Talks & Presentations. I’ve compiled a list of some recent talks I’ve given– in particular, ones with interactive slides or videos associated with them. Apologies if you find any mistakes/typos/silly things in the notes or slides; feel free to contact me about them.

For an exhaustive list of talks I’ve given / conferences I’ve attended, see Conferences/Workshops/Seminars Attended.

Derived Categories and Rational Points for a class of toric Fano varieties
from November 15, 2021 @ Derived, Birational, and Categorical Algebraic Geometry, BIRS.

##### : A video of this talk is available from 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.

##### : A video of this talk is available on Youtube!

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

##### : A video of this talk is available on Vimeo!

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?