At a time when mathematicians are rethinking the foundations of their field, it is worth recalling an earlier crisis and how it was dealt with. In the late 17th century the Royal Society championed an experimental approach to the study of nature as the proper path to the advancement of knowledge and the preservation of civic peace. Mathematics was admired, but also feared, as dangerously dogmatic and coercive. In response John Wallis, the leading mathematician in the group, developed a radical new approach: Whereas traditional mathematics prided itself on irrefutable deductive proofs, Wallis’ methodology relied on material intuition, inductive reasoning, and truth-claims founded on agreement, not coercion. It was a new mathematics, designed to align the field with the Society’s empirical and consensus-driven approach.

Since the early 20th century, philosophers have debated the relationship between informal proofs of the sort mathematicians have produced in their work for centuries, and the formalized proofs that appear in the branch of logic known as "proof theory". In the early 21st century, these questions grew more pressing with the development of computer verification systems like Isabelle, Mizar, and Lean, which enabled mathematicians to more efficiently (but still slowly) translate informal proofs into formalized ones. Over the past year AI systems have suddenly upended the relationship of mathematicians to both formalized and informal proofs, by generating new informal proofs (like OpenAI's announcement of May 20 of a new bound for the planar unit distance problem) and new formalizations of long and difficult informal proofs into formalized proofs (like Anthropic's announcement on Sept. 4 of a formalization of Wiles's proof of Fermat's Last Theorem), and the combination (like OpenAI's announcement on Sept. 8 of a novel formalized proof of a counterexample to existence and smoothness of solutions to the Navier-Stokes equation). In this talk, I will discuss the relevance of the work of various philosophers of mathematics to the novel situation we are confronted with, and the questions this raises about the goal of mathematics as a human endeavor.

One of the early non-numerical, experimental uses of digital computers was to generate music. In this talk I explore a somewhat unusual case and its aftermath from 1960s Hungary. Miklós Havass, one of Hungary’s earliest computer science graduates, defended his thesis on “Musical Composition with Electronic Calculators” in 1963. While the country’s contemporary music scene was lacking, it had a strong tradition in folk music education due to the strenuous efforts of composers and ethnomusicologists Béla Bartók and Zoltán Kodály. This led Havass to use one of the first computers in Hungary to compose “epigone” folk songs based on Kodály’s easily available sheet music booklets. The project not only achieved public notice, but caught the attention of the elderly Kodály himself, who invited Havass to participate in his Folk Music Research Group.\nAt the time the Group was preparing catalogues to include all Hungarian folk songs. The organization of the vast amount of melodies in a way that enabled easy search in the collection, finding similar melodies (for example for historical or cross-cultural comparisons), and analysis were all quite challenging. Kodály called for Havass’ help in computerizing the process. The first technological achievements of the Group were presented and demonstrated at the annual conference of the International Folk Music Council in 1964.\nSadly, this early attempt at computational ethnomusicology ended relatively soon. The computers were not yet ready to process such a large amount of information, Havass was offered a demanding software developer position and had to leave the research group a couple years later, and the dynamic leader of the group, Kodály, died in 1967. Ethnomusicology also took an anthropological turn in the 1970s, which made the field less perceptive to such attempts for the coming decades.

Throughout the years between his first engagement with mathematics under Isaac Barrow and the production of the manuscript that evolved into Book I of the Principia, Newton was only sporadically engaged by issues of motion, whether earthly or astronomical. His first concerted interest involved collisions, which he modeled on the material circumstances of a springlike mechanism. In doing so, and in a subsequent turn to circular motion, the young Newton presumed without comment what would later be formalized as the Principia's three laws of motion. He was hardly the only one, at least in England, to make such presumptions. We will discuss this early work in order to unpack what transpired when, in 1679, Newton responded to a letter from Robert Hooke, his former critic in optics. We'll then consider what in consequence took place following the famous visit to Newton by Edmond Halley in 1684. We’ll conclude with a brief examination of the interesting reaction of the mathematician John Wallis to Newton's manuscript account of fall under resistance, sent to Wallis for comment by Halley two years before the Principia.

