<p>See abstract. </p>

In the first paper, we prove that for a closed Legendrian submanifold L of dimension n&gt;2 with a loose chart of size η, any Legendrian isotopy starting at L can be C0-approximated by a Legendrian isotopy with energy arbitrarily close to η/2. This in particular implies that the displacement energy of loose displaceable Legendrians is bounded by half the size of its smallest loose chart, which proves a conjecture of Dimitroglou Rizell and Sullivan. In the second paper, we show that the Legendrian lift of an exact, displaceable Lagrangian has vanishing Shelukhin-Chekanov-Hofer pseudo-metric by lifting an argument due to Sikorav to the contactization. In particular, this proves the existence of such Legendrians, providing counterexamples to a conjecture of Rosen and Zhang. After completion of the manuscript, we noticed that Cant (arXiv:2301.06205) independently proved a more general version of our main result.

<p>See abstract. </p>

No description available.

<p>Smith theory provides powerful tools for understanding the geometry of singular sets of group actions on spheres. In this thesis, tools from Smith theory and spectral sequences are brought together to study the singular sets of elementary abelian groups acting locally linearly on S4. It is shown that the singular sets of such actions are homeomorphic to the singular sets of linear actions. A short review of the literature on group actions on S4 is included.</p> / Master of Science (MSc)

<p>Let $\Theta_n$ be the group of $h$-cobordism classes of homotopy spheres, i.e. closed smooth manifolds which are homotopy equivalent to $S^n$, under connected sum. A homotopy sphere $\Sigma^n$ which is not diffeomorphic to $S^n$ is called ``exotic.'' For an oriented smooth manifold $M^n$, the {\bf inertia group} $I(M)\subset\Theta_n$ is defined as the subgroup of homotopy spheres such that $M\#\Sigma$ is orientation-preserving diffeomorphic to $M$. This thesis collects together a number of results on $I(M)$ and provides a summary of some fundamental results in Geometric Topology. The focus is on dimension $7$, since it is the smallest known dimension with exotic spheres. The thesis also provides two new results: one specifically about $7$-manifolds with certain $S^1$ actions, and the other about the effect of surgery on the homotopy inertia group $I_h(M)$.</p> / Master of Science (MSc)

<p>Let $p>5$ be a prime and $X_0$ a simply-connected $4$-manifold with boundary the Poincar\'e homology sphere $\Sigma(2,3,5)$ and even negative-definite intersection form $Q_=\text_8$ . We obtain restrictions on extending a free $\bZ/p$-action on $\Sigma(2,3,5)$ to a smooth, homologically-trivial action on $X_0$ with isolated fixed points. It is shown that for $p=7$ there is no such smooth extension. As a corollary, we obtain that there does not exist a smooth, homologically-trivial $\bZ/7$-equivariant splitting of $\#^8 S^2 \times S^2=E_8 \cup_ \overline$ with isolated fixed points. The approach is to study the equivariant version of Donaldson-Floer instanton-one moduli spaces for $4$-manifolds with cylindrical ends. These are $L^2$-finite anti-self dual connections which asymptotically limit to the trivial product connection.</p> / Doctor of Philosophy (PhD)

Our thesis explores aspects of the geometrical work and thought of Isaac Newton in order to better understand and re-evaluate his approach to geometry, and specifically his synthetic methods and the organic description of plane curves. In pursuing this research we study Newton's geometrical work in the context of the changing view of geometry between the late 16th and early 18th centuries, a period defined by the responses of the early modern geometers to a new Latin edition of Pappus' Collectio. By identifying some of the major challenges facing geometers of this period as they attempted to define and practice geometry we are able to contrast Newton's own approach to geometry. The themes emerging from the geometrical thought of early modern geometers provide the mathematical context from which to understand, interpret and re-evaluate the approach taken by Newton. In particular we focus on Newton's profound rejection of the new algebraic Cartesian methods and geometrical philosophies, and the opportunity to focus more clearly on some of his most astonishing geometrical contributions. Our research highlights Newton's geometrical work and examines specific examples of his synthetic methods. In particular we draw attention to the significance of Newton's organic construction and the limitations of Whiteside's observations on this subject. We propose that Newton's organic rulers were genuinely original. We disagree with Whiteside that they were inspired by van Schooten, except in the loosest sense. Further, we argue that Newton's study of singular points by their resolution was new, and that it has been misunderstood by Whiteside in his interpretation of the transformation effected by the rulers. We instead emphasise that it was the standard quadratic transformation. Overall we wish to make better known the importance of geometry in Newton's scientific thought, as well as highlighting the mathematical and historical importance of his organic description of curves as an example of his synthetic approach to geometry. This adds to contemporary discourse surrounding Newton's geometry, and specifically provides a foundation for further research into the implications of Newton's geometrical methods for his successors.

No description available.

In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. We discuss how to construct a line bundleover the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reducethe prequantum line bundle we employ real polarization such that the system does notbreak Heisenberg’s uncertainty principle anymore. From the prequantum bundle and thepolarization we construct the sought after Hilbert space.