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Well-Distributed Sequences: Number Theory, Optimal Transport, And Potential Theory, Louis Max Brown
Well-Distributed Sequences: Number Theory, Optimal Transport, And Potential Theory, Louis Max Brown
Yale Graduate School of Arts and Sciences Dissertations
The purpose of this dissertation will be to examine various ways of measuring how uniformly distributed a sequence of points on compact manifolds and finite combinatorial graphs can be, providing bounds and novel explicit algorithms to pick extremely uniform points, as well as connecting disparate branches of mathematics such as Number Theory and Optimal Transport. Chapter 1 sets the stage by introducing some of the fundamental ideas and results that will be used consistently throughout the thesis: we develop and establish Weyl's Theorem, the definition of discrepancy, LeVeque's Inequality, the Erdős-Turán Inequality, Koksma-Hlawka Inequality, and Schmidt's Theorem about Irregularities of …
Construction Of The Universal Second Motivic Chern Class Using Cluster Varieties, Oleksii Kislinskyi
Construction Of The Universal Second Motivic Chern Class Using Cluster Varieties, Oleksii Kislinskyi
Yale Graduate School of Arts and Sciences Dissertations
Denote by $B\Gamma$ the classifying space of an algebraic group $\Gamma(\mathbb{C})$. In this thesis we will take Milnor's realization of it -- the classifying simplicial space $B_{\Gamma\bullet}$. Consider the weight two motivic complex $ \mathbb{Z}_{\mathcal{M}}(2)$ for a regular algebraic variety $X$, placed in degrees $[1,4]$: $$ B_2(\mathbb{C}(X)) \to \bigwedge^2 \mathbb{C}(X)^* \to \bigoplus\limits_{D\in div(X)}\mathbb{C}(D)^* \to \bigoplus\limits_{D\in cod_2( X)}\mathbb{Z}.$$ Let $G$ be a split semisimple simply connected algebraic group over $\mathbb{Q}$. Then the degree 4 cohomology $H^4(B_{G\bullet}, \mathbb{Z}_{\mathcal{M}}(2))$ is known to be isomorphic to $\mathbb{Z}$. This can be deduced, for example, from the main result of Brylinski and Deligne \cite{BD}, although they …
Dynamics Of Unipotent Subgroups On Infinite Volume Space, Minju Lee
Dynamics Of Unipotent Subgroups On Infinite Volume Space, Minju Lee
Yale Graduate School of Arts and Sciences Dissertations
This thesis consists of five separate projects. They are organized into the following sections: 1. Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends. In joint work with Oh, we establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d, 1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d, 1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb H^d$ is a convex cocompact manifold with Fuchsian ends. 2. Topological proof of Benoist-Quint. Let $G=\mathrm{SO}^\circ(d,1)$, $\Delta
Three Families Of Lie Algebras Of Exponential Growth From Vertex Operator Algebras, Gabriel Legros
Three Families Of Lie Algebras Of Exponential Growth From Vertex Operator Algebras, Gabriel Legros
Yale Graduate School of Arts and Sciences Dissertations
We study three families of infinite-dimensional Lie algebras defined from Vertex Operator Algebras and their properties. For $N=0$ VOAs, we review the construction of the Fock space $V_L$ from an even lattice $L$ and provide an algebraic description of the Lie algebra $g_{II_{25,1}}$ from the perspective of $24$ different Niemeier lattices $N$ via the decomposition $II_{25,1} = N \oplus II_{1,1}$ using the no-ghost theorem. For $N=1$ SVOAs we review the construction of the Fock space $V_{NS}$ and provide an explicit basis for the spectrum-generating algebra of the Lie algebra $g_{NS}$. For $N=2$ SVOAs, we describe the structure of $g^{(2)}_{NS}$ explicitly …
On The Galois Action On Motivic Fundamental Groups Of Punctured Elliptic And Rational Curves, Nikolay Malkin
On The Galois Action On Motivic Fundamental Groups Of Punctured Elliptic And Rational Curves, Nikolay Malkin
Yale Graduate School of Arts and Sciences Dissertations
The main motive of this thesis is to study the action of the motivic Galois group on the motivic fundamental group of an algebraic curve X punctured at a finite set of points. The algebraic, geometric, and analytic aspects of this action are examined in two cases: for X=P1 and for X an elliptic curve. To study this action, we rely on motivic correlators, canonical elements in the fundamental Lie coalgebra of the category of mixed motives over a number field. We trace three themes: (1) The Lie coalgebra structure on the motivic correlators. Using combinatorial arguments with an injection …