results from classical mechanics in a generalized setting through the language of differential forms. We will observe how a natural symplectic form may be given to the cotangent bundle of a smooth manifold and how to recover a generalization of Lagrangian mechanics under a certain restriction.
Title: Equivariance and coherence
Abstract: After reviewing the metaplectic representation, I will explain how we can think about the projectivity of the symplectic action as an "anomaly". Then we will consider 3 and 4-dimensional finite gauge theory, where the automorphism groups are now higher groups. The group cocycle capturing the projectivity in the metaplectic case will be replaced by similar cohomological data characterizing these higher automorphism group(oid)s.
Title: Higher Rank Uniformization
Abstract: The uniformization theorem tells us that there is a unique hyperbolic hermitian metric on each Riemann surface of genus at least 2. Because the automorphism group of the hyperbolic plane is PSL(2,R), this ends up giving an identification of the Teichmuller space of a surface S (the space of isotopy classes of complex structures on S) with a component of the variety of PSL(2,R) local systems on S. I will explain this story in more detail, and talk about a conjectural generalization to SL(n,R) that I'm working on, together with Kydonakis, Nolte, and Thomas.
Title: An unperturbed approach to Morse homology: Morse-Bott Homology
Abstract: In this seminar I will introduce the Morse-Bott approach to Morse homology, which doesn’t necessitate to perturb a real valued function to be a Morse function. Why is that important? In the spirit of “transversality is the opposite of symmetry”, Morse functions almost always destroy the inner symmetries of a space, i.e. don’t take them into account. Morse-Bott functions instead are a little bit more flexible in that regard and sometimes let us preserve these symmetries. I will spend 10 minutes recalling what Morse homology is (but you will appreciate the talk more if you are familiar with that already) and then talk about different approaches to Morse-Bott homology. Time permitting I will mention a couple of applications of Morse Bott homology (in symplectic geometry for example) but do not expect anything too detailed since I’m currently learning it myself :)
Title: Lagrangian and Heegaard-Floer homologies
Abstract: A Floer homology theory is, roughly speaking, a homology theory derived by running the machinery of Morse homology on a high- or infinite-dimensional space associated to a manifold in some way. These theories do not generally produce homology groups isomorphic to singular homology, but instead give information about structures on the manifold in question depending on the space associated. In this talk, we'll discuss the original formulation given by Andreas Floer in 1988, now known as Lagrangian Floer homology and used in Floer's proof of the Arnold conjecture in symplectic geometry, as well as the newer theory of Heegaard-Floer homology and its applications to 3- and 4-manifold topology.
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Title: Quasiconformal Maps and the MRMT
Abstract: Quasiconformal (qc) maps are generalizations of conformal maps in the following sense: infinitesimal circles are mapped to infinitesimal ellipses in a bounded way. After a review on conformal maps, we start the talk by making sense of this. Through this notion, one can formulate the so-called measurable Riemann mapping theorem (MRMT), a helpful tool in surface topology, for instance. We’ll discuss some heuristics to prove the MRMT. Finally, we will close with some applications of qc maps in geometry/topology. In spite of the MRMT, no background in measure theory or PDE will be assumed.
Title: Extrinsic curvature flows and classical geometric problems
Abstract: The goal of this presentation is to explain how to understand geometric flows from the perspective of parabolic PDEs and how they are linked to minimal surfaces and surfaces with constant mean curvature. We will consider two parabolic flows: the mean curvature flow (MCF) and the volume preserving mean curvature flow (VPMCF). Both flows will be interpreted as suitable gradient flows and will be linked to corresponding classical geometric problems, i.e., the MFC to minimal surfaces and the VPMCF to the isoperimetric problem. Lastly, we will discuss (in a rather general perspective) how to address the study of long time behavior of solutions to these two problems.
Title: The Geometry of Ambiguity
Abstract: Following an eponymous survey by Mathieu Anel, I will describe how classical tangent spaces in algebraic geometry can be generalized to tangent complexes which recover the tangent space for smooth varieties but provide a better linearization for singular spaces (whose classical tangent spaces have too high a dimension). I will restrict my exposition to singular spaces that arise from taking non-transverse intersections or quotients of smooth spaces by non-free group actions. This is intended to serve as a concrete answer to the age-old question “Why should anyone care about derived algebraic geometry?”