NTSGrad Spring 2022/Abstracts
This page contains the titles and abstracts for talks scheduled in the Spring 2022 semester. To go back to the main GNTS page, click here.
Jan 25
Jerry Yu Fu 
Canonical lifting and isogeny classes of Abelian varieties over finite field 
I will give a brief introduction from SerreTate's canonical lifting, the GrothendieckMessing theory and their applications to class group and estimation of size of isogeny classes of certain type of abelian varieties over finite fields.
I will present some recently proved results by me and some with my collaborator. 
Feb 1
TBA 
Feb 8
Di Chen 
A nontrivial bound on 5torsion in class groups. 
I will discuss A. Shankar and J. Tsimerman’s recent work on a nontrivial bound on 5torsion in class groups of imaginary quadratic fields. I focus on ideas of proofs and assume several black boxes without proofs. This is a good application of elliptic curves and Galois cohomology. 
Feb 15
John Yin 
Bertini Theorems over Finite Fields/Poonen Sieve 
Consider the question: What's the probability that a projective plane curve of degree d over F_q is smooth as d approaches infinity? Assuming some sort of independence, this should be something like the product over closed points in P^2 of the proportion of plane curves which are smooth at the closed point. A version of this turns out to be true, and it is proven through the Poonen Sieve. 
Feb 25
TBA 
Mar 1
TBA 
Mar 8
TBA 
Mar 15
TBA 
Mar 22
TBA 
Mar 29
Tejasi Bhatnagar 
The theorem of Honda and Tate 
In this talk, we aim to understand the classification of abelian varieties over finite fields, up to isogeny. To every abelian variety, we can associate a certain algebraic number. This is called a Weilq number. The Theorem of Honda and Tate tells us that, up to isogeny this association is a bijection. We won’t necessarily prove the entire theorem, but we will see bits and pieces to understand whatever we can and mostly try and get an understanding of the important objects that we’ll come across. 
Apr 5
Hyun Jong Kim 
CohenLenstra for imaginary quadratic function fields 
A CohenLenstra type statement is one which claims or states that certain objects are distributed inversely proportional to the size of their automorphism groups. Originally stated for class groups of quadratic number fields, Ellenberg, Venkatesh, and Westerland showed that an analogue for imaginary quadratic function fields over finite fields hold. I will introduce the CohenLenstra heuristics for number fields and outline the proof to Ellenberg, Venkatesh, and Westerland's theorem. 
Apr 12
Qiao He 
Does rigid analytic varieties has Hodge symmetry? 
I will survey a paper by our last week’s speaker Alexander Petrov. It is well known that there is a symmetry between the Hodge numbers of a Kahler Manifold (in particular, projective variety). In the padic world, we have similar analytic space: rigid analytic variety. Then it is a natural question to ask whether the Hodge number of rigid analytic variety still has a symmetry. It turns out that the answer is no! I will try to explain how to construct a counterexample in the talk.

Apr 19
TBA 

Apr 26
TBA 

May 3
Jerry Yu Fu 
Canonical lifting and size of isogeny classes 
I will give a brief review from SerreTate's canonical lifting theorem, the GrothendieckMessing theory and their applications to class group and isogeny classes of certain type of abelian varieties over finite fields.
I will present some recently proved results by me and some with my collaborator.
