Past Junior Geometry and Topology Seminar

28 May 2015
16:00
Antonio De Capua
Abstract

A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.

  • Junior Geometry and Topology Seminar
21 May 2015
16:00
Matthias Wink
Abstract

In this talk we're going to discuss Hamilton's maximum principle for the Ricci flow. As an application, I would like to explain a technique due to Boehm and Wilking which provides a general tool to obtain new Ricci flow invariant curvature conditions from given ones. As we'll see, it plays a key role in Brendle and Schoen's proof of the differentiable sphere theorem.

  • Junior Geometry and Topology Seminar
14 May 2015
16:00
Carlos Alfonso Ruiz
Abstract
I will present a model theoretic point of view of algebraic geometry based on certain objects called Zariski Geometries. They were introduced by E. Hrushovski and B. Zilber and include classical objects like compact complex manifolds, algebraic varieties and rigid analytic varieties. Some connections with non commutative geometry have been found by B. Zilber too. I will concentrate on the relation between Zariski Geometries and schemes. 
  • Junior Geometry and Topology Seminar
7 May 2015
16:00
Bruce Bartlett
Abstract

The Ising model is a well-known statistical physics model, defined on a two-dimensional lattice. It is interesting because it exhibits a "phase transition" at a certain critical temperature. Recent mathematical research has revealed an intriguing geometry in the model, involving discrete holomorphic functions, spinors, spin structures, and the Dirac equation. I will try to outline some of these ideas.

  • Junior Geometry and Topology Seminar
12 March 2015
16:00
Abstract

Quiver varieties and their quantizations feature prominently in
geometric representation theory. Multiplicative quiver varieties are
group-like versions of ordinary quiver varieties whose quantizations
involve quantum groups and $q$-difference operators. In this talk, we will
define and give examples of representations of quivers, ordinary quiver
varieties, and multiplicative quiver varieties. No previous knowledge of
quivers will be assumed. If time permits, we will describe some phenomena
that occur when quantizing multiplicative quiver varieties at a root of
unity, and work-in-progress with Nicholas Cooney.

  • Junior Geometry and Topology Seminar
5 March 2015
16:00
Pavel Safronov
Abstract

I will explain the basics of deformation quantization of Poisson
algebras (an important tool in mathematical physics). Roughly, it is a
family of associative algebras deforming the original commutative
algebra. Following Fedosov, I will describe a classification of
quantizations of (algebraic) symplectic manifolds.
 

  • Junior Geometry and Topology Seminar
26 February 2015
16:00
Siran Li
Abstract

In this talk I shall discuss some classical results on isometric embedding of positively/nonegatively curved surfaces into $\mathbb{R}^3$. 

    The isometric embedding problem has played a crucial role in the development of geometric analysis and nonlinear PDE techniques--Nash invented his Nash-Moser techniques to prove the embeddability of general manifolds; later Gromov recast the problem into his ``h-Principle", which recently led to a major breakthrough by C. De Lellis et al. in the analysis of Euler/Navier-Stokes. Moreover, Nirenberg settled (positively) the Weyl Problem: given a smooth metric with strictly positive Gaussian curvature on a closed surface, does there exist a global isometric embedding into the Euclidean space $\mathbb{R}^3$? This work is proved by the continuity method and based on the regularity theory of the Monge-Ampere Equation, which led to Cheng-Yau's renowned works on the Minkowski Problem and the Calabi Conjecture. 

    Today we shall summarise Nirenberg's original proof for the Weyl problem. Also, we shall describe Hamilton's simplified proof using Nash-Moser Inverse Function Theorem, and Guan-Li's generalisation to the case of nonnegative Gaussian curvature. We shall also mention the status-quo of the related problems.

  • Junior Geometry and Topology Seminar
12 February 2015
16:00
Agnese Bissi
Abstract

 In this talk I will present a basic introduction to conformal symmetry from a physicist perspective. I will talk about infinitesimal and finite conformal transformations and the conformal group in diverse dimensions. 

  • Junior Geometry and Topology Seminar
5 February 2015
16:00
Lucas Branco
Abstract

The moduli space of G-Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory.

We shall begin the talk by first introducing G-Higgs bundles for reductive Lie groups and the associated Hitchin fibration, and sketching how to realize Langlands duality through spectral data. We shall then look at particular types of branes (BAA-branes) which correspond to very interesting geometric objects, hyperholomorphic bundles (BBB-branes). 

The presentation will be introductory and my goal is simply to sketch some of the ideas relating these very interesting areas. 

  • Junior Geometry and Topology Seminar

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