We study analytic functions on the open unit p-adic poly-disk centered at the multiplicative identity and prove that such functions only vanish at finitely many n-tuples of roots of unity unless they vanish along a translate of the formal multiplicative group. (Note that a root of unity lies on the...
Using Eynard-Orantin topological recursion, we prove here a result concerning the equivariant Gromov-Witten invariants for the projective line equipped with the standard action of the 2-torus. Our result is that the genus g, n point Gromov-Witten potential with arbitrary primary insertions may be written as a sum over certain genus...
Algebras and their bimodules form a 2-category in which 2-morphisms are certain zero-th Hochschild cohomology groups. When we derive this structure (i.e., use Hochschild cochains instead of HH^0 for 2-morphisms), we find that algebras form a category in dg cocategories. The Hochschild-Kostant-Rosenberg theorem and non-commutative calculus give a rich algebraic...
We define multi-indexed Deligne extensions and multi-indexed log-variations of Hodge structures in the category of (filtered) logarithmic D-modules, via the idea of Bernstein– Sato polynomials and Kashiwara–Malgrange filtrations, generalizing the Deligne canonical extensions of flat vector bundles. We also obtain many comparison results with perverse sheaves via the logarithmic de...
This work is concerned with the Laudau-Ginzburg $A$-model, or the Fukaya-Seidel category, associated with a Laurent polynomial $f: (\C^*)^n \ o \C$. We use constructible sheaves on a real $n$-dimensional torus to describe the Lagrangian thimbles associated to $f$. Then we discuss the application to Homological Mirror Symmetry for smooth...
Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along...
This dissertation contains three results related to modular forms and Galois representations of low weight. In chapter 1, we prove that the Galois pseudo-representation valued in a Hecke algebra which acts faithfully on a space of weight one Katz modular forms of level prime to p is unramified at p....
In this thesis, we advocate for the use of slice spheres, a common generalization of representation spheres and induced spheres, in parameterized homotopy theory. First, we give an algebraic characterization of the layers of the Hill-Hopkins-Ravenel slice filtration.
Next, we explore the homology of parameterized symmetric powers from this point...
We compute Lawson homology groups and semi-topological K-theory for certain "degenerate" varieties. "Degenerate" varieties are those smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. Our main method of study makes use of a technique of Bloch and...
This thesis is devoted to the study of A-branes on symplectic tori and to the Mirror Symmetry conjecture. Using a method called Seidel's mirror map, we are able to reconstruct the homogeneous coordinate ring of a complex abelian variety using Lagrangian intersection theory on the mirror symplectic torus. Moreover, we...
We prove the L<SUP>2</SUP>-convergence of polynomial ergodic averages of multiple commuting transformations for totally ergodic systems. We show that for each set of polynomials, each average is controlled by a particular characteristic factor introduced by Host and Kra, which is an inverse limit of nilsystems. We then investigate for which...
Homotopy Gerstenhaber structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a B-infinity algebra by an associative algebra. Actions of B-infinity algebras on associative and B-infinity algebras are analyzed, extensions of B-infinity algebras by associative...
The Satake category is the category of perverse sheaves on the affine Grassmannian of a complex reductive group G. The global cohomology functor induces a tensor equivalence between the Satake category and the category of finite-dimensional representations of the split form of the Langlands dual group of G. We give...
Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X, denoted as $H^{p,q}(X)$ where q and p are integers with q nonnegative, can be represented in the triangulated category of motives over the field of complex numbers, denoted as $DM^{eff,-}_{Nis}$....
This dissertation addresses the structure of the group of interval exchange transformations. The two primary topics considered are:
a) the classification of interval exchange actions for certain groups; and b) properties of the interval exchange group which are reflected in the dynamics of interval exchange maps.
In Chapter 3 a...
The homotopy groups of bo^tmf are shown to be isomorphic to the homotopy groups of a wedge of suspensions of spectra related to integral Brown-Gitler spectra. We will then restate Mahowald's proof of the topological splitting of bo^bo and subsequently apply similar techniques to construct a map that realizing the...
This paper covers three main topics. The first is addressing the question of interpolating between disparate index theorems on noncommutative two-tori. The second is to compute Hochschild cohomology for quantum special linear and special unitary groups. The third is producing an orthonormal basis for the vector space of matrix corepresentations...
The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
We compare two different methods to compute the mod 2 homology of an infinite loop space. One method is to approximate the infinite loop functor using functor calculus. The other is to approximate the spectrum using an Adams resolution. We show that these two ways lead to isomorphic spectral sequences....
Let X,Y be algebraic varieties defined over the reals. Assume Y is smooth and X is Gorenstein. Suppose f:X -> Y is a flat R-morphism such that all the fibers have rational singularities. We show that the pushforward of any smooth, compactly supported measure on X has a continuous density...
We address the problem of efficient maintenance of the answer to a new type of query: Continuous Maximizing Range-Sum (Co-MaxRS) for moving objects trajectories. The traditional static/spatial MaxRS problem finds a location for placing the centroid of a given (axes-parallel) rectangle $R$ so that the sum of the weights of...
