Events

Research Area: Algebraic Geometry/Number Theory

Cynthia Dai

"Introduction to Naive Height"

In this seminar, we will be focusing on three results: Mordell-Weil theorem, Falting theorem, and potentially Vojta conjecture. If time permits, we will also try to cover Manin's conjecture for toric varieties. We will start slow and spend a one or two talks on naive heights on projective space, then define Weil heights for projective varieties, and study their properties. After this, we will focus on abelian varieties, and once we are familar with those objects, we introduce Neron-Tate heights, and finally prove the first main result we want to cover (actually, I think this will be all we can do this term).

For today's talk, I will review some algebraic number theory, then define Naive heights. 

MC 5417

Jérémy Champagne

"Weyl's equidistribution theorem in function fields"

Finding a proper function field analogue to Weyl's theorem on the equidistribution of polynomial sequences is a problem that was originally considered by Carlitz in 1952. As noted by Carlitz, Weyl's classical differencing methods can only handle polynomials with degree less than the characteristic of the field. In this talk, we discuss some recent methods which avoid this "characteristic barrier", and we show the existence of polynomials with extremal equidstributive behaviour. 

This is joint work with Yu-Ru Liu, Thái Hoàng Lê and Trevor D. Wooley.

MC 5403

AJ Fong

"Galois representations of the Picard groups of surfaces"

Algebraic geometry provides a natural framework to study solutions of Diophantine equations. I will sketch why the Picard group of a surface is interesting from the perspective of finding rational points.

MC 5403

Speaker: William Gollinger

"The Adams Spectral Sequence"

In this second lecture of the series we illustrate the spectral sequence formalism by computing some examples of the Leray-Serre spectral sequence. This tool was introduced in Serre's thesis to compute the cohomology of fibre bundles, and is much simpler to conceptualize and execute than the Adams spectral sequence. We will emphasise multiplicativity and naturality as useful tools for performing these calculations. 

MC 5417

Lucia Martin Merchan

"A Grassmannian bundle over a Spin(7) manifold"

Abstract: In this talk we study the geometry of the fiber bundle G(2,M) of oriented 2-planes on a Riemannian manifold (M,g) with a Spin(7) structure. More precisely, we construct an almost complex structure and we discuss how to compute its torsion when the holonomy of g is contained in Spin(7).

MC 5417