Large language models have recently demonstrated advanced capabilities in solving IMO and Putnam problems; yet their role in research mathematics has remained fairly limited. The key difficulty is verification: suggested proofs may look plausible, but cannot be trusted without rigorous checking. We present a framework, called LLM+CAS, and an associated tool, O-Forge, that couples frontier LLMs with a computer algebra systems (CAS) in an In-Context Symbolic Feedback loop to produce proofs that are both creative and symbolically verified. Our focus is on asymptotic inequalities, a topic that often involves difficult proofs and appropriate decomposition of the domain into the "right" subdomains. Many mathematicians, including Terry Tao, have suggested that using AI tools to find the right decompositions can be very useful for research-level asymptotic analysis. In this paper, we show that our framework LLM+CAS turns out to be remarkably effective at proposing such decompositions via a combination of a frontier LLM and a CAS. More precisely, we use an LLM to suggest domain decomposition, and a CAS (such as Mathematica) that provides a verification of each piece axiomatically. Using this loop, we answer a question posed by Terence Tao: whether LLMs coupled with a verifier can be used to help prove intricate asymptotic inequalities. More broadly, we show how AI can move beyond contest math towards research-level tools for professional mathematicians.

Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra systems (CASs) provide support for solving ODEs by first simplifying them, in particular through the use of Lie point symmetries. Finding these symmetries is, however, itself a difficult problem for CASs. Recent works in symbolic regression have shown promising results for recovering symbolic expressions from data. Here, we adapt search-based symbolic regression to the task of finding generators of Lie point symmetries. With this approach, we can find symmetries of ODEs that existing CASs cannot find.

We tackle the problem of neural machine translation of mathematical formulae between ambiguous presentation languages and unambiguous content languages. Compared to neural machine translation on natural language, mathematical formulae have a much smaller vocabulary and much longer sequences of symbols, while their translation requires extreme precision to satisfy mathematical information needs. In this work, we perform the tasks of translating from LaTeX to Mathematica as well as from LaTeX to semantic LaTeX. While recurrent, recursive, and transformer networks struggle with preserving all contained information, we find that convolutional sequence-to-sequence networks achieve 95.1% and 90.7% exact matches, respectively.

In this article, I will cover a new Neural Network approach to solving 1st and 2nd order Ordinary Differential Equations, introduced in Guillaume Lample and François Charton (Facebook AI Research)'s ICLR 2020 spotlight paper, "Deep Learning for Symbolic Mathematics"¹. This paper tackles symbolic computation tasks of integration and solving 1st & 2nd order ODEs with a seq2seq Transformer, we will focus on the latter today. To give context to this paper, although Neural Network methods have seen great success in clearly structured statistical pattern recognition tasks -- e.g. Not only does Symbolic Computation require AI to infer complex mathematical rules, they also require a flexible, contextual understanding of abstract mathematical symbols in relation to each other. At the time of authoring, Computer Algebra Systems (CAS) (such as Matlab, Mathematica) held state-of-the-art performance on symbolic mathematics tasks, driven by a backend of complex algorithms such as the 100-page long Risch algorithm for indefinite integration.

Wolfram Alpha has been a huge hit with students. Whether in college or high school, Wolfram Alpha has become a ubiquitous way for students to get answers. But it's a one-shot process: a student enters the question they want to ask (say in math) and Wolfram Alpha gives them the (usually richly contextualized) answer. It's incredibly useful--especially when coupled with its step-by-step solution capabilities. But what if one doesn't want just a one-shot answer? What if one wants to build up (or work through) a whole computation?

A machine learning tool called Northstar lets users play with data visually. Yang-Hui He, a mathematical physicist at the University of London, is an expert in string theory, one of the most abstruse areas of physics. But when it comes to artificial intelligence (AI) and machine learning, he was naïve. "What is this thing everyone is talking about?" he recalls thinking. Then his go-to software program, Mathematica, added machine learning tools that were ready to use, no expertise required.

Wolfram Research today announced free access to the engine that powers its technology stack. The Wolfram Engine is available to developers for free, assuming it is used for non-production development. Wolfram Research is best known for creating the modern technical computing system Mathematica and the computational knowledge engine Wolfram Alpha (stylized Wolfram Alpha). Founded by computer scientist Stephen Wolfram, the company celebrated the 10-year anniversary of Wolfram Alpha just last week. "The Wolfram Engine is the heart of all our products," Stephen Wolfram explains.

These codes, typically designed by the quantitative analysts at investment banks, help determine prices for investment products, make trading decisions, and assess and control financial risk. The rate of growth in this area is striking. For example, the volume of the parent industry, custom ("overthe-counter") derivative securities trading, has increased 12-fold from 1990 to 2000 to 80 trillion dollars. Spending for modeling software is close to a billion dollars a year, with an expected growth rate of about 10 percent. Scholes equation described in figure 1.

Denise Garcia knows that her students sometimes cheat, but the situation she unearthed in February seemed different. A math teacher in West Hartford, Connecticut, Garcia had accidentally included an advanced equation in a problem set for her AP Calculus class. Yet somehow a handful of students in the 15-person class solved it correctly. Those students had also shown their work, defeating the traditional litmus test for sussing out cheating in STEM classrooms. Garcia was perplexed, until she remembered a conversation from a few years earlier.

For those who are looking for something a little less costly, Udacity also offers a number of free machine learning courses ranging from 10 weeks to four months. Lynda from LinkedIn is a leading online learning platform that helps anyone learn a wide range of skills, including machine learning. To help grasp the basics of technology and data mining, Alison is a Galway-based e-learning platform offering a number of free online courses on software development, data science and machine learning. Udemy offers thousands of online courses with which to upskill, including a number of machine learning courses.