mathematics

Canonical correlation analysis is a widespread technique for discovering linear relationships between two sets of variables. In high dimensions, however, standard estimates of the canonical directions cease to be consistent without assuming further structure. In this setting, a possible solution consists in leveraging the presumed sparsity of the solution: only a subset of the covariates span the…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism topos, homotopy topos type theory, homotopy type theory …

Unlike Caesar, each letter is replaced by a different letter according to an arbitrary mapping (not a simple shift). We solve it by comparing the letter frequency of the ciphertext to the known letter frequency in English, and guessing short words ( the , and , a ...). Platform: picoGym Category: Cryptography Points: 150 pts Difficulty: Beginner Technique: Monoalphabetic substitution, frequency a…

OpenAI is at the center of another controversy over machines tackling big math problems. This one’s about the twin prime conjecture.

Over the last few months, the mathematical capabilities of LLMs have improved dramatically, to the point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the math…

UMBC mathematician Matthew Kvalheim has received an Air Force Office of Scientific Research Young Investigator Program Award to develop better ways of simplifying hybrid systems, like a drone swarm, humanoid robot, or aircraft. Kvalheim aims to create mathematical tools that make hybrid dynamics—which govern almost all real-world systems—easier to study and control, giving researchers a clearer “…

Researchers devised the dice to solve a tricky math problem: When playing a game, who goes first?

Mathematicians now have a formidable tool for breaking open challenging problems, but working out how best to use it isn't going to be easy

Nature Communications, Published online: 11 September 2026; doi:10.1038/s41467-026-77737-0 In a deterministic, non-chaotic system, the initial state fixes the future completely, yet that future may be impossible to forecast from it. Using a cellular automaton of communicating cells, this study shows that predictability is instead built by the dynamics: collective topological structures self-organ…

Following is copy of this question from Math Stackexchange posted by QuantumFieldMedalist . A bounty was placed on the question on that website by me, which resulted in no new comments or answers. Hence given the subject of question, I am posting the same question here verbatim: I have been studying some aspects of Ricci flow, namely existence, uniqueness, finite time extinction, the preservation…

Scientific Reports, Published online: 11 September 2026; doi:10.1038/s41598-026-70795-w Numerical analysis of complete period-adding structures in a novel chaotic system for message encoding

The Hodge conjecture asserts that every rational class of type (p,p) on a smooth complex projective variety is a rational combination of classes of algebraic cycles; it is known for p≤ 1 and p≥ n-1 and open in the middle degrees of every dimension n≥ 4. Reductions of the conjecture typically restate it or split it into further open hypotheses, such as the existence of an algebraic representative …

This monograph introduces a novel architectural paradigm at the intersection of non-equilibrium quantum dynamics, analytic number theory, and synthetic genomics to resolve the cellular metabolic energy-budget paradox. While traditional biophysics models treat non-coding genomic sequences as stochastic, ergodic polymers subjected to random thermal fluctuations, this framework establishes that macr…

I have baseline data on 100 participants from an ongoing longitudinal study. At the screening visit each participant answered sociodemographic, clinical and lifestyle questions and completed a ...

Famous and challenging problems in mathematics are falling to AI at a rapid pace. Where does this leave mathematicians, and what will the field look like in coming years?