**Talks & Presentations.** Here are some recent talks I've given-- in particular, ones with interactive slides or videos associated with them.
For an exhaustive list of talks I've given and conferences I have attended, see Conferences/Workshops/Seminars Attended.

**Root systems, moduli interpretations, and their derived categories**

from March 19, 2023 @ Canadian Western Algebraic Geometry Symposium

**Abstract. **Based on ongoing work with Aaron Bertram, we will explore properties of toric varieties constructed from root systems, their moduli space interpretations due to Losev-Manin and Batyrev-Blume, and decompositions of their bounded derived categories of coherent sheaves.

**Global generation of test ideals in mixed characteristic and applications**

from October 22, 2022 @ *AMS Special Session*

**Abstract. **(*Part of AMS Special Session on Recent Advances in Algebraic Geometry and Commutative Algebra in or Near Characteristic p*) I will discuss recent joint work with Christopher Hacon and Karl Schwede in which we define a notion of test ideals for rings of finite type over a complete local Noetherian ring that commutes with localization. I will also discuss applications of our definition.

**Derived Categories and Rational Points for a class of toric Fano varieties **

from November 15, 2021 @ Derived, Birational, and Categorical Algebraic Geometry, BIRS.

**Abstract.** I will discuss forthcoming work with Matthew Ballard on using the derived category of coherent sheaves to detect the existence of rational points on a particular family of arithmetic toric Fano varieties. More precisely, I will explain how we show that a member of this family of varieties is rational if and only if its bounded derived category of coherent sheaves admits a full étale exceptional collection.

**Derived Cats, Arithmetic, and Rationality**

from April 23, 2021 @ UCSC AG & NT Seminar

**Abstract.** When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety \(X\), to what extent can \(D^b(X)\) be used as an invariant to answer rationality questions? In particular, what properties of \(D^b(X)\) are implied by \(X\) being rational, stably rational, or having a rational point? On the other hand, is there a property of \(D^b(X)\) that implies that \(X\) is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Mattew Ballard, Alexander Duncan, and Patrick McFaddin.

**Derived Categories, Arithmetic, and Rationality**

from March 16, 2021 @ the Derived Seminar.

**Abstract.** When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety \(X\), to what extent can \(D^b(X)\) be used as an invariant to answer rationality questions? In particular, what properties of \(D^b(X)\) are implied by \(X\) being rational, stably rational, or having a rational point? On the other hand, is there a property of \(D^b(X)\) that implies that X is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Mattew Ballard, Alexander Duncan, and Patrick McFaddin.

**What is a Toric Variety?**

from January 7, 2021 @ What is… A Seminar?

**Abstract.** I love toric varieties. In this talk I’ll tell you how I think about them, and why I love them.

**Derived Categories, Arithmetic, and Rationality**

from December 8, 2020 @ CMS Winter Meeting

**Abstract.** When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety \(X\), to what extent can \(D^b(X)\) be used as an invariant to answer rationality questions? In particular, what properties of \(D^b(X)\) are implied by \(X\) being rational, stably rational, or having a rational point? On the other hand, is there a property of \(D^b(X)\) that implies that X is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Mattew Ballard, Alexander Duncan, and Patrick McFaddin.

**Exceptional Collections of toric varieties associated to root systems**

from Fall 2018 @ UofSC AG Seminar

**Abstract.** Given a root system \(R\), one can construct a toric variety \(X(R)\) by taking the maximal cones of \(X(R)\) to be the Weyl chambers of \(R\). The automorphisms of \(R\) act on \(X(R)\), and a natural question arises: can one decompose the derived category of coherent sheaves on \(X(R)\) in a manner that is respected by \(Aut(R)\)? Recently, Castravet and Tevelev constructed \(Aut\)-stable full exceptional collections for \(D^b(X(R))\) when \(R\) is of type \(A_n\). In this talk, we discuss progress towards answering this question in the case where \(R\) is of type \(D_n\), with emphasis on the interesting case of \(R=D_4\).

**Examples of Spherical Varieties**

from Summer 2018 @ UofSC

**Abstract.** We discuss various examples of spherical, horospherical, and wonderful varieties, as well as related definitions and theorems that have not yet been introduced in the seminar.

**Pure motives as a universal cohomology theory**

from Spring 2018 @ MaSC

**Abstract.** In this talk, we attempt to answer the following questions: Why did we build the category of pure motives over \(k\)? What is a Weil cohomology theory? Does the category of pure motives over \(k\) give us what we want?