Past Junior Geometry and Topology Seminar

30 May 2013
12:00
Vittoria Bussi
Abstract
This talk is not a detailed and precise exposition on DAG, but it is 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 they might reveal interesting new insights also on classical objects. We select some of the several motivations for introducing it (non-representability of moduli problem and non-naturality of the obstruction theory), and then we will go through the homotopy theory of simplicial commutative algebras and their cotangent complex. We will introduce the category of derived schemes and we will describe their relation with classical schemes. A good amount of time will be dedicated to examples.
• Junior Geometry and Topology Seminar
23 May 2013
15:00
Rafael Torres
Abstract
"Among the first successes of the h-cobordism theorem was the classification of simply connected closed 5-manifolds. Dimension five is sufficiently large to be able to implement the tools of surgery theory, yet low enough to allow an explicit classification of the manifolds. These traits make dimension five interesting in terms of existence results of geometric structures, like Riemannian metrics of positive Ricci/nonnegative sectional/positive sectional curvature, Einstein metrics, contact structures, Sasakian structures, among others. The talk will be a limited survey of the five-dimensional symbiosis between topology and geometry"
• Junior Geometry and Topology Seminar
16 May 2013
15:00
Robert Kropholler
Abstract
I will be taking us on a journey through low dimensional topology, starting in 2 dimensions motivating handles decompositions in a dimension that we can visualize, moving onto to a brief of note of what this means in 3 dimensions and then moving onto the wild world of 4 manifolds. I will be showing a way in which we can actually try and view a 4 manifold before moving onto a way of manipulating these diagrams to give diffeomorphic 4 manifolds. Hopefully, I will have time to go into some ways in which Kirby calculus has been used to show that certain potential exotic 4 spheres are not exotic and some results on stable diffeomorphims of 4 manifolds.
• Junior Geometry and Topology Seminar
9 May 2013
15:00
Alejandro Betancourt
Abstract
Based on ideas from Eells and Sampson, the Ricci flow was introduced by R. Hamilton in 1982 to try to prove Thurston's Geometrization Conjecture (a path which turned out to be successful). In this talk we will introduce the Ricci flow equation and view it as a modified heat flow. Using this we will prove the basic results on existence and uniqueness, and gain some insight into the evolution of various geometric quantities under Ricci flow. With this results we will proceed to define Perelman's $\mathcal{F}$ and $\mathcal{W}$ entropy functionals to view the Ricci flow as a gradient flow. If time permits we will briefly sketch some results from Cheeger and Gromov's compactness theory, which, along with the entropy functionals, alow us to blow up singularities.This is meant to be an introductory talk so I will try to develop as much geometric intuition as possible and stay away from technical calculations.
• Junior Geometry and Topology Seminar
2 May 2013
15:00
Subhojoy Gupta
Abstract
We shall introduce complex projective structures on a surface, and discuss a new result that relates grafting, which are certain geometric deformations of these structures, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any Fuchsian representation of a surface-group, the set of projective structures with that as holonomy, is dense in moduli space.
• Junior Geometry and Topology Seminar
7 March 2013
15:00
Benjamin Volk
Abstract
This talk will give a quick and dirty introduction to orbifold bordism. We will start by briefly recalling some basic properties and definitions of orbifolds and sketch (very roughly) how orbifolds can be defined in the language of $C^\infty$-stacks due to Joyce (after introducing these). We will then review classical bordism theory for manifolds (in some nonstandard way) and discuss which definitions and results generalize to the orbifold case. A word of warning: this talk is intended to be an introduction and wants to give an overview over the subject, so it is likely that we will be sloppy here and there.
• Junior Geometry and Topology Seminar
28 February 2013
15:00
Abstract
In this first of two talks, I shall introduce the Virtual Haken Conjecture and the major players involved in the proof announced by Ian Agol last year. These are the special cube complexes studied by Dani Wise and his collaborators, with a large supporting cast including the not-inconsiderable presence of Perelman’s Geometrization Theorem and the Surface Subgroup Theorem of Kahn and Markovic. I shall sketch how the VHC follows from Agol’s result that, in spite of the name, specialness is entirely generic among non-positively curved cube complexes.
• Junior Geometry and Topology Seminar
21 February 2013
15:00
Thomas Wasserman
Abstract
<p><span>Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.</span><br /><span>One can do better and compute the cohomology in terms of this function using the so-called Witten complex.</span><br /><span>Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.</span><br /><span>I will explain Witten's (heuristic) arguments and describe the Witten complex.</span></p>
• Junior Geometry and Topology Seminar
14 February 2013
15:00
Antonio De Capua
Abstract
• Junior Geometry and Topology Seminar
7 February 2013
15:00
Jakob Blaavand
Abstract
This talk is a basic introduction to the wonderful world of Higgs bundles on a Riemann Surface, and their moduli space. We will only survey the basics of the theory focusing on the rich geometry of the moduli space of Higgs bundles, and the relation to moduli space of vector bundles. In the end we consider small applications of Higgs bundles. As this talk will be very basic we won't go into any new developments of the theory, but just mention the areas in which Higgs bundles are used today.
• Junior Geometry and Topology Seminar