Past Junior Geometry and Topology Seminar

9 February 2012
13:00
Hemanth Saratchandran
Abstract
I will give a brief introduction into how Elliptic curves can be used to define complex oriented cohomology theories. I will start by introducing complex oriented cohomology theories, and then move onto formal group laws and a theorem of Quillen. I will then end by showing how the formal group law associated to an elliptic curve can, in many cases, allow one to define a complex oriented cohomology theory.
  • Junior Geometry and Topology Seminar
2 February 2012
13:00
Chris Hopper
Abstract
I will give an introduction to the variational characterisation of the Ricci flow that was first introduced by G. Perelman in his paper on "The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159. The first in a series of three papers on the geometrisation conjecture. The discussion will be restricted to sections 1 through 5 beginning first with the gradient flow formalism. Techniques from the Calculus of Variations will be emphasised, notably in proving the monotonicity of particular functionals. An overview of the local noncollapsing theorem (Perelman’s first breakthrough result) will be presented with refinements from Topping [Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055.]. Some remarks will also be made on connections to implicit structures seen in the physics literature, for instance of those seen in D. Friedan [Ann. Physics 163 (1985), no. 2, 318–419].
  • Junior Geometry and Topology Seminar
26 January 2012
13:00
Jakob Blaavand
Abstract
In this talk we will discuss geometric quantization. First of all we will discuss what it is, but shall also see that it has relations to many other parts of mathematics. Especially shall we see how the Hitchin connection in geometric quantization can give us representations of a certain group associated to a surface, the mapping class group. If time permits we will discuss some recent results about these groups and their representations, results that are essentially obtained from geometrically quantizing a moduli space of flat connections on a surface."
  • Junior Geometry and Topology Seminar
20 January 2012
12:00
to
13:45
Vittoria Bussi
Abstract
This is the second of two talks about Derived Algebraic Geometry. We will go through the various geometries one can develop from the Homotopical Algebraic Geometry setting. We will review stack theory in the sense of Laumon and Moret-Bailly and higher stack theory by Simpson from a new and more general point of view, and this will culminate in Derived Algebraic Geometry. We will try to point out how some classical objects are actually secretly already in the realm of Derived Algebraic Geometry, and, once we acknowledge this new point of view, this makes us able to reinterpret, reformulate and generalize some classical aspects. Finally, we will describe more exotic geometries. In the last part of this talk, we will focus on two main examples, one addressed more to algebraic geometers and representation theorists and the second one to symplectic geometers.
  • Junior Geometry and Topology Seminar
19 January 2012
12:00
to
13:45
Vittoria Bussi
Abstract
This is the first of two talks about Derived Algebraic Geometry. Due to the vastity of the theory, the talks are conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as it might reveal interesting new insights also on classical objects, rather than a detailed and precise exposition. We will start with an introduction on the very basic idea of this theory, and we will expose some motivations for introducing it. After a brief review on the existing literature and a speculation about homotopy theories and higher categorical structures, we will review the theory of dg-categories, model categories, S-categories and Segal categories. This is the technical part of the seminar and it will give us the tools to understand the basic setting of Topos theory and Homotopical Algebraic Geometry, whose applications will be exploited in the next talk.
  • Junior Geometry and Topology Seminar
1 December 2011
12:00
Martin Palmer
Abstract
After recalling some definitions and facts about spectra from the previous two "respectra" talks, I will explain what Thom spectra are, and give many examples. The cohomology theories associated to various different Thom spectra include complex cobordism, stable homotopy groups, ordinary mod-2 homology....... I will then talk about Thom's theorem: the ring of homotopy groups of a Thom spectrum is isomorphic to the corresponding cobordism ring. This allows one to use homotopy-theoretic methods (calculating the homotopy groups of a spectrum) to answer a geometric question (determining cobordism groups of manifolds with some specified structure). If time permits, I'll also describe the structure of some cobordism rings obtained in this way.
  • Junior Geometry and Topology Seminar
17 November 2011
12:00
Michael Gröchenig
Abstract
This is the first in a series of $\geq 2$ talks about Stable Homotopy Theory. We will motivate the definition of spectra by the Brown Representability Theorem, which allows us to interpret a spectrum as a generalized cohomology theory. Along the way we recall basic notions from homotopy theory, such as suspension, loop spaces and smash products.
  • Junior Geometry and Topology Seminar
10 November 2011
12:00
Tim Adamo
Abstract
Chern-Simons theory is topological gauge theory in three dimensions that contains an interesting class of operators called Wilson lines/loops, which have connections with both physics and pure mathematics. In particular, it has been shown that computations with Wilson operators in Chern-Simons theory reproduce knot invariants, and are also related to Gauss linking invariants. We will discuss the complex generalizations of these ideas, which are known as holomorphic Chern-Simons theory, Wilson operators, and linking, in the setting of Calabi-Yau three-folds. This will (hopefully) include a definition of all three of these holomorphic analogues as well as an investigation into how these ideas can be translated into simple homological algebra, allowing us to propose the existence of "homological Feynman rules" for computing things like Wilson operators in a holomorphic Chern-Simons theory. If time permits I may say something about physics too.
  • Junior Geometry and Topology Seminar
3 November 2011
12:00
Benjamin Volk
Abstract
We will give an introduction to the theory of d-manifolds, a new class of geometric objects recently/currently invented by Joyce (see http://people.maths.ox.ac.uk/joyce/dmanifolds.html). We will start from scratch, by recalling the definition of a 2-category and talking a bit about $C^\infty$-rings, $C^\infty$-schemes and d-spaces before giving the definition of what a d-manifold should be. We will then discuss some properties of d-manifolds, and say some words about d-manifold bordism and its applications.
  • Junior Geometry and Topology Seminar

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