Events

Speaker: Thomas Bray

"The Birkhoff-Kakutani Theorem"

I will introduce the basic concepts pertaining to topological groups. After this, I will show one way of proving the Birkhoff-Kakutani theorem. Time permitting, I will demonstrate how one can use Birkhoff-Kakutani to build a complete metric on a Polish group.

MC 5403

Speaker: William Gollinger

Topology Learning Seminar: The Adams Spectral Sequence

The Adams Spectral Sequence was introduced by Frank Adams in his paper "On The Structure and Application of the Streenrod Algebra" (1958) with applications to the stable homotopy groups of spheres and the Hopf-Invariant One problem. In the context of the stable homotopy category it was soon upgraded to the general problem of computing the coefficients of extraordinary cohomology theories.  In this series of lectures we will outline a construction of the Adams Spectral Sequence following Ravenel's "Green Book", and give applications including computations of some stable homotopy groups of spheres as well as certain Madsen-Tillmann bordism groups which have recently been of interest in the theory of TQFTs. 

The seminar assumes some basic knowledge of algebraic topology (in particular homotopy theory and ordinary homology theory) but is aimed to be expository, introducing the audience to important topological concepts such as stable homotopy theory and cohomology operations. The topics presented will be roughly in the following order: examples of the Leray-Serre spectral sequence; the Stable Homotopy Category; construction of the Adams Spectral Sequence; the Steenrod Algebra and its dual; computations.

MC 5417

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

Algebraic Geometry/Number Theory

Speaker: Cynthia Dai

"Local Heights"

In this talk, we will wrap up heights on projective space by prove Northcott's theorem for algebraic numbers. Next, we will local heights with respect to a Cartier divisor.

MC 5417

Speaker: Owen Sharpe

"An Application of Equidistribution Modulo One to Computer Graphics."

We discuss algorithms for generating pseudorandom, uniformly distributed points on the unit circle and the unit sphere. We then apply these algorithms to texture and terrain generation, using ideas from the Perlin noise scheme.

MC 5403

Speaker: Alex Pawelko

"Symmetry Reduction and the Quest for G2 Moment Maps"

We present an overview of the classical theory of moment maps from symplectic geometry and their use within the Marsden-Weinstein-Mayer theory of symplectic reduction, with an emphasis on the Lie theoretic considerations that arise. If time permits, we will then discuss some attempts to generalize moment maps to the setting of G2 manifolds.

MC 5417