Imagine the hands on a clock. For every complete the minute hand makes, the seconds hand makes 60, while the hour hand only goes one twelfth of the way. We may think of the hour hand as generating a group such that when we ``move'' twelve times then we get back to where we […]
For a finite group G, a G-module M, and a field F, an element u in H^d(G,M) is negligible over F if for each field extension L/F and every continuous group homomorphism from Gal(L^{sep}/L) to G, u is in the kernel of the induced homomorphism H^d(G,M) to H^d(L,M). Negligible cohomology was first introduced by Serre […]
What are the possible angles between two integer vectors in R^n? If we fix one such possible angle and one integer vector x, is there always another integer vector y […]
In order to understand a topological space X, it is often easier to understand X in terms of an action by a group G. When X is a compact complex […]
Biquandle module enhancements are invariants of knots and links generalizing the classical Alexander module invariant. A quiver categorification of these invariants was introduced in 2020. In this work-in-progress (joint with […]
A modification of the cellular resolution of the diagonal given by Bayer-Popescu-Sturmfels gives a virtual resolution of the diagonal for smooth projective toric varieties and toric Deligne-Mumford stacks which are […]
Around 20 years ago, physicists Michael Faux and Jim Gates invented Adinkras as a way to better understand Supersymmetry. These are bipartite graphs whose vertices represent bosons and fermions and […]
I will introduce two general problems and explain how they surprisingly connect with each other and with other aspects of mathematics (for a glimpse, Sato—Tate, hypergeometric functions, moduli spaces of […]
In the field of commutative algebra, the principal object of study is (unsurprisingly) commutative algebras. A somewhat unintuitive fact is that results about commutative algebras can be gleaned from an associated non-commutative algebra whose generators are very analytic in nature. This object is called the ring of differential operators, often denoted by D. In a sense gives […]
In this talk, we explore sequences and their autocorrelation functions. Knowing the autocorrelation function of a sequence is equivalent to knowing the magnitude of its Fourier transform. Resolving the lack of phase information is called the phase problem. We say that two sequences are equicorrelational to mean that they have the same aperiodic autocorrelation function. […]
The classical Noether-Lefschetz Theorem states that a suitably general algebraic surface S of degree d ≥ 4 in complex projective 3-space P3 contains no curves besides complete intersections, that is, curves of the form S ∩ T where T is another surface. After discussing briefly Noether’s non-proof of this theorem and hinting at the idea […]
Integer partitions are ubiquitous in mathematics, arising in subjects as disparate as algebraic combinatorics, algebraic geometry, number theory, representation theory, to mathematics physics. Many of the deepest results on partitions have their origin in the work of Ramanujan. In this lecture, we will describe a completely new and unexpected role for partitions that also arises […]
