In a recent study of large nonnull sample covariance matrices, a new sequence of functions generalizing the Gaussian unitary ensemble (GUE) Tracy-Widom distribution of random matrix theory was obtained. This article derives Painlevé formulas of these functions and uses them to prove that they are indeed distribution functions. Applications of these new distribution functions to last-passage percolation, queues in tandem, and totally asymmetric simple exclusion process are also discussed. As a part of the proof, a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented

We prove that any compact self-dual Einstein $4$-orbifold of positive scalar curvature whose isometry group contains a $2$-torus is, up to an orbifold covering, a quaternion Kähler quotient of $(k-1)$-dimensional quaternionic projective space by a $(k-2)$-torus for some $k\ge 2$. We also obtain a topological classification in terms of the intersection form of the $4$-orbifold

Let $M$ be a compact Riemannian manifold in which $Y$ is an embedded hypersurface separating $M$ into two parts. Assume that the metric is a product on a tubular neighborhood $N$ of $Y$. Let $\Delta$ be a Laplace-type operator on $M$ adapted to the product structure on $N$. Under certain additional assumptions on $\Delta$, we establish an asymptotic expansion for the logarithm of the regularized determinant $\det \Delta$ of $\Delta$ if the tubular neighborhood $N$ is stretched to a cylinder of infinite length. We use the asymptotic expansions to derive adiabatic splitting formulas for regularized determinants

In dimension $d\ge 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schrödinger operators have neither resonances nor eigenvalues. If $d=2$, we show that there are potentials with no resonances or eigenvalues away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on ${\mathbb{R}}^d$. In odd dimensions $d\ge 3$, we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is superexponentially decaying in time

In this article, we study the topology of compact manifolds with positive isotropic curvature (PIC). There are many examples of nonsimply connected compact manifolds with PIC. We prove that the fundamental group of a compact Riemannian manifold of dimension at least $5$ with PIC does not contain a subgroup isomorphic to the fundamental group of a compact Riemann surface. The proof uses stable minimal surface theory

We prove that a translation surface that has two transverse parabolic elements has a totally real trace field. As a corollary, nontrivial Veech groups that have no parabolic elements do exist.

The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms

We investigate conditions for simultaneous normalizability of a family of reduced schemes; that is, the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective $\mathbb{C}$-varieties $(X_y){}_{y\in Y}$ with normal parameter space $Y$—algebraic or analytic—admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure $\stackrel{̲}{{\mathcal{O}}_{X_y}}$ is locally independent of $y$. When the $X_y$ are curves, projectivity is not needed, and the statement reduces to the well-known $\delta$-constant criterion ofTeissier. The proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta and forth) to more abstract algebraic ones

Using the theory of nonnegative intersections, duality of the Schubert varieties, and a Pieri-type formula for the varieties of maximal, totally isotropic subspaces, we get an upper bound for the canonical dimension $\mathrm{cd}({\mathrm{Spin}}_n)$ of the spinor group ${\mathrm{Spin}}_n$. A lower bound is given by the canonical $2$-dimension ${\mathrm{cd}}_2({\mathrm{Spin}}_n)$, computed in [10]. If $n$ or $n+1$ is a power of $2$, no space is left between these two bounds; therefore, the precise value of $\mathrm{cd}({\mathrm{Spin}}_n)$ is obtained for such $n$. We also produce an upper bound for canonical dimension of the semispinor group (giving the precise value of the canonical dimension in the case when the rank of the group is a power of $2$) and show that spinor and semispinor groups are the only open cases of the question about canonical dimension of an arbitrary simple split group possessing a unique torsion prime