In this work we explore a connection between some high dimensional asymptotic problems and random matrix theory. In the first part, we establish a link between the Wishart ensemble and random critical points of holomorphic sections over complex projective space and use this to establish asymptotics on the average number...
The moduli stack of bundles on a smooth complete curve over a field, is an immensely rich geometric object and is of central importance to the Geometric Langlands program. This thesis represents a contribution towards a motivic, in the sense of Voevodsky and Morel-Voevodsky, understanding of this stack. Following the...
This thesis is naturally split into two parts. In the first part, we develop the theory of multi- linear algebra for Tate objects over exact categories endowed with an exact tensor product. We study all possible choices of tensor product and we give a geometric interpretation of the results. In...
In this thesis, we study arithmetic phenomena exhibited by polynomial dynamical systems on the projective line. Specifically, given a number field $K$, we are interested in the arithmetic of orbits of points $\alpha\in K$ under polynomials $\phi\in K[z]$. Given such a polynomial $\phi$ of degree $d\ge2$, we prove a lower...
In this paper, we show almost-Gelfand property of connected symmetric pairs (G, H) over finite fields of large characteristics by showing almost-sigma-invariant property of double coset H\G/H where sigma is the associated anti-involution combining with epsilon-version of Gelfand's trick
The holomorphic sigma-model is a field theory that exists in any complex dimension that describes the moduli space of holomorphic maps from one complex manifold to another. We introduce the general notion of a holomorphic field theory, which is one that is sensitive to the underlying complex structure of the...
homotopy theory studies a parametrization of stable homotopy theory in terms of algebraic objects called formal groups. Transchromatic homotopy theory is specifically concerned with the behavior of spaces and cohomology theories as these formal groups change in height. We pursue a central transchromatic object, the K(n − 1)- localization of...
We prove a uniform scalar curvature bound for solutions of the conical Kahler-Ricci flow when the twisted canonical bundle is semiample and the cone divisor is obtained from the associated Iitaka-Kodaira fibration. In the course of the proof we establish uniform bounds for the potential of the metric and its...
In fish, caudally propagating waves of neural activity produce muscle bending moments. These moments, coupled with forces due to the body's elastic properties and forces due to fluid-body interactions, determine the deformation kinematics for swimming. Fully resolved simulations of neurally-activated swimming can be used to decode activation patterns underlying observed...
In this thesis we study minimal measures for Lagrangian systems on compact manifolds. This thesis consists of three parts which are closely related.
The first part is Chapter 3 and Chapter 4. In Chapter 3 and 4, we consider geodesic flows on compact surfaces with higher genus. We show that...