The Grothendieck constant \(K_G\) is a fundamental quantity measuring the worst-case gap between a particular class of integer quadratic optimization problem and its efficiently solvable semidefinite programming relaxation. It has found diverse applications in fields such as functional analysis, combinatorial optimization, theoretical computer science, and quantum information theory. Its value has remained open since its introduction by Alexander Grothendieck over 70 years ago, despite sustained interest from the mathematical community. Our work determines the tenths digit for the first time, proving \(6\pi/11 < K_G < \pi/(2\log(1+\sqrt{2})) - 10^{-4}\). We will describe the technical details of these improvements, which relies on conceptually new approaches to both sides of the problem: new asymptotic rounding schemes for the upper bound, and a lower-bound method based on proving fundamental limitations of entire classes of rounding schemes rather than constructing individual hard instances. We will describe the AI research system we built, and the roles played by both AI and human researchers throughout the process. Finally, we will discuss what this experience suggests about the importance of human research judgement and the future of human-AI collaboration.

Noncrossing arc diagrams are a visual representation of permutations. They naturally contain complete information on lattice congruences and lattice quotients on the weak order of type A. They are unreasonably effective when considering another partial order of type A, the shard intersection order. This talk will provide an overview of the existing results on noncrossing arc diagrams for permutations and present analogous results (joint with Barnard and Reading) on arc diagrams for signed permutations (type B). These results elucidate sublattice relationships between shard intersection orders on different finite Coxeter groups.

The talk will present an original interpretation of Zeno's paradox in the light of Aristotle's definitive solution of it. While arguing that this solution dissolves the paradoxical aspect of Zeno's argument, it also contextualizes its relevance in the perspective of Aristotle's notions of (potential) infinity and continuity.

The Hungarian High School Mathematics Journal, with a rich history since its foundation in 1893, has been an important factor in the country’s mathematical talent nourishment and management. Paul Erdös was not only an eminent problem solver in the journal’s year-long competitions, but contributed to the journal from time to time throughout his life. Erdös published his first “research article” in the journal while still in high school, proposed problems for the competitions, had his results reported on in the pages of the journal and contributed a few expository articles in his recognizable, unique style. The aim of this talk is twofold. First, to give a brief overview of the Hungarian High School Mathematics journal and its history. Second, to show how Erdös engaged with the youth, as seen through his engagement with the journal.

How well can people, computers and other creatures coordinate with their network neighbors? The answer will involve ideas and techniques from groups, cooperative games, probability and more.

We will explain some exciting connections between card shuffling and probability. Card shuffling is an old subject that combines probability with modern algebra. We will explore different shuffles through group theoretic and linear-algebraic points of view. Finally, we will investigate a model for casino card shuffling machines, and present some new research in this area.

Consider a rope with two ends attached to poles, and a knot in the middle. You are asked to undo the knot without untying the ends of the rope from the poles. When is this even possible? And if it is possible, how hard is it to undo the knot? These turn out to be very interesting math questions! In this talk, I will discuss current research dealing with this type of problem, including research being done by USC undergrads Nathan Derhake and Jinn Chung under my supervision.

The Pythagorean Theorem and similar triangles are two fundamental mathematical results. They have been known for at least 3800 years. Nonetheless, mathematicians still find new ways of combining these principles. We discuss two Ancient Greek proofs of the Pythagorean theorem. High school student A. Jackson and C. Johnson recently published several surprising proof of the Pythagorean theorem. We will discuss one of their proofs that uses infinitely many similar triangles.

This event is not hosted by the math department, and to attend this event you must fill out this ☞ RSVP form. USC’s Risk Management Organization is hosting an actuarial night. The speaker will share insights into the actuarial insdustry and career prospects.

Current USC math faculty and grad students will discuss their experience applying to grad school, and participate in Q&A with attendees.

In this talk, we discuss Rosenboom’s forthcoming book Propositional Music, a study of generative approaches in music and survey of his over sixty-year career as a composer and performer working across electronic music, minimalism, jazz, pop, and contemporary music.

John von Neumann is well-known for his foundational contributions to many fields, from pure mathematics, to physics, economics, computer science, operations research, and nuclear policy. Among the most impactful was a report, drafted in Spring 1945, sketching a design proposal for an electronic digital stored-program computer that would become the blueprint for virtually all electronic computer projects developed in the years immediately following World War II. In this talk, I will track two separate strands that led to and shaped von Neumann's interest in computation: one, through his conversations and collaborations with David Hilbert, Kurt Godel, Max Newman, and Alan Turing during the 1920s and 1930s; and the other through his work on operations research and partial differential equations during World War II. These two strands intersected at a surprising juncture: a series of discussions, beginning in January 1945, with Warren McCulloch and Walter Pitts, who had developed the first mathematical model of a neural network in 1943. I will describe how von Neumann took from McCulloch and Pitts the idea that both biological neurological systems and electronic computers could be seen as Turing machines, and how this led him to investigate the connections between automated computation, agentic systems, and ultimately the possibility of self-reproducing systems.

