Department of Mathematics

Around 1915, Polya and Hilbert independently speculated that self-adjoint operators could prove the Riemann Hypothesis for zeros of L-functions, and roughly 70 years later, Colin de Verdière advanced this idea by suggesting a physics-oriented way to make inhomogeneous equations into homogeneous ones. Making this rigorous, Bombieri and Garrett in the 2010’s showed that the simplest instance of Colin de Verdière’s idea does not succeed because at most 94% of the zeros of zeta can participate in the spectrum of such operators. That is, the discrete spectrum is small. This motivates understanding analogous operators with a large discrete spectrum. To do so, we build on the well-known identification of the universal enveloping algebra of a Lie algebra with the algebra of left invariant differential operators on the group, recall an intrinsic characterization of the Laplacian/Casimir on spaces of automorphic forms, and discuss how we can attempt to understand the spectrum of related automorphic differential operators.

Composition of functions usually depends on the order in which it is done: $f(g(x))$ usually is not \(g(f(x))\). Sometimes, however,
\(f(g(x)) = g(f(x))\) for all \(x\). We then say \(f(x)\) and \(g(x)\) commute.

In this talk, we will see how polynomials can satisfy \(f(g(x)) = g(f(x))\). It turns out that there are only two interesting ways this can happen, and one way
has an unexpected relationship to trigonometry.

Note: Free refreshments. The talk starts at 5:40.

The 20th century saw both the birth and rapid, widespread adoption of one of the most ubiquitous organizational tools available in modern mathematics: category theory. Category theory provides a unifying language in which one may formally express relationships and analogies within and between very different classes of objects, whether algebraic, geometric, analytic, or otherwise; it is structured enough to be meaningful, and it is flexible enough to be expressive. However, while already a major success story, the categorical phenomenon has continued to expand into higher-dimensional variants– so-called higher categories. These 21st-century generalizations come in many different forms and often serve vastly different purposes. In this talk, we will take a peek at three kinds of higher categories: double categories, \((\infty, 1)\)-categories, and \(A_\)-categories. As demonstrations of the utilities of these gadgets, we will briefly discuss applications in dynamical systems theory, derived algebraic geometry, and homological mirror symmetry.

Abstract: Nematic liquid crystals are classified as partially-ordered materials due to the tendency of their molecules to align along preferred directions. This partial alignment, combined with fluid-like properties, render these materials useful in a range of applications, including their widespread use in optical displays. When confined to thin channels, the fluid flow extends the applications of nematics further to optofluidic devices and guided micro-cargo transport. We examine the intrinsic coupling between the fluid motion and the orientation of nematic molecules by considering steady unidirectional flows in thin channels. We employ a reduced Beris-Edwards framework which leads to a system of nonlinear and coupled differential equations. Our analytical and numerical results highlight the universality of order reconstruction solutions, characterized by distinct orientational sub-domains that are separated by domain walls.