We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector ...

Motivated by the analysis of high-dimensional neuroimaging signals located over the cortical surface, we introduce a novel Principal Component Analysis technique that can handle functional data located over a two-dimensional ...

We study random walks on the groups $\Bbb F^d_p \rtimes$ SL$_d$($\Bbb F_p$). We estimate the spectral gap in terms of the spectral gap of the projection to the linear part SL$_d$($\Bbb F_p$). This problem is motivated by ...

Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we ...

We study the mean-field limit of the Kuramoto model of globally coupled oscillators. By studying the evolution in Fourier space and understanding the domain of dependence, we show a global stability result. Moreover, we ...

In this thesis, the stability of the family of subextremal Kerr-Newman space- times is studied in the case of linear scalar perturbations. That is, nondegenerate energy bounds (NEB) and integrated local energy decay ...

We prove Wise’s W-cycles conjecture: Consider a compact graph Γ′ immersing into another graph Γ. For any immersed cycle Λ : S¹ → Γ, we consider the map Λ′ from the circular components S of the pullback to Γ′. Unless Λ′ is ...

In recent years, Sparse Principal Component Analysis has emerged as an extremely popular dimension reduction technique for highdimensional data. The theoretical challenge, in the simplest case, is to estimate the leading ...

Mendelian randomization is an epidemiological method for using genetic variation to estimate the causal effect of the change in a modifiable phenotype on an outcome from observational data. A genetic variant satisfying ...

A fundamental problem in Bayesian nonparametrics consists of selecting a prior distribution by assuming that the corresponding predictive probabilities obey certain properties. An early discussion of such a problem, although ...

In this thesis we study the self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ ...

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^*$($M$) of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised ...

This paper shows that, in dimensions two or more, there are no holomorphic isometries between Teichüller spaces and bounded symmetric domains in their intrinsic Kobayashi metric.

The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure ...

I investigate the $K_2$ groups of the quotients of Fermat curves given in projective coordinates by the equation $F_n:X^n+Y^n=Z^n$. On any quotient where the number of known elements is equal to the rank predicted by ...

The current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was ...

We show that the augmented base locus coincides with the exceptional locus (i.e., null locus) for any nef $\Bbb R$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness ...

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) ...