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1. Subconvexity in Inhomogeneous Vinogradov Systems

   https://doi.org/10.1093/qmath/haac027  

   Wooley, Trevor D. (August 2022, The Quarterly Journal of Mathematics)

   Abstract When k and s are natural numbers and $${\mathbf h}\in {\mathbb Z}^k$$, denote by $$J_{s,k}(X;\,{\mathbf h})$$ the number of integral solutions of the system $$ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\leqslant j\leqslant k), $$ with $$1\leqslant x_i,y_i\leqslant X$$. When $$s\lt k(k+1)/2$$ and $$(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$$, Brandes and Hughes have shown that $$J_{s,k}(X;\,{\mathbf h})=o(X^s)$$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov’s mean value theorem, we obtain an asymptotic formula for $$J_{s,k}(X;\,{\mathbf h})$$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation. 

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2. Rates of convergence for regression with the graph poly-Laplacian

   https://doi.org/10.1007/s43670-023-00075-5  

   Trillos, Nicolás García; Murray, Ryan; Thorpe, Matthew (November 2023, Sampling Theory, Signal Processing, and Data Analysis)

   Abstract In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$$ ${\left\{x_i\right\}}_{i=1}^n$and a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$ ${\left\{y_i\right\}}_{i=1}^n\subset R$we let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$ $u_n:{\left\{x_i\right\}}_{i=1}^n\to R$be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$ $y_i=g\left(x_i\right)+{\xi }_i$, for iid noise$$\xi _i$$ ${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$ $u_n$togin the large data limit$$n\rightarrow \infty $$ $n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model. 

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3. Stabilization of the cohomology of thickenings

   Bhatt, Bhargav; Blickle, Manuel; Lyubeznik, Gennady; Singh, Anurag; Zhang, Wenliang. (April 2019, American journal of mathematics)

   For a local complete intersection subvariety $X = V (I)$ in $P^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $$X$$, the cohomology of vector bundles on the formal completion of $P^n$ along $$X$$ can be effectively computed as the cohomology on any sufficiently high thickening $$X_t = V (I^t)$$; the main ingredient here is a positivity result for the normal bundle of $$X$$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $$X_t$$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $$X$$, and the main new ingredient is a version of the Kodaira- Akizuki-Nakano vanishing theorem for $$X$$, formulated in terms of the cotangent complex. 

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4. A uniform treatment of Grothendieck's localization problem

   https://doi.org/10.1112/S0010437X21007715  

   Murayama, Takumi (January 2022, Compositio Mathematica)

   Let $$f\colon Y \to X$$ be a proper flat morphism of locally noetherian schemes. Then the locus in $$X$$ over which $$f$$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $$X$$ , the same property holds for other local properties of morphisms, even if $$f$$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $$F$$ -rationality, and the ‘Cohen–Macaulay and $$F$$ -injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero. 

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5. Uncoupled isotonic regression via minimum Wasserstein deconvolution

   https://doi.org/10.1093/imaiai/iaz006  

   Rigollet, Philippe; Weed, Jonathan (April 2019, Information and Inference: A Journal of the IMA)

   Abstract Isotonic regression is a standard problem in shape-constrained estimation where the goal is to estimate an unknown non-decreasing regression function $$f$$ from independent pairs $$(x_i, y_i)$$ where $${\mathbb{E}}[y_i]=f(x_i), i=1, \ldots n$$. While this problem is well understood both statistically and computationally, much less is known about its uncoupled counterpart, where one is given only the unordered sets $$\{x_1, \ldots , x_n\}$$ and $$\{y_1, \ldots , y_n\}$$. In this work, we leverage tools from optimal transport theory to derive minimax rates under weak moments conditions on $$y_i$$ and to give an efficient algorithm achieving optimal rates. Both upper and lower bounds employ moment-matching arguments that are also pertinent to learning mixtures of distributions and deconvolution. 

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