In the last few years, neural networks devoted to image recognition have become more and more reliable.

In this talk, I will describe (simplified) versions of these networks as somehow general templates for classification, not just of images but of more general objects.

Still, I will show that even working with the most sophisticated image recognition models, one can still fool them on purpose and sometimes the resulting "hallucination" is stable under perturbation. One talks in this case about "hallucinations" because a human eye would have no problem in recognizing the fooling image. This of course presents a problem for "mission critical" applications.

I will address the use of these general templates for classification in Mathematics. In this case, two challenges are readily available: one is the capture of the features of the mathematical object of interest in terms of a numerical tensor (an array); the other is to have a large enough set of objects of different type on which to perform the training. Both problems are easily solvable in the case of image recognition. If time permits, I will address the challenges one faces to implement this scheme in recognizing an integrable Hamiltonian system vs a non-integrable one (a binary classification).

Mathematical proving and mathematical proofs are the heart of mathematics, but mathematical proving and mathematical proofs are generally only taught in depth at post-secondary institutions. This disjuncture between student learning of mathematics in K-12 and post-secondary systems impact the development of mathematical thinking and abilities of youths in the United States. The advent of generative artificial intelligence is opening up possibilities of more structured ways for youths to learn mathematics in a PK-20 context, until they enter the workforce. This project conducts a framework synthesis of evidence of the use of artificial intelligence in teaching of mathematical proving and mathematical proofs in universities in American, European, and Asian universities. The educational experiences of Fields Medalists (1990-2023) augment the findings. The source of evidence includes conventional academic publications used in a typical evidence process; coupled with a systematic scanning and review of the mathematical proving and proofs courses held in the universities in these three regions. This project pays particular attention to the integration of artificial intelligence in the teaching of mathematical proving and mathematical proofs. In addition, a PK-20 framework of teaching mathematical proving and proofs is proposed based on the syntheses of insights.

We will explore how methods from the theory of invariants can be used in deep learning. Conversely, we will show how to use machine learning techniques to get new results on invariants of binary forms.

Many delicate problems in number theory have only recently become amenable to numerical exploration due to the painfully slow convergence rate. Quantities associated to elliptic curves often converge at the scale of the logarithm of the conductor; thus while we may have millions of curves with conductors at most 10^{2}0, this translates to less than 50 (and one would never conjecture on properties of primes from integers up to 50!). Improvements in computing power have led to larger data sets, which in conjunction with machine learning (ML) techniques have found new behavior. We discuss the recent successes of He, Le and Oliver who used ML to distinguish numerous curves based on standard invariants, and discovered oscillatory behavior in the coefficients of the associated L-functions, which agrees with recently developed theoretical models. We report on work of the author and his colleagues on lower order terms in coefficients in families, describing an open conjecture where the "nice" term is hard to extract due to large, fluctuating terms, in the hopes of forming collaborations with audience members.

We discuss how problems from discrete mathematics can tackled with reinforcement learning (RL), a subfield of machine learning. While machine learning techniques are typically stochastic and the neural networks underlying them are notoriously difficult to interpret, the actions taken by a reinforcement learning agent can be used to unambiguously verify solutions proposed by the RL algorithm, and are sometimes interpretable by domain experts. We will illustrate this by applying RL to detect whether a given knot is the unknot, and whether a given knot is ribbon. The latter was used to rule out many proposed potential counterexamples to the smooth Poincare Conjecture in 4D.

Studying arithmetic properties of the moduli space of algebraic curves is a classical problem with many implications in arithmetic geometry and applications to cryptography. We explore how to use methods of deep learning to better understand the rational points of the moduli space. Several examples of such methods, from the moduli space of genus two curves, will be described in detail illustrating our approach. The talk will be accessible to a general audience.

Reduction of integer binary forms is a classical problem in mathematics. It basically is the idea of picking a coordinate system such that the binary form has "small" coefficients. However, the only case that is fully understood is for quadratics. In 1917, in the first part of his thesis, Gustav Julia suggested a very interested reduction method for an arbitrary degree binary form. It is based on the idea of defining a quadratic (Julia quadratic) J_f which is covariant under the action of the modular group via coordinate changes. This quadratic is a positive definite quadratic and therefore has only one root in the upper-half complex plane H_2, say alpha_f. Since J(f) is an SL(2,Z)-covariant, then bringing alpha_f to the fundamental domain F of SL(2,Z) by a matrix M in SL(2,Z), induces an action f → f^M on binary forms. In this talk we explore the idea of a neural network which performs Julia reduction for binary forms of degree d > 2. The talk will be accessible to mathematicians, computer scientists, and graduate students who are familiar to basic concepts of machine learning.

We present a novel automatic architecture search algorithm capable of handling non-layered architectures. The algorithm iteratively prunes connections and adds neurons to optimize the network’s complexity while meeting a specified error threshold. In our approach, an architecture is represented as an arbitrary oriented graph with weights, biases, and an activation function, allowing for flexibility in structure.

We validate the effectiveness of our algorithm through experiments on two test problems. The first task involves predicting the brightness of subsequent points in a standard test image based on previous points. The second task focuses on approximating the bivariate function that determines the brightness of a black and white image. In both scenarios, our optimized networks consistently outperform standard neural network architectures by a significant margin.