NTSGrad Spring 2021/Abstracts
This page contains the titles and abstracts for talks scheduled in the Spring 2021 semester. To go back to the main GNTS page, click here.
Contents
Jan 26
Eiki Norizuki 
$p$adic groups and their representations 
This will be a prep talk for Thursday's NTS talk.
We will talk about subgroups and decompositions of $p$adic groups as well as the BruhatTits tree of $\text{SL}_2$. We try to understand the right class of representations for $p$adic groups which turn out to be smooth admissible representations.

Feb 2
Qiao He 
Supersingular locus of Unitary Shimura variety 
I will give a summary of supersingular locus of Unitary Shimura variety. This description is really the first and an important step to understand the structure of Unitary Shimura variety. Turns out that the description of such locus will boil down to certain linear algebra. The final result will be the supersingular locus have a stratification, and the incidence relation will be closely related with the BruhatTits building of unitary group. Also, each strata is closely related with affine Deligne Lustig variety. The Dieudonne module theory will be summarized. Take it for granted, all the remaining material can follow easily! 
Feb 9
Ivan Aidun 
Simple Sieving 
The idea of sieving out primes is among the oldest in mathematics. However, it has proven incredibly fruitful, and now sieve techniques lie behind some of the most striking results in modern number theory, such as the results of Zhang, Maynard, and the Polymath project on bounded gaps between primes. In this talk, I will develop some of the basic sieve constructions, from Eratosthenes and Legendre to Brun, and hint at some of the developments that lie beyond. This talk will be accessible to a general mathematical audience. 
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Feb 16
Asvin G 
F_un with F_1 
You have probably heard of a field with one element in various places and might have been, very understandably, confused. How can there be a field with one element and even if there is, how could it possible be interesting? I will try and explain the philosophy behind why this is a reasonable thing to wish for and various mathematical facts that *should* be interpreted through this lens.
The talk will just be a bunch of examples of the various manifestations of the field with one element throughout mathematics! 
Feb 23
Yu Fu 
CM liftings of Abelian Varieties 
This will be a introductory talk to introduce the CM liftings of Abelian Varieties.
HondaTate theory tells us every abelian variety over a finite field can be lifted to an abelian variety with smCM in characteristic 0. There are various lifting problems if you drop/change some of the conditions, i.e. Is it an isogeny or residue class field extension necessary? Can we lift any abelian variety over a finite field to a normal domain up to isogeny? Etc.etc. Let's explore with some fun examples! 
Mar 2
Will Hardt 
Linear Relations Among Galois Conjugates 
In 1986, Smyth asked, and conjectured an answer to, the question of what can be the coefficients of a linear relation among Galois conjugates over Q. That is, for which (a_1,...,a_n) in Z^n do there exist Galois conjugates \gamma_1, ..., \gamma_n such that \sum_{i=1}^n a_i \gamma_i = 0? I will talk about joint work with John Yin in which we answer the analogous question over the function field F_q(t). We also formulate what we think is the right generalization of Smyth's Conjecture over a general number field. 
Mar 9
Di Chen 
Negative Pell Equations 
I will review negative Pell equations and introduce Stevenhagen’s conjecture briefly. Then I discuss its relation with 2^krank of class groups and introduce basic tools like genus theory, Artin pairing, Redei matrices and Redei reciprocity. 
Mar 16
Hyun Jong Kim 
Comparison of A1degrees 
I recently talked about the GrothendieckWitt ring and some A1enriched enumerations, such as degrees, during my specialty exam. I will go into some more detail on when and how some of A1enumerations, such as Morel's A1 Brouwer degree, the local A^1 Brouwer degree, the enriched Euler number and the A1degree of maps of more general maps of schemes, are defined. 
Mar 23
Asvin Gothandaraman 
Computational number theory 
I will talk about computational number theory. It will all be pretty elementary and I will cover topics like how to factor integers quickly using number fields or elliptic curves and some related topics. 
Mar 30
Ruofan Jiang 
Galois theory over k(x) 
When k=C, this is the very classical theory of Riemann surface; for other k, especially when k is char p, the Galois theory of k(x) becomes much wilder. One way to study it is via rigid geometry, which enable us to talk about “analytical patching” in a much general context....

Apr 6
Jiaqi Hou 
Hecke algebras for padic groups 
Given a smooth representation of some padic group, we can associate it with modules over Hecke algebras. We will introduce the Satake transform which identifies the spherical Hecke algebra of a reductive group w.r.t a special maximal compact subgroup with a commutative ring of Weyl group invariants. The Satake isomorphism can help us understand spherical representations.

Apr 13
Yunus Tuncbilek 
Three of the Biggest Questions That Science Can’t Answer 
I will present three problems that I really enjoyed working on. The problems will be combinatorial in nature and will look very simple, but their actual difficulties will range from challenging to extremely difficult, slash possibly impossible. The talk assumes no background — literally. I presented one of the problems to my Calc 2 class today.

Apr 20
Yu Fu 
Chabauty method at bad reduction 
This is an introduction talk to the ChabautyColeman method at a prime of bad reduction. I will talk about results, progress and also include a proof by using intersection theory. Let's also take a look at some examples to estimate if the bound is sharp or we can obtain a better one.

Apr 27
Sang Yup Han 
Arithmetic Invariant Theory 
This talk will give a very general overview of arithmetic invariant theory of Bhargava, Gross, and Wang, including background motivation and examples.

May 4
Dionel Jaime 
Tiling in Vector Spaces Over Finite Fields 
In the setting of vector spaces over finite fields, we will look at results connecting the question of whether a subset tiles to whether this set possesses an orthogonal basis of characters. In particular, properties of cyclotomic polynomials and properties if the discrete Fourier transform will be highly relevant.