A famous and consequential open problem in computer science is to design algorithms that multiply n x n matrices in (nearly) n^2 operations. For more than 50 years, the quest for such an "exponent 2" algorithm for matrix multiplication has captured the imagination of computer scientists and mathematicians alike, and it continues to do so today. In this talk I will describe how this algorithmic problem is cast as a mathematically appealing question about tensor rank, and describe a novel approach that imports the problem into the domain of group theory and representation theory. I'll discuss generalizations to algebraic objects beyond groups, connections to problems in combinatorics and other areas of math, and give a sense of the current state-of-the-art in this program aimed at finding an exponent 2 algorithm for matrix multiplication.

In 1874, Brill and Noether gave the name Riemann-Roch theorem to a statement about the number of linearly independent meromorphic functions satisfying suitable conditions on orders of poles on what are now called Riemann surfaces in terms of `topological' information; results in this direction were first proposed by Riemann and improved by his student Roch. Over the next 80 years, Riemann-Roch theorems served as objects of persistent fascination, and acted as prisms refracting themes of interest in mainstream mathematics, reaching a zenith in Alexander Grothendieck's formulation in the mid 1950s. I will trace the question of why Riemann-Roch style theorems played this role, situating this question within an analysis of how different generations reflected on questions of importance.

This talk centers on the first (and perhaps only) doctorate in the theory of correlations --- granted by University College London in 1899 to Alice Lee. The production, reception, and subsequent applications of Lee's research clarify the aims of biometricians, the perceived legitimacy of arguments based on regression analysis (by those who did and did not understand the mathematics), and ways in which bias was measured and assessed.

In this talk, we trace the development of musical minimalism in the early 1960s through its interactions with mathematicians and mathematical ideas. We also discuss some related notions of continuity and construction appearing across math, art, and musical minimalism.

Symmetry has been a structuring device and motif for music in many cultures, and probably everyone who has taken music lessons as a child can call to mind the time translation symmetry encoded by the regular beats of a metronome. Nonetheless, the idea of fractal symmetry in music seems much less common. In this talk, I'll describe recent research investigating the presence of fractal structures in Africanist music, in response to a hypothesis of choreographer Reggie Wilson. I'll describe both the project and "scientific" findings, as well as the broader context of my unexpected and ongoing engagement with Reggie Wilson and his Fist and Heel Performance Group at the interplay of math and dance. This is joint work with Claudio Gómez-Gonzáles, Sidhanth Raman and Siddharth Viswanath.

The probability that a uniformly random permutation on \(2n\) letters has exactly \(\alpha\) transpositions is equal to the probability that a uniformly random isometry of the \(n\)-cube has exactly \(\alpha\) pairs of faces that are fixed. In this talk, I will describe generalizations of this relationship, show how this gives information about the expected value of letters of permutations with a given number of \(k\)-cycles, and propose several open questions related to families of permutation statistics.

This talk will present joint research with Brenda Danilowitz on Geometry for Artists, a workshop that Max Dehn developed in the late 1940s at Black Mountain College in North Carolina. An unusual blend of projective and descriptive geometries, the course reflected the experimental college’s avant-garde late modernist legacy as well as Dehn’s ongoing philosophical investigations into the common roots of mathematics and ornamentation.

Some of my favorite combinatorial proofs have been discovered with undergraduates. We'll present clever counting arguments and beautiful bijections involving binomial coefficients, squares, cubes, and Fibonacci numbers.

Scientists and mathematicians apply the continuum hypothesis to model fluid flow, e.g. the flow of substances like air or water. Models of Newtonian fluids in the real world often result in a system of dissipative equations. In particular, these equations have an internal diffusion term that allows them to dissipate energy, but the way in which this happens in general is non-trivial and chaotic; in a word, turbulent.  As a result, for many of these systems one has exponential divergence of trajectories that start infinitesimally close to one another, making them highly unpredictable. As a result, the first challenge of many to accurately simulate turbulent flows we directly observe in the real world (like the climate) is that the initial state of our system trajectory is unknown, and guessing a good initial state has been a subject of intense study for decades. Data assimilation addresses this issue by continually incorporating observed data into the model equations. In this talk, I will discuss my work using a theoretically simple yet rigorous approach to data assimilation in order to accurately model fluid flows.