Mathematics
For a discrete topological space $D$, let $\mathrm{Met}(D)$ denote the set of metrics on $D$ that are compatible with the discrete topology, equipped with the topology induced by the supremum distance. Ishiki's Conjecture 5.1 asserts that…
This paper addresses the integrated Master Surgical Scheduling Problem and Surgical Case Assignment Problem under weekend bed capacity constraints. We propose a joint optimization framework that simultaneously determines the assignment of…
Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like $k$-means. Such methods are strongly connected to quantization problems, which aim to approximate…
The study of coordinator polynomials of Weyl group lattices was initiated by Conway and Sloane. In 2013, by using a trigonometric substitution approach, Wang and Zhao proved the real-rootedness of the type $D$ coordinator polynomials.…
In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to $O(n)\times O(n)$ symmetric Simons cone for $n \geq 5$, and lies on one side of the cone has to have a unique asymptotics in the…
We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of…
For a given $d$-regular graph $G$, a Maker-Breaker degree game is played by two players who alternately claim previously unclaimed edges of $G$. In the standard variant, the goal of Maker is to maximize the maximum degree of their induced…
A classical problem in lattice path enumeration counts paths that remain on one side of a boundary line. We study several classes of paths where this boundary is porous and show that they are related through a single half-turn rotation…
In this paper, we will prove a result about the structure of polytopes whose floating bodies are not smooth. This will then be used to prove a result about the combinatorial structure of minimizers of the volume product.
We prove sharpness of the avalanche phase transition in the one-dimensional Bak--Sneppen model, for both the one-sided and isotropic update rules. In the one-sided model, the avalanche range is described by a nonlinear self-interacting jump…
We introduce sequential analogues of directed (parametrized) topological complexity, in the context of motion planning problems requiring a system to traverse a prescribed sequence of intermediate states while respecting directed dynamics…
We prove a double-sum analogue of Whipple's ${}_3F_2$ summation formula. As an application, we establish a restricted sum formula for multiple binomial sums, which arise as special cases of Igarashi's parametrized multiple zeta series.
For $n\ge2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on $\mathbb R^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no…
We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of…
We establish a sharp prime orbit theorem for every homoclinic class of a $C^\infty$ diffeomorphism on a closed surface with positive topological entropy. Let $\mathcal{H}$ be a homoclinic class with topological entropy $h > 0$. Then there…
In this article we show that an $\mathbb{A}^1$ bundle map or a vector bundle map $p: X \to Y$ induces trivial local fibration $\underline{Sing}(X) \to \underline{Sing}(Y)$. Using this, we first show that for Korus Russel threefolds of first…
We show that any $d$-dimensional local ring $A$ with a dualizing complex, $\mathrm{depth} A=d-1$, and cyclic deficiency module $K^{d-1}(A)$ admits a maximal Cohen--Macaulay module. It is constructed as the unique nonzero cohomology module…
We develop a general framework for studying generic homeomorphisms of compact metric spaces using combinatorial amalgamation methods inspired by Fra\"iss\'e theory. Our approach combines Rosendal's criterion for comeager conjugacy classes…
Decision diagrams (DDs) have become a powerful tool for discrete optimization, supporting a wide range of algorithms that span cut-generation procedures, decomposition methods, and specialized branch-and-bound searches. Despite this growth,…
In this paper, we study compactness, rigidity, and related geometric properties of complete K\"ahler Ricci shrinkers through the polarized Fano fibration structure.