Scientific

I will discuss a lemma which is usually attributed to Atkinson, is very useful and apparently has been rediscovered multiple times. The lemma says that if (X, B, mu, T) is an ergodic system and f is in L^1(mu) with mean zero, then the Birkhoff sums of f do not diverge almost surely. Moreover the sums switch signs in a wide sense, infinitely many times. I will give the proof and discuss some applications of this result. Then I will discuss higher dimensional analogues, where the situation is significantly more complicated.

Quantum phase transitions occur when a quantum system undergoes a sharp change in its ground state, e.g. between a ferro- and para-magnet. I will present a remarkable set of transitions, called quantum critical, that are described by conformal field theories (CFTs). I will focus on 2 and 3 spatial dimensions, where the conformal symmetry is powerful yet less constraining than in 1 dimension. We will probe these scale-invariant theories via the structure of their quantum entanglement. The methods will include large-N expansions, the AdS/CFT duality from string theory, and large-scale numerical simulations. Finally, we’ll see that certain quantum Hall states, which are topological in nature, possess very similar entanglement properties. This hints at broader principles that relate very different quantum states.

For other events in this series see the quanTA events website.

Transfer systems are discrete objects that encode the homotopy theory of N∞ operads, i.e., the operads whose algebras are homotopy commutative monoids with a class of equivariant transfer (or norm) maps. They have a rich combinatorial structure defined in terms of the subgroup lattice of the group of equivariance, G. Indeed, if G is a cyclic p-group, there are Catalan-many transfer systems that assemble into the Tamari lattice (i.e., associahedron). In this talk, I will show that when G is finite Abelian, transfer systems are in natural bijection with weak factorization systems on the poset category of subgroups of G. This leads to a novel involution on the lattice of transfer systems, generalizing an observation of Balchin–Bearup–Pech–Roitzheim for cyclic groups of squarefree order. I will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg–Hill saturation conjecture.

This is joint work with Angélica Osorno and a team of Reed College undergraduates: Evan Franchere, Usman Hafeez, Peter Marcus, Weihang Qin, and Riley Waugh (the Electronic Collaborative Mathematics Research Group, or eCMRG).

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g., K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss ongoing work with Keegan Boyle trying to understand the equivariant 4-genus.

A triangulation of a surface is a way to divide it into a finite number of triangles. Let us pick a random triangulation uniformly among all those with a fixed size and genus. What can be said about the behaviour of these random geometric objects when the size gets large? We will investigate three different regimes: the planar case, the regime where the genus is not constrained, and the one where the genus is proportional to the size. Based on joint works with Baptiste Louf, Nicolas Curien and Bram Petri.