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We establish a Rademacher type theorem involving Hamiltonians $H(x,p)$ under very weak conditions in both of Euclidean and Carnot-Carath\'eodory spaces. In particular,$H(x,p)$ is assumed to be only measurable in the variable $x$, and to be…

Let $p(Y_1, \dots, Y_d, Z_1, \dots, Z_e)$ be a self-adjoint noncommutative polynomial, with coefficients from $\mathbb{C}^{r \times r}$, in the indeterminates $Y_1, \dots, Y_d$ (considered to be self-adjoint), the indeterminates $Z_1,…

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our…

Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…

We show a statistical version of Taylor's theorem and apply this result to non-parametric density estimation from truncated samples, which is a classical challenge in Statistics \cite{woodroofe1985estimating, stute1993almost}. The…

We prove that for any degree d, there exist (families of) finite sequences a_0, a_1,..., a_d of positive numbers such that, for any real polynomial P of degree d, the number of its real roots is less than or equal to the number of the…

Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log…

We consider large linear and nonlinear fixed point problems, and solution with proximal algorithms. We show that there is a close connection between two seemingly different types of methods from distinct fields: 1) Proximal iterations for…

Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the…

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular…

Let $(X,d)$ be a metric space and $f: X \rightarrow X$ be a homeomorphism. We say that a dynamical system $(X,f)$ is \emph{expansive}, with constant of expansivity $c \in \mathbb{R{^+}}$, if for all $x,y \in X$ , $x \neq y$, exists $n \in…

We study the algebraic $K$-theory of rings of the form $R[x]/x^e$. We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for $R$ a perfectoid…

The polynomial Szemer\'{e}di theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of…

Let $G = (V, E)$ be a graph. A set $S \subseteq V$ is a restrained dominating set (RDS) if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V - S$. The restrained domination number of $G$, denoted by $\gamma_r(G)$,…

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space $X$ and a continuous surjective map $T : X \to X$. By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…

Let $f_1,\dots,f_k \in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an $n<x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on an earlier…

We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports…

We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D>=2, whose points may be occupied by $c$ distinct…