Thesis Format
Alternative Format
Degree
Doctor of Philosophy
Program
Mathematics
Supervisor
Pinsonnault, Martin
Abstract
This thesis is divided into two parts: in the first part we study some geometric aspects of the Hofer metric on the group $\Ham(M,\om)$ associated to a symplectic manifold. More specifically, we investigate how the centralizer $C(T)$ of a toric action sits inside $\Ham(M,\om)$, and compare this with the maximal torus in a Lie group, which is always a flat and totally geodesic submanifold with respect to any bi-invariant metric. We also study the intrinsic geometry of the centralizer with respect to the metric induced by the Hofer metric.
In the second part of this document, we study the embedding space of two disjoint standard symplectic balls $B^4(c_1)\sqcup B^4(c_2)$ of capacities $c_1$ and $c_2$ in $S^2\times S^2$ with respect to any symplectic form. The set of admissible capacities is subdivided into polygonal regions in which the homotopy type of the embedding space is constant. We find the set of all stability chambers and compute the homotopy type of the relevant embedding spaces in some of these chambers.
Summary for Lay Audience
Imagine we have a flat piece of paper (in 2D space) and we want to place it inside a book (3D space). The paper can fit smoothly into the book without any folds or tears, this is like embedding one space (the paper) into another (the book). The paper retains all its original properties (like being flat) while now existing inside a larger space. Now imagine that each surface above is endowed with different rules of measuring area. A symplectic embedding (in 2D space) refers to fitting the paper into the book without messing up the way these areas are measured.
In higher dimensions (4D or higher), symplectic embeddings are more complex. The first pioneering discovery in this direction was Gromov's famous non-squeezing theorem showing that if we try embed a ball $B^{2n}(b)$ (the paper) into an infinite cylinder $B^2(c)\times R^{2n-2}$ (the book) while preserving all 2 dimensional areas, then the radius of the ball (i.e. b) must be smaller than the radius of the cylinder (i.e. c). In other words, even if the volume of the cylinder is infinite, there is no way to squeeze a big ball into it without deforming the areas of 2 dimensional objects within it. His proof uses the theory of $J$-holomorphic curves, which opened up this problem for $4D$ spaces and ever since, symplectic embeddings in dimension $4$ have been widely studied. A higher complexity question is whether given two different papers embedded inside a book can one move (smoothly) from the first paper to the second one. This question was affirmatively answered by McDuff in dimension 4.
In this thesis, we undertake a similar problem of understanding the embedding space of two balls inside the product of two spheres $S^2\times S^2$. The embedding for one ball was studied by Pinsonnault, we build upon this work by first laying a general framework, and then we find all different ways of embedding 2 balls inside $S^2 \times S^2$ (essentially there are 8 different ways). We also find a topological description of the space of embeddings after having fixed “one way” to embed. We find that in some cases the space of embeddings does in fact have the homotopy type of a finite dimensional space. But in some other cases, its homotopy type is non-trivial in the sense that it has non-trivial holes in arbitrary dimensions.
The other part of this thesis studies Hofer geometry, which presents a way to measure how much energy it takes to move things around in a special kind of space called a symplectic manifold, which describes systems like planetary motion and the motion of a pendulum. Imagine, if we move a shape in a way that preserves some of its special properties, like the ruler we used to measure 2-dimensional areas. Hofer geometry tells us how hard or "costly" it is to make such a move. It's like finding the shortest or most efficient path to move such a shape without changing these rules. In simpler terms, it helps us figure out the "distance" between two different configurations of a system by measuring the effort needed to transform one into the other.
We study these spaces under the presence of another form of symmetry, which we call group actions. This space was discovered to be flat w.r.t the measurement above in the 90s by Bialy and Polterovich, we extend this result to the case when there are symmetries. We study the diameter of this equivariant group, when the metric (the ruler) is restricted to this equivariant group.
Recommended Citation
Singh, Prakash, "Maximal Torus in Hofer Geometry and Embeddings of two balls in $S^2\times S^2$" (2024). Electronic Thesis and Dissertation Repository. 10598.
https://ir.lib.uwo.ca/etd/10598
