# Past Junior Geometry and Topology Seminar

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.

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.

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.

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.

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.

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.

I will introduce simple homotopy theory and then discuss relations between some conjectures in 2 dimensional simple homotopy theory and the 3 and 4 dimensional Poincaré conjectures.

Since its genesis in 1915, General Relativity has proven to be one of the most successful physical theories ever invented. Providing a description of the large scale structure of the universe it continues to be in agreement with all experimental tests to high accuracy. By merging Classical Mechanics and Electrodynamics to a consistent geometrical theory of space-time it has become one of the two pillars of modern theoretical physics alongside Quantum Mechanics. This talk aims to give an introduction to the ideas and concepts of General Relativity. After briefly reviewing Classical (Newtonian) Mechanics and experiments in contradiction with it the framework and axioms of General Relativity will be introduced. This will be followed by a survey on major implications of the (new) geometrical description of gravity. Finally an outlook on physics beyond General Relativity will be provided.

Lagrangian Floer cohomology categorifies the intersection number of (half-dimensional) Lagrangian submanifolds of a symplectic manifold. In this talk I will describe how and when can we define Lagrangian Floer cohomology. In the case when Floer cohomology cannot be defined I will describe an alternative invariant known as the Fukaya (A-infinity) algebra.