The study of symmetries in the setting of group actions via “invariant” polynomials is called invariant theory. Throughout the 20th century, invariant theory has had a profound influence in several fields in mathematics, most notably those that fall under the broad purview of algebra. Over the last two decades, new directions in invariant theory have emerged out of connections to computational complexity. Particularly exciting is the Geometric Complexity Theory (GCT) program that has uncovered connections between invariant theory and foundational problems in complexity such as identity testing and the celebrated P vs NP problem.

In this talk, I will discuss recent advances in invariant theory concerning matrix invariants and semi-invariants, (non-commutative) identity testing, null cones and orbit closures. Towards the end, I will discuss some very promising directions for the future in this rapidly expanding field. Based on several joint works with Harm Derksen, Ankit Garg, Rafael Oliveira, and Avi Wigderson.

Local cohomology modules are fundamental tools in commutative algebra, due to the algebraic and geometric information they carry. For instance, they can help determine the number of equations necessary to define an affine variety. Unfortunately, however, the application of local cohomology is limited by the fact that these modules are typically very large (e.g., not finitely generated), and can be difficult to determine explicitly. In this talk, we discuss new techniques developed to understand the structure of local cohomology (e.g., coming from invariant theory).  We also describe recently- discovered "connectedness properties" of spectra that local cohomology encodes.

Algebraic geometry is concerned with algebraic varieties, which can be understood as solution sets of polynomial equations. At the heart of research is the classification of algebraic varieties, and a geometric solution is provided in the form of a moduli space. Roughly speaking, a moduli space is itself an algebro-geometric object whose points represent equivalence classes of algebraic varieties of a fixed type. This talk begins with the moduli space of curves, which parametrizes equivalence classes of complex algebraic curves (i.e. Riemann surfaces) of a fixed genus. This moduli space, like most moduli spaces appearing in algebraic geometry, is not a compact space. A celebrated result of Deligne and Mumford provides a geometric way to compactify this space. The goal of this talk is to discuss recent progress towards compactifying moduli spaces of higher dimensional complex algebraic varieties (e.g. complex algebraic surfaces).

Active learning is, in essence, any teaching strategy that encourages students to take an active part in the learning process.  Research suggests that active learning is a beneficial addition to the STEM curriculum—it promotes a deeper level of understanding and may help improve the retention rates of STEM majors.  To demonstrate how active learning components can be utilized in UCI mathematics courses, I present materials I developed and implemented for upper and lower division courses at UCI.  These materials span an assortment of media and are designed to facilitate student engagement in a variety of ways.  I also discuss ideas for future active learning opportunities in UCI mathematics courses, and I summarize potential undergraduate projects, which will allow students to explore advanced mathematics and applications (including work with real world data) in a hands-on exploratory fashion.