In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\CM^{ss}}$ is uniformized by a formal scheme $\CN$ which is called Rapoport-Zink space. We show that the irreducible components of the induced...
The Picard group is an important invariant of the $K(n)$-local category. If the prime $p$ is relatively large compared to the height $n$, the Picard group of the $K(n)$-local category is purely algebraic. In \cref{chapter:finitetype}, we describe the necessary and sufficient numerical condition when an element $X$ in the Picard...
The structural aspects of biological systems are tightly paired with their functions. This understanding has been demonstrated over a broad range of length scales, spanning the ultrastructure of a cell to the macroscopic architecture of organs. Connecting structure and function relies on the integration of physical and biological sciences to...
We use Goerss-Hopkins theory to show that if E is a p-local Landweber exact homology theory of height n and p > n^2 + n + 1, then there exists an equivalence hSpE ≃ hD(E∗E) between homotopy categories of E-local spectra and differential E∗E-comodules, generalizing Bousfield’s and Franke’s results to...
This thesis contains results in mathematical quantum ergodicity in a probabilistic or a complex analytic setting. For the former, we show that a random orthonormal basis of spherical harmonics is almost surely quantum ergodic, in which the randomness is induced by the generalized Wigner ensemble. For the latter, we show...
This dissertation concerns the probabilistic aspects of diffusion processes generated by a family of differential operators, which is similar to the family of hypoelliptic Laplacian operators, acting on the tangent bundle of a compact Riemannian manifold. By lifting the processes to the product of the frame bundle and the euclidean...
In this thesis, we study pushforwards of canonical and log-pluricanonical bundles on projective log canonical pairs over the complex numbers. We partially answer a Fujita-type conjecture proposed by Popa and Schnell in the log canonical setting. Built on Kawamata’s result for morphisms that are smooth outside a simple normal crossing...
The Brink-Schwarz superparticle is a one-dimensional analogue of the Green-Schwarz superstring. In this thesis, we use the Batalin-Vilkovisky formalism to study the superparticle. After proving a vanishing result for its Batalin-Vilkovisky cohomology, we explain the sense in which the superparticle exhibits general covariance in the world-line. Using techniques from rational...
This dissertation addresses the property of amenability of discrete groups and their actions. In Chapter 2, following the introduction, all necessary definitions are given to introduce amenable groups, elementary amenable groups, random walks, topological full groups, Thompson's group $F$ and to show connections between them. The chapter also briefly covers...
The spatial autoregressive model has been widely applied in science, in areas such as economics, public finance, political science, agricultural economics, environmental studies and transportation analyses. The classical spatial autoregressive model is a linear model for describing spatial correlation. In this work, we expand the classical model to include time...
Arising from the study of multiple ergodic averages, nilsequences and multiple correlation sequences lie at the crossroads of ergodic theory, combinatorics and number theory. We study these types of sequences along various subsequences of integers, and provide applications to ergodic theory and harmonic analysis. Our first result involves multiple correlation...
We study plurisubharmonic functions and their applications to K\"ahler geometry. We begin by studying regularity of envelopes of plurisubharmonic functions, particularly when the reference form is degenerate. This is then applied to show regularity of geodesic of K\"ahler metrics on singular varieties, as well as regularity of certain geodesic rays....
In this thesis, we study applications of the theory of perverse sheaves and their enhancements to problems in birational geometry. In the first application, we give positive results towards a conjecture of Batyrev about the nonnegativity of stringy Hodge numbers. In particular, we prove the nonnegativity of $(p,1)$-stringy Hodge numbers...
The variation of entropy in a family of dynamical systems is a natural indication of the bifurcations that the family undergoes. In the context of one-dimensional dynamics, Milnor's monotonicity of entropy conjecture (now a theorem of Bruin and van Strien) asserts that for polynomial interval maps with real critical points...
In this dissertation we study the connections over principal bundles in dimension four with bounded Yang-Mills energy, and present a new result on the existence a global Coulomb gauge with estimate in optimal space. To be precise, let $A$ be a $W^{1,2}$-connection on a principal $\text{SU}(2)$-bundle $P$ over a smooth...
We study the symmetry groups with respect to various equivalence relations defined on subshifts, and more generally, on Cantor systems. Two basic notions of equivalence for dynamical systems are conjugacy and flow equivalence. In this dissertation, we focus on the well-studied automorphism group, which is the group of self-conjugacies, and...
This thesis provides an introduction to decay rates for the damped wave equation on compact manifolds. It also gives a proof of a sharp decay rate for solutions to the damped wave equation on the torus with damping of a particular polynomial form. Finally it gives a proof of a...
In this thesis, we study the geometry of planar shapes and their harmonic caps. Specifically, given a compact continuum $P$, we are interested in constructing a planar cap $\hat P$ such that $P$ and $\hat P$ can be glued together along their boundary to form a topological sphere with prescribed...
We study modular forms, Jacobi forms, and hermitian formal Fourier-Jacobi series over imaginary quadratic fields. In the first section, we prove that the ring of classical Jacobi forms of a fixed genus g, varying index m and weight k is generated by theta functions. From this result we show that...