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.

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.

: 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.

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.

: 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.

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

: 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.

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 \(R=D_4\).

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.

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?