 Award ID(s):
 2055118
 NSFPAR ID:
 10404468
 Date Published:
 Journal Name:
 Canadian Mathematical Bulletin
 Volume:
 66
 Issue:
 1
 ISSN:
 00084395
 Page Range / eLocation ID:
 243 to 258
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integervalued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on ${\mathbb {C}}^2$ . For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty $ . For t hooks, we prove distributions that are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1\xi^{1}(\xi q)^n\right). \end{align*} $$more » « less

Abstract Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in ${\mathbb{R}}^{n1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$ such that each manifold $Z_d$ is diffeomorphic to a component of the zero set on $\Gamma$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $\Gamma$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $\Gamma$ is bounded by a polynomial in $\deg (p)$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$ containing a subset $D$ homeomorphic to a disk, and a family of polynomials $\{p_m\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that $(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n1}, Z_{d_m}),$ i.e. the zero set of $p_m$ in $D$ is isotopic to $Z_{d_m}$ in ${\mathbb{R}}^{n1}$. This says that, up to extracting subsequences, the intersection of $\Gamma$ with a hypersurface of degree $d$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $0 \leq k \leq n2$ and every sequence of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$, a subsequence $\{a_{d_m}\}_{m\in \mathbb{N}}$ and homogeneous polynomials $\{p_{m}\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $b_k$ denotes the $k$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $\Gamma$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $d$, of the set $\Sigma _{d_m,a, \Gamma }$ of polynomials verifying (0.1) is positive, but there exists a constant $c_\Gamma$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $a$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $\Gamma$, for most polynomials a Bézouttype bound holds for the intersection $\Gamma \cap Z(p)$: for every $0\leq k\leq n2$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n1}{2}}}.\end{equation*}$$

Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $K$ and $L$ be star bodies in ${\mathbb R}^n,$ let $0<k<n$ be an integer, and let $f,g$ be nonnegative continuous functions on $K$ and $L$, respectively, so that $\g\_\infty =g(0)=1.$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{nk}n}K^{\frac kn}} \le \frac n{nk} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $K$ stands for volume of proper dimension, $C$ is an absolute constant, the maximum is taken over all $(nk)$dimensional subspaces of ${\mathbb R}^n,$ and $d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$intersection bodies in ${\mathbb R}^n.$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem.more » « less

Abstract We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $n\geq 2$. In dimension $n=1$, we construct an elliptic singular integral operator $H_{\flat } $ for which the energy conditions are not necessary for boundedness of $H_{\flat }$. The convolution kernel $K_{\flat }\left ( x\right ) $ of the operator $H_{\flat }$ is a smooth flattened version of the Hilbert transform kernel $K\left ( x\right ) =\frac{1}{x}$ that satisfies ellipticity $ \vert K_{\flat }\left ( x\right ) \vert \gtrsim \frac{1}{\left \vert x\right \vert }$, but not gradient ellipticity $ \vert K_{\flat }^{\prime }\left ( x\right ) \vert \gtrsim \frac{1}{ \vert x \vert ^{2}}$. Indeed the kernel has flat spots where $K_{\flat }^{\prime }\left ( x\right ) =0$ on a family of intervals, but $K_{\flat }^{\prime }\left ( x\right ) $ is otherwise negative on $\mathbb{R}\setminus \left \{ 0\right \} $. On the other hand, if a onedimensional kernel $K\left ( x,y\right ) $ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [30], the $T1$ theorem holds for such kernels on the line. This paper includes results from arXiv:16079.06071v3 and arXiv:1801.03706v2.

We study the \emph{offline reinforcement learning} (offline RL) problem, where the goal is to learn a rewardmaximizing policy in an unknown \emph{Markov Decision Process} (MDP) using the data coming from a policy $\mu$. In particular, we consider the sample complexity problems of offline RL for the finite horizon MDPs. Prior works derive the informationtheoretical lower bounds based on different datacoverage assumptions and their upper bounds are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the \emph{Adaptive Pessimistic Value Iteration} (APVI) algorithm and derive the suboptimality upper bound that nearly matches $ O\left(\sum_{h=1}^H\sum_{s_h,a_h}d^{\pi^\star}_h(s_h,a_h)\sqrt{\frac{\mathrm{Var}_{P_{s_h,a_h}}{(V^\star_{h+1}+r_h)}}{d^\mu_h(s_h,a_h)}}\sqrt{\frac{1}{n}}\right). $ We also prove an informationtheoretical lower bound to show this quantity is required under the weak assumption that $d^\mu_h(s_h,a_h)>0$ if $d^{\pi^\star}_h(s_h,a_h)>0$. Here $\pi^\star$ is a optimal policy, $\mu$ is the behavior policy and $d(s_h,a_h)$ is the marginal stateaction probability. We call this adaptive bound the \emph{intrinsic offline reinforcement learning bound} since it directly implies all the existing optimal results: minimax rate under uniform datacoverage assumption, horizonfree setting, single policy concentrability, and the tight problemdependent results. Later, we extend the result to the \emph{assumptionfree} regime (where we make no assumption on $ \mu$) and obtain the assumptionfree intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL.more » « less