Mathematics
Modular Functions Decode Monster Group Exponents for Primes >3
Duncan and Swisher introduce a modular function-based method to derive the exponents in the prime factorization of the Monster group's order, specifically for primes larger than 3. This approach leverages properties of modular functions to explain these exponents, connecting number theory and group …
ADE Bundles: Bridging Surface Geometry and Lie Theory
This review paper explores the construction and properties of ADE bundles over surfaces. It leverages the inherent connections between surface geometry and ADE Lie theory to provide a comprehensive overview of related problems.
Extremal Problems in Subgraph Density for Ordered and Colored Graphs
This paper investigates three distinct extremal problems related to subgraph densities. It corrects a prior result on k-edge star maximization, determines the extrema for k-edge stars in ordered graphs, and presents novel complete results for the maximum density of a specific 3-edge-colored complete…
Inducibility of Rainbow K-Cliques for k ≥ 11 Determined
This paper proves that the inducibility of a rainbow k-clique (R) is k!/(k^k-k) for k ≥ 11. This addresses a generalization of a problem by Erdős and Sós, posed by Huang. The extremal construction is a balanced recursive blow-up of R. The authors also establish a broader result, showing that k-verte…
Arctic Circle Theorem: Central Disorder in Aztec Diamond Domino Tilings Converges to Circle
In random domino tilings of large Aztec diamonds, the central region of coexisting tile orientations forms a shape arbitrarily close to a circle of radius n/sqrt(2), while outer regions exhibit uniform alignment. This Arctic Circle Theorem is proven using interacting particle systems, specifically a…
Pi and Tau in Binary
In binary representation, the digits of Pi and Tau are identical, with Tau's digits being a left-shifted version of Pi's. This mathematical curiosity arises from the fundamental relationship between the two constants (Tau = 2 * Pi) and the nature of binary representation.
Profinite Criterion for Primitive Words in One-Relator Groups with Torsion
This paper introduces a method for identifying surface subgroups within specific one-relator groups that possess torsion. This discovery leads to the derivation of a profinite criterion. This criterion ascertains whether a given word in a free group is primitive, offering a novel tool for analyzing …
Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G₂-Laplacian Flow
This paper presents a proof of the uniqueness of asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G₂-structures. The authors extend existing arguments for Ricci solitons to the context of G₂-Laplacian flow. The research also investigates the inheritance of sym…