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 Db(X) be used as an invariant to answer rationality questions? In particular, what properties of Db(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of Db(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 Matthew Ballard, Alexander Duncan, and Patrick McFaddin.
Derived Categories, Arithmetic, and Rationality Questions Sponsored by the Meyer Fund
Oct. 30, 2019 4pm (MATH 350)
Grad Student Seminar
Howie Jordan (CU Boulder)
X
Would you like to play a game? It's the day before Halloween, so let's get a little spooky. Open this door with the key of imagination, and step into a world beyond your usual understanding. A world where the axiom of choice fails, but not horrendously so. A world where every set of real numbers is Lebesgue measurable. A world where we can't turn a single sphere into two with the same volume or well-order every set. But it will all start with a simple game
The goal of this talk is to provide an introduction to an alternative axiom of set theory, one which contradicts the axiom of choice but shares some of that axiom’s implications, as well as its own unique, interesting, and even paradoxical consequences. We will compare this axiom to choice, see how they contradict, and if time permits we may even discuss implications of our considerations for foundations of mathematics.
You Have No Choice; It is Already Determined
Oct. 30, 2019 5pm (Math 350)
Math Club
Sean O'Rourke (CU Boulder)
X
How many times do you have to shuffle a deck of cards to mix it up? The answer depends on what we mean by shuffling. I will discuss this question and explain how shuffling a deck of cards can be modeled mathematically.
How many times should you shuffle a deck of cards?
Derived Categories, Arithmetic, and Rationality Questions