arXiv subjects

Shengtao Guo

Publications and source records attributed to Shengtao Guo.

9 recordsLinked to original sources

arXiv2026

A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere

We construct an explicit one-parameter family of smooth metrics on the Gromoll-Meyer exotic seven-sphere, converging in $C^\infty$ to a fixed further Cheeger deformation of the Eschenburg-Kerin metric and having strictly positive sectional curvature for all sufficiently small positive parameter values. The first perturbation preserves the totally geodesic flats of one zero-plane family while making curvature positive near the other; the second removes the remaining zero curvature. The metric and the proof were discovered by the Odin Automatic AI Research Agent.

math.DG↗

arXiv2026

Vector Balancing via Directional Total Variation

Our main result is a $3\sqrt{2π}$ bound for the Komlós signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $κ\ge0$, if a bounded open convex set supports a probability density with directional total variation at most $κ$ in every unit direction, then its open-set Banaszczyk transform supports another such density with the same $κ$, provided the translation vector $v$ satisfies $κ\|v\|_2\le1/3$. As a consequence, every finite set system in which each element belongs to at most $t$ sets, where $t\ge1$ is an integer, admits a two-coloring whose imbalance in each set is less than $3\sqrt{2πt}$. This gives the square-root dependence predicted by the Beck-Fiala conjecture. The proof was discovered by the Odin Automatic AI Research Agent.

math.CO↗

arXiv2026

An Improved Lower Bound for the Complex Grothendieck Constant

We prove $K_G^{\mathbb{C}}>1.35584631827168$ for the classical complex Grothendieck constant, closing more than one quarter of the gap between Davie's lower bound and Haagerup's upper bound. The numerical part of the proof is rigorously verified by interval arithmetic. The lower bound and the proof were discovered by the Odin Automatic AI Research Agent.

math.FA↗

arXiv2026

A Metric with Positive Sectional Curvature on $S^3\times S^3$

We construct a smooth Riemannian metric with strictly positive sectional curvature on $S^3\times S^3$. Since $S^3\times S^3$ is even-dimensional and has Euler characteristic zero, our result disproves the positive-curvature case of Hopf's sign conjecture. The metric and the proof were discovered by the Odin Automatic AI Research Agent.

math.DG↗

arXiv2026

On Unavoidable Faces of High-Dimensional Polytopes

Kalai's cube--simplex conjecture asserts that for all positive integers $\ell,k$, there is an integer $f(\ell,k)$ such that every polytope of dimension at least $f(\ell,k)$ has either a simplex $\ell$-face or a cube $k$-face; let $f_s(\ell,k)$ denote the threshold restricted to simple polytopes. Finiteness of $f(\ell,k)$ is known only for $\ell,k \leq 2$. In addition, Kalai proved that $f_s(2,k) \leq 2k^2$. Here we prove that $f_s(\ell,k)$ is finite for all $\ell \geq 2$ and $k \geq 3$, the first such result beyond $\ell = 2$, with $f_s(2,k) \leq 2k^2-1$ and $f_s(\ell,k) \leq \tfrac{1}{2}k^2\ell\,2^k$ for $\ell \geq 3$. In the opposite direction, we obtain the lower bounds $f(\ell,k) \geq (5\lfloor \ell/2 \rfloor + (\ell \bmod 2) - 1)(k-1)+1$ and $f_s(\ell,k) \geq \max\{4,\,2(\ell-1)\}(k-1)+1$. A companion question asks for the minimum possible size of a 3-face within a higher-dimensional polytope. Meisinger, Kleinschmidt and Kalai proved that every rational $d$-polytope with $d \geq 9$ has a $3$-face with fewer than $78$ vertices or fewer than $78$ facets. Here we improve their bound: every convex polytope of dimension at least $15$ has a $3$-face with at most $13$ facets. One step of our proof requires an explicit exact rational certificate or identity on flag numbers. This certificate is computed using linear programming.

math.CO↗

arXiv2026

A Complex Structure on $S^2\times S^4$

We show that $S^2\times S^4$ admits a complex structure. Starting from a modular family of complex two-tori associated with the $(3,4,\infty)$ triangle group and the compactification constructed in [Alpöge 2026], we replace the period lattice by its unique monodromy-invariant index-two superlattice and compactify the resulting family. In the integral Mayer--Vietoris calculation, a primitive local class whose double is the class of a cusp component and the unique nonzero torsion class contributed by the multiplicity-four fibre restrict to the same class of order two on the common boundary. We then prove that the resulting compact complex threefold is diffeomorphic to $S^2\times S^4$. The proof is discovered by the Odin Automatic AI Research Agent.

math.GM↗

arXiv2026

A Metric with Positive Sectional Curvature on $S^2\times S^3$

We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric whose zero-curvature planes are the horizontal lifts of the tangent planes to those tori. We then perturb this metric by the real part of a global complex-valued symmetric $2$-tensor. Differentiation along the circle fibers produces a trace-free first variation of the second fundamental form on local horizontal lifts of the flat tori. The Gauss equation converts this into a positive second-order curvature term that dominates as the fibers shrink. A quantitative lower bound for the Hessian in directions normal to the set of zero-curvature planes extends this positivity to nearby planes. The metric and the proof are discovered by the Odin Automatic AI Research Agent.

math.DG↗

arXiv2026

Weak-Type Bounds for Convolution on the Boolean Hypercube

Let $G$ be the Boolean hypercube which carries uniform measure $λ$, and let $T_μ$ denote convolution by a finite positive measure $μ$ on $G$. For $ψ_μ(u)=\sup\{uλ(\{T_μf\geq u\}):f\geq 0,\|f\|_1=1\},$ we prove Talagrand's convolution conjecture (Talagrand, 1989): if $μ_a=((1+a)δ_1/2+(1-a)δ_{-1}/2)^{\otimes n}$ and $0 1$ and $n\geq1$, where $C_a$ depends only on $a$. The proof utilizes the reverse-heat and Boolean-bridge framework of Chen (2025) and the localized terminal-discrepancy method of Xiang and Zhang (2026). We introduce a new power coupling: each reverse edge ratio is split into two geometric powers. This choice produces a switched exponential weight which restores the exact reverse jump rate of the perturbed coordinate. The resulting endpoint comparison yields an anti-concentration profile estimate without the iterated-logarithmic factor. The proof was discovered by the Odin Automatic AI Research Agent.

math.PR↗

arXiv2025

A Challenging Benchmark of Anime Style Recognition

Given two images of different anime roles, anime style recognition (ASR) aims to learn abstract painting style to determine whether the two images are from the same work, which is an interesting but challenging problem. Unlike biometric recognition, such as face recognition, iris recognition, and person re-identification, ASR suffers from a much larger semantic gap but receives less attention. In this paper, we propose a challenging ASR benchmark. Firstly, we collect a large-scale ASR dataset (LSASRD), which contains 20,937 images of 190 anime works and each work at least has ten different roles. In addition to the large-scale, LSASRD contains a list of challenging factors, such as complex illuminations, various poses, theatrical colors and exaggerated compositions. Secondly, we design a cross-role protocol to evaluate ASR performance, in which query and gallery images must come from different roles to validate an ASR model is to learn abstract painting style rather than learn discriminative features of roles. Finally, we apply two powerful person re-identification methods, namely, AGW and TransReID, to construct the baseline performance on LSASRD. Surprisingly, the recent transformer model (i.e., TransReID) only acquires a 42.24% mAP on LSASRD. Therefore, we believe that the ASR task of a huge semantic gap deserves deep and long-term research. We will open our dataset and code at https://github.com/nkjcqvcpi/ASR.

cs.CV↗