Events

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

Speaker: Nicolas Chavarria

"T-minimal Theories and Dimension II"

We continue looking at Will Johnson's work on the dimension of definable sets in t-minimal theories.

MC 5403

Speaker: Paul Marriott

"Statistics and Geometry: We don't talk any more."

George Bernard Shaw once said Britain and America are two counties separated by a common language. Perhaps the same can be said for Statistics and Geometry. This talk gives a high-level overview of a recent graduate course which explored the relationship between Statistics and Geometry. It looks at what the disciplines have in common but also where there are points of substantive difference. The talk will review the long history of geometric tools finding a place in statistical practice and will highlight modern developments using ideas from convex, differential and algebraic geometry and showing applications in Neuroscience.

MC 5417

Speaker: Cynthia Dai

"Global Heights"

In this talk, we will wrap up local heights with respect to a presentation of a line bundle, then define global heights. If time permits, we will also talk about Weil heights.

MC 5417

Speaker: Manuel Fernandez, Georgia Tech

"On the $\ell_0$-Isoperimetry of Measurable Sets"

Gibbs-sampling, also known as coordinate hit-and-run (CHAR), is a random walk used to sample points uniformly from convex bodies. Its transition rule is simple: Given the current point p, pick a random coordinate i and resample the i'th coordinate of p according to the distribution induced by fixing all other coordinates. Despite its use in practice, strong theoretical guarantees regarding the mixing time of CHAR for sampling from convex bodies were only recently shown in works of Laddha and Vempala, Narayanan and Srivastava, and Narayanam, Rajaraman and Srivastava. In the work of Laddha and Vempala, as part of their proof strategy, the authors introduced the notion of the $\ell_0$ isoperimetric coefficient of a measurable set and provided a lower bound for the quantity in the case of axis-aligned cubes. In this talk we will present some new results regarding the $\ell_0$ isoperimetric coefficient of measurable sets. In particular we pin down the exact order of magnitude of the $\ell_0$ isoperimetric coefficient of axis-aligned cubes and present a general upper bound of the $\ell_0$ isoperimetric coefficient for any measurable set. As an application, we will mention how the results give a moderate improvement in the mixing time of CHAR.

MC 5501