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.