Undergraduate Seminar Talks

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.

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.

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.