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.

Adaptive methods are used in almost all disciplines of applied and computational mathematics, where the problem solving procedure adapts to the feedback from the quantities of interest. Inspired by this methodology, we will discuss the practice of the adaptive education, and the adaptive design of classes for the data science specialization. With the help of the smart classroom technologies at UC Irvine, we can better execute the instruction of programming in mathematics and data science to make our students learn to be more competent for both industry and graduate school.  Lastly, we share some ideas of future plans to further strengthen our strong undergraduate program and serve our community.