# PACM Graduate Students

# PACM Graduate Students

**On the distribution of heat in fibered magnetic fields**

**Hezekiah Grayer, Princeton**

We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for "nearly integrable" magnetic fields up to a "critical" size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and is related to the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul-Hudson-Helander.

**Vector-Valued Concentration on the Symmetric Group**

**Miriam Gordin,Princeton**

Concentration inequalities for real-valued functions are well understood in many settings and are classical probabilistic tools in theory and applications -- however, much less is known about concentration phenomena for vector-valued functions. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group. Furthermore, we discuss the implications of this result regarding the distortion of embeddings of the symmetric group into Banach spaces, a question which is of interest in metric geometry and algorithmic applications. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a question of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel

**A few steps towards the Erdős–Hajnal conjecture**

** Tung Nguyen, Princeton**

A cornerstone of Ramsey theory says that every graph contains a clique or independent set of logarithmic size, which is asymptotically optimal for almost all graphs. The Erdős–Hajnal conjecture from 1977 predicts a very different situation in graphs with forbidden induced subgraphs; more precisely, the conjecture asserts that for every graph $H$, there exists $c=c(H)>0$ such that every $n$-vertex graph with no induced copy of $H$ has a clique or independent set of size at least $n^c$. This conjecture remains wide open, and we will discuss recent progress on it in the talk.