We review some simple techniques based on monotone-mass transport that allow us to obtain transport-type inequalities for any log-concave probability measures, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp–Lieb variance inequality.

We prove the consistency of

$$~~\text{add}\left( \mathcal{N} \right)<\operatorname{cov}\left( \mathcal{N} \right)<\mathfrak{p}\text{=}\mathfrak{s}\text{=}\mathfrak{g}< \text{add}\left( \mathcal{M} \right)=\text{cof}\left( \mathcal{M} \right)<\mathfrak{a}=\mathfrak{r}=\text{non}\left( N \right)=\mathfrak{c}$$

$\text{ZFC}$

We study metric Diophantine approximation for function fields, specifically, the problem of improving Dirichlet's theorem in Diophantine approximation.

We study a canonical ${{\text{C}}^{*}}$-algebra, $\text{S}\left( \Gamma ,\mu \right)$, that arises from a weighted graph $\left( \Gamma ,\mu \right)$, specific cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of $\text{S}\left( \Gamma ,\mu \right)$, and study the structure of its positive cone. We then study the $*$-algebra, $\mathcal{A}$, generated by the generators of $\text{S}\left( \Gamma ,\mu \right)$, and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements$x\,\in \,{{M}_{n}}\left( \mathcal{A} \right)$ have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

We explicitly find regulators of an infinite family $\{{{L}_{m}}\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring ${{\mathbb{F}}_{q}}\left( t \right)$, where ${{\mathbb{F}}_{q}}$ is the finite field of order $q$ with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family $\{{{L}_{m}}\}$ and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain $\{{{L}_{m}}\}$ as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field ${{\mathbb{F}}_{q}}\left( t \right)$ in $\{{{L}_{m}}\}$.

Let $\mathcal{A}$ and $\mathcal{B}$ be large subsets of $\{1,\,.\,.\,.\,,\,N\}$. We study the number of pairs $\left( a,b \right)\,\in \,\mathcal{A}\,\times \,\mathcal{B}$ such that the sum of binary digits of $a\,+\,b$ is fixed.

We complete Mori's program with symmetric divisors for the moduli space of stable seven-pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. We also give a moduli theoretic description of the first flip, which has not appeared in the literature.

We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an “almost disjointness preserving” operator is well approximable by a disjointness preserving one. However, this approximation is not always possible, as our examples show.

A theorem of Ostrowski implies that $\log \left( x \right),\,\log \left( x+1 \right),\,.\,.\,.$ are algebraically independent over $\mathbb{C}\left( x \right)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In this paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply them to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form ${{\mathbb{N}}^{n}}\,\times \,\mathbb{Z}/{{n}_{1}}\mathbb{Z}\,\times \,.\,.\,.\,\mathbb{Z}/{{n}_{r}}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.