Difference between revisions of "NTSGrad Spring 2020/Abstracts"

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I'll present some of the formalization of Shimura varieties, with a strong emphasis on examples so that we can all get a small foothold whenever someone says the term ''Shimura variety''.  
 
I'll present some of the formalization of Shimura varieties, with a strong emphasis on examples so that we can all get a small foothold whenever someone says the term ''Shimura variety''.  
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== Feb 11 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Hardt and John Yin'''
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| bgcolor="#BCD2EE"  align="center" | ''Primality Tests Arising From Counting Points on Elliptic Curves Over Finite Fields''
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We will give some background on counting the rational points on an elliptic curve over a finite field. Then we will apply this theory to a couple of specific elliptic curves and explain how it results in (impractical) primality tests.
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== Feb 25 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ivan Aidun'''
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| bgcolor="#BCD2EE"  align="center" | ''Golomb Topologies and Infinitely Many Irreducibles''
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In 1955, Furstenberg gave a proof of the infinitude of primes by imposing a topology on Z. Under this topology, all open sets are infinite, but if you assume only finitely many primes then {1} is open. A new, similar, topology was introduced by Golomb in 1959, which turned Z^+ into a connected Hausdorff space. A general Golomb topology on a domain R was introduced by Knopfmacher and Porubský in 1997. I will draw on a paper of Clark, Lebowitz-Lockard, and Pollack, and discuss interesting properties of these topologies, and how they relate to properties of the domain R.
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== Mar 3 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''
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|-
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| bgcolor="#BCD2EE"  align="center" | ''Perspectives on Rational Points''
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|-
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This talk is going to be an all-you-can-eat smorgasbord of techniques for finding rational points on curves.
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== Mar 24 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Boggess'''
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|-
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| bgcolor="#BCD2EE"  align="center" | ''Squares in Arithmetic Progressions''
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We will see how results about rational points on curves can say something about integers in arithmetic progressions.
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[[Media:mar_24_slides.pdf | Slides]]
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== Mar 31 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hyun Jong Kim'''
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| bgcolor="#BCD2EE"  align="center" | ''A1 Homotopy Degree''
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Morel and Voevodsky's A1 Homotopy Theory (c. 1998-1999) develops a homotopy theory for algebraic geometry that is analogous to the more familiar homotopy theory from algebraic topology - here, the unit interval [0,1] is replaced with the affine line A1. One concept that emerges from Morel and Voevodsky's theory is A1 Homotopy Degree, which can be defined for maps from the A1 homotopy sphere to itself. I will focus on how to compute the A1 Homotopy Degrees of such maps in a nice case, which is tractable due to work by Kass and Wickelgren (2016).
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== Apr 7 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Peter Wei'''
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|-
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| bgcolor="#BCD2EE"  align="center" | '' An introduction to l-adic Galois representations''
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The goal of this talk is to introduce l-aidc Galois representations in various aspects, mostly related to the ones appearing in geometry through l-aidc étale cohomology of varieties. I will focus on Galois representations in two cases,  finite fields and local fields.
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Latest revision as of 07:59, 13 April 2020

This page contains the titles and abstracts for talks scheduled in the Spring 2020 semester. To go back to the main GNTS page, click here.

Jan 21

Qiao He
Representation theory and arithmetic geometry

In this talk I will talk about the relation between representation theory and arithmetic geometry. In particular, I will try to discuss several examples that connect representation theory and arithmetