Paraproducts are one of the essential tools of harmonic analysis, used to decompose a product of two functions. Motivated by a similar question in complex analysis, Pott, Reguera, Sawyer and Wick studied the composition of "paraproduct-type operators", to which classical paraproducts belong. Their goal was to find joint conditions on the symbol functions for the composition to be bounded. They classified many paraproduct-type compositions. One of the operators to remain unclassified was the composition of two classical dyadic paraproducts. In this talk, we will discuss the joint conditions for the boundedness of two paraproducts as well as certain weighted inequalities.

In this talk we discuss the existence and the importance of asymptotic colengths for families of \(m\)-primary ideals in a Noetherian local ring \((R,m)\).  We explore various families such as weakly graded families, weakly \(p\)-families and weakly inverse \(p\)-families and discuss a new analytic method to prove the existence of limits. Additionally, if time permits we will talk about Minkowski type inequalities, positivity results, and volume = multiplicity formulas for these families of ideals. This talk is based on a joint work with Cheng Meng.

Parseval frames provide redundant encodings, with equal-norm Parseval frames offering optimal robustness against one erasure. However, constructing such frames can be challenging. The Paulsen Problem asks to determine how far an ε-nearly equal-norm Parseval frame is from the set of all equal-norm Parseval frames. In this talk, I will present an approach to the Paulsen’s problem based on quiver invariant theory

Let \(R\) be a commutative Noetherian domain with identity and finite Krull dimension \(d\). If \(R\) is non-singular, then for every ideal \(I \subseteq R\) and \(n \in \mathbb{N}\), the symbolic power \(I^{(dn)}\) is contained in the ordinary power \(I^n\). This property is known as the Uniform Symbolic Topology Property, reflecting a uniform comparison between symbolic and ordinary powers of ideals in \(R\).

Historically, this property was first established for rings over complex numbers by Ein, Lazarsfeld, and Smith, then extended to non-singular rings containing a field by Hochster and Huneke. Later, Ma and Schwede proved it for reduced ideals in excellent non-singular rings of mixed characteristic, and Murayama extended it to all regular rings, even non-excellent ones. Their proofs have profound connections with the subjects of multiplier/test ideal theory, closure operations, and constructions of big Cohen-Macaulay algebras.

The study of symbolic powers becomes significantly more challenging in the presence of singularities. In this talk, we focus on prime characteristic strongly \(F\)-regular singularities that arose from Hochster and Huneke's tight closure theory. We show that an \(F\)-finite strongly \(F\)-regular domain satisfies the Uniform Symbolic Topology Property, meaning there exists a constant \(C\) such that \(I^{(Cn)} \subseteq I^n\) for all ideals \(I \subseteq R\) and \(n \in \mathbb{N}\). We will explore the role of splitting ideals and Cartier linear maps in establishing this result.

This talk considers provable methods for cryo-electron microscopy, which is an increasingly popular imaging technique for reconstructing 3-D biological macromolecules from a collection of noisy and randomly oriented projection images, with applications in e.g., drug design. The talk will present two uniqueness guarantees for recovering these structures from the second moment of the projection images, as well as two associated numerical algorithms. Mathematically, the results boil down to ensuring unique solutions to highly structured non-linear equations.

The Briançon-Skoda Theorem states that \(\overline{J^{n + \lambda-1}} \subseteq J^{\lambda}\) for all integers \(\lambda\geq 1\) and \(J=(f_1,\ldots, f_n)\subseteq \mathbb{C}[x_1,\ldots, x_m],\) thus giving a relationship between the usual powers of an ideal and the integral closure of its powers. Unfortunately, this result does not hold for arbitrary rings. However, there have been numerous generalizations up to a closure operation on \(J^{\lambda}\). For example, Hochster and Huneke proved generalizations using tight closure and +-closure. In this talk, I’ll talk about the history of the Briançon-Skoda theorem and some of its generalizations, as well as present a generalization using sufficiently functorial choices of BCM-algebras. This is joint work with Karl Schwede.

Systems that can be classified as oscillatory media consist of small oscillating elements that interact with each other via some form of coupling. Interest in these systems stems in part from their ability to generate beautiful structures, including target patterns and spiral waves.  When coupling between oscillators occurs over long spatial scales, and for certain parameter values, the set of unstable waven umbers that generate these patterns is no longer constrained to a narrow band. This then leads to interesting new structures called spiral chimeras, which are solutions that look like spiral waves in the far field but have an incoherent core where the 'oscillators' are no longer in synchrony with the rest of the pattern. As a first step in understanding this phenomenon, we rigorously study the existence of 'standard' spiral waves in an oscillating chemical reaction where the source of the nonlocal coupling is due to a fast-diffusing component. Our approach is based on the method of multiple-scales and Lyapunov-Schmidt reduction, which allow us to rigorously derive a nonlocal amplitude equation for these patterns. 

Examples include oscillating chemical reactions like the Belusov-Zhabotinsky reaction, colonies of yeast cells, and under certain assumptions heart and brain tissue.

I will motivate the subject of (Newton-) Okounkov bodies from the perspective of intersection theory by presenting  the statement of Theorem 4.9 in Kaveh and Khovanskii's 2012 Annals paper. Time-permitting, I will discuss examples: a notable special case of this result is Kushnirenko's theorem on the degree of a projectively-embedded toric variety.