Mathematician Bjoern Bringmann has inspired the development of a proof-of-concept tool designed to automate the verification of asymptotic estimates. These estimates are inequalities that aim to describe relationships between mathematical quantities, especially for very large values of parameters, while allowing for an unspecified multiplicative constant loss. The focus here is on simpler inequalities involving a finite number of positive real numbers combined through basic arithmetic operations like addition, multiplication, division, exponentiation, and comparisons like minimum and maximum. Core Mathematical Problem and Initial Approach Consider the weak arithmetic mean-geometric mean inequality: [ \max(a, b, c) \lesssim (abc)^{1/3} ] where ( a, b, c ) are positive real numbers, and the symbol ( \lesssim ) indicates that the inequality holds up to a multiplicative constant. To verify this inequality, one can split the problem into cases based on the relative sizes of ( a, b, ) and ( c ). For instance, if ( \max(a, b, c) = a ), then the inequality simplifies to: [ a \lesssim (abc)^{1/3} ] which, after taking logarithms, becomes a positive linear combination of the hypotheses ( b \lesssim a ) and ( c \lesssim a ). This simplified form is a standard linear programming problem, and various software packages can solve it. Development of the Tool With the increasing demand for automating such tasks, especially in scenarios involving numerous inequalities, the author embarked on a project to develop a Python-based tool. Leveraging Large Language Models (LLMs) to generate initial code and autocomplete, the author spent about four hours coding a basic verifier. The tool can handle inequalities like the one mentioned above and provide detailed, automated proofs. For the inequality ( \max(a, b, c) \lesssim (abc)^{1/3} ), the Python code is straightforward: python a = Variable("a") b = Variable("b") c = Variable("c") assumptions = Assumptions() assumptions.can_bound((a * b * c) ** (1/3), max(a, b, c)) The output of the tool splits the problem into several cases and verifies each one, ultimately confirming the inequality's validity. Handling Complex Expressions While the tool currently handles simpler inequalities effectively, it is poised to evolve. One potential upgrade is to manage more complex hypotheses involving expressions like ( \epsilon )-dependent terms or norms of functions. For instance, the tool aims to verify inequalities such as: [ \log \left( \frac{\max(A, B, C)}{\min(A, B, C)} \right) \lesssim \log \left( \frac{\max(a, b, c)}{\min(a, b, c)} \right) ] for any ( A, B, C ) defined in terms of ( a, b, c ) under specific constraints. The author acknowledges that the current code is inefficient and inelegant but sees it as a stepping stone towards a more sophisticated tool. Applications and Future Directions Such a tool has significant practical applications, particularly in fields like Partial Differential Equations (PDE) and Harmonic Analysis, where verifying complex multilinear expressions can be both time-consuming and error-prone. For example, the author once needed to establish: [ \max(a_1 + b_1, ..., a_n + b_n) \leq (1 + o(1)) (\max(a_1, ..., a_n) + \max(b_1, ..., b_n)) ] for sequences ( a_i ) and ( b_i ) under certain conditions. This required checking multiple inequalities, which the tool could automate, potentially saving substantial time and effort. Collaboration and Enhancement The future of this project hinges on collaboration between mathematicians and expert programmers. Features that mathematicians might find valuable include the ability to input an expression to estimate, along with a fixed set of bounding tools (like splitting integrals, integrating by parts, or applying Hölder and Sobolev inequalities), and having the tool optimize the bounds with a verifiable proof certificate. One long-term goal is to integrate AI for suggesting possible splittings of sums or integrals, thus improving the efficiency of the verification process. Additionally, the tool could output proof certificates in formal proof assistant languages such as Lean, enhancing its utility in rigorous mathematical research. Industry Insights and Company Profiles Industry experts, including Heather Macbeth, have noted that the integration of automated tools changes the way mathematicians approach proof writing. Formal proof assistants, for instance, require meticulous and structured proofs, a shift that could benefit from automated support. Projects like this often thrive in collaborative environments, such as within platforms like SageMath, which combine symbolic computation with a user-friendly interface for mathematicians. By automating the verification of asymptotic estimates, this tool could significantly streamline research processes, reduce manual labor, and minimize errors. The potential for such a tool in academic and industrial settings highlights the growing importance of interdisciplinary efforts in advancing mathematical software.