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Small resolutions of closures of K-orbits in flag varieties
(2020-05)
We construct explicit proper morphisms \(\mu\,\colon Z\to Y\), where \(Y\) is the closure of a \(K\)-orbit in a flag variety. Of particular interest is when \(\mu\) is a resolution of singularities and when it is a small ...
Zeros of random trigonometric polynomials with dependent coefficients
(2021-05)
It is well known that the expected number of real zeros of a random cosine polynomial (of degree $ n $)\begin{equation*} V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \quad x \in (0,2\pi) , \end{equation*} where the coefficients ...
Mahler measure and its behavior under iteration
(2021-05)
For an algebraic number $\alpha$ we denote by $M(\alpha)$ the Mahler measure of $\alpha$. Mahler measure is a height function on polynomials with integer coefficients. Moreover, as $M(\alpha)$ is again an algebraic number ...
Weyl's Law for cusp forms of arbitrary Archimedean type
(2022-07)
We generalize the work of E. Lindenstrauss and A. Venkatesh establishing Weyl's Law for cusp forms from the spherical spectrum to arbitrary Archimedean type. Weyl's law for the spherical spectrum gives an asymptotic formula ...
Poset shifted matroids and graphs
(2021-07)
We generalize the notion of shiftedness in simplicial complexes, matroids, and graphs. Using this generalization, called $P$-shiftedness, we give a condition for a matroid to be transversal in terms of a poset $P$. Our ...
Proper maps and involutions of unit balls in Euclidean Levi-flat spaces
(2021-07)
As models of strictly pseudoconvex domains, we consider holomorphic functions on the unit ball $\ball{n}=\{z\in\C^n:|z|<1\}$. In particular, we focus on proper holomorphic maps $\ball{n}\to\ball{N}$. In the equidimensional ...
Divisor labelling of staircase diagrams and fiber bundle structures on Schubert varieties
(2023-05)
Let Gr(𝑟, 𝑛) denote the Grassmannian of 𝑟-dimensional subspaces of the 𝑛-dimensional vector space ℂⁿ over the field of complex numbers. Each Gr(𝑟, 𝑛) contains a unique codimension 1 Schubert subvariety called the ...
Stability and blowup problems on the magnetohydrodynamic equations and Boussinesq equations
(2020-07)
Fluid mechanics is the study of the behavior of fluids and deformation of the fluid under the influence of shearing forces. One of the most widely used and studied system of equations in fluid mechanics are the Navier-Stokes ...
Fibers as normal and spun-normal surfaces in link manifolds
(2023-05)
The technique of inflating ideal triangulations developed by Jaco and Rubinstein gives a procedure starting with an ideal triangulation 𝒯* of the interior of a compact 3-manifold 𝑀, that constructs a triangulation 𝒯⌄𝛬 ...
Investigation into the development of a quantitatively based summation conception of the definite integral
(2021-07)
Research has shown the majority of students who have completed a university calculus course reason about the definite integral primarily in terms of prototypical imagery or in purely algorithmic and non-quantitative ways. ...