After Mordell’s conjecture for curves was proved by Faltings, attentions turn to the distribution of rational and integral points on higher dimensional varieties, which is encoded in the celebrated Vojta’s conjecture. Along this line we proved a subspace type inequality, improving the result of Ru-Vojta, on surfaces. Meanwhile, we obtain a sharp criterion of when some certain surfaces admit a non Zariski-dense set of integral points. Joint with Huang, Levin.
Diophantine Approximation on Surfaces and Distribution of Integral Points
Let A be a finite simple non-abelian Mal'cev algebra (e.g. a group, loop, ring), and let B be the countable atomless Boolean algebra. We show that the automorphism groups of filtered Boolean powers of A by B have ample generics, which gives a new proof of our previous results that these groups have the small index property, uncountable cofinality and the Bergman property. This is joint work with Nik Ruskuc.
Generic automorphisms of Boolean powers
Tue, Sep. 23 2:30pm (MATH 2…
Grad Algebra/Logic
Charlotte Aten (CU Boulder)
X
We will continue covering background material pertaining to the recent paper "A categorical perspective on constraint satisfaction: The wonderland of adjunctions" by Hadek, Jakl, and Oprsal. Topics will include the discrete Grothendieck construction, Kan extensions, and the nerve of a functor.
Categorical methods for constraint satisfaction (Part 2)
Tue, Sep. 23 2:30pm (MATH 3…
Topology
Howy Jordan
X
A topological space is stratified when it is equipped with a nice decomposition into nice subsets. In the classical case, the nice subsets are manifolds. When the decomposition is nice enough, one can equip the stratified space with a stratified tangent bundle.
"Is there a category of stratified spaces whose fiber bundles include these stratified tangent bundles?" Motivated by this question, we will investigate categories of stratified spaces. In particular, we will consider cases where the underlying stratification can vary and cases where there is a fixed base stratification and find interesting topologies and group objects along the way.
Diophantine Approximation on Surfaces and Distribution of Integral Points