Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs (X,D). Here X is a normal variety and D is a reduced divisor on X. Integral to the definition of a rational pair is the ...
In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent ...
In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's ...
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are <italic>universal</italic>. We probe the edges of universality by studying ...
The goal of this thesis is to develop one of the threads of what is known in random matrix theory as universality, which essentially is that a large class of matrices generalizing the Gaussian matrices (certain Wigner ...
We consider the inverse problem of determining the electromagnetic material parameters of a body from information obtainable only at the boundary of the body; such information comes in the form of a boundary map which we ...
Inverse problems arise in various areas of science and engineering including medical imaging, computer vision, geophysics, solid mechanics, astronomy, and so forth. A wide range of these problems involve elliptic operators. ...
Let $\Gamma$ be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of $\Gamma ,$ have no cycles, and no finite connected components are called essential spanning forests. The set ${\cal ...
We study Crouzeix's conjecture: for any polynomial p and any square matrix A, the spectral norm of the matrix p(A) is at most double of the supremum norm of the polynomial p on the numerical range of the matrix A.
Finite Difference (FD) schemes have been used widely in computing approximations for partial differential equations for wave propagation, as they are simple, flexible and robust. However, even for stable and accurate ...
A new methodology was proposed in Finkelstein and Kastner (2007,2008) to derive finite-difference (FD) schemes in the joint time-space domain to reduce dispersion error. The key idea is that the true dispersion relation ...
This thesis studies bootstrap percolation, a problem in probability, as well as several topics in the application of sums of squares to combinatorial optimization. In the chapter on percolation, we bound the critical ...
We study the macroscopic geometry of first-passage competition on the integer lattice <bold>Z</bold><super>d</super>, with a particular interest in describing the behavior when one species initially occupies the exterior ...
We define graded group schemes and graded group varieties and develop their theory. We give a generalization of the result that connected graded bialgebras are graded Hopf algebra. Our result is given for a broader class ...
Let <italic>k</italic> be a field and <italic>B</italic> either a finitely generated free <italic>k</italic>-algebra, or a regular <italic>k</italic>-algebra of global dimension two with at least three generators, generated ...
We study Hopf algebras of finite Gelfand-Kirillov dimension. By analyzing the free pointed Hopf algebra F(t), we show the existence of certain Hopf subalgebras of pointed Hopf domains H whose GK-dimension are finite and ...
Inverse problems arise in different disciplines including exploration geophysics, medical imaging and nondestructive evaluation. In some settings, a single modality displays either high contrast or high resolution but not ...
We study hybrid imaging techniques that aim to overcome the ill-posedness of the Electric Impedance Tomography using the acousto-electric coupling. For the isotropic case, we consider the problem of reconstructing the ...
This thesis studies the <italic>hydrodynamic limit</italic> and the <italic>fluctuation limit</italic> for a class of interacting particle systems in domains. These systems are introduced to model the transport of positive ...
In this work, we study the stability aspect of two inverse boundary-value problems (IBVPs) on an infinite slab with partial data. The uniqueness aspects of these IBVPs were considered and studied by Li and Uhlmann for the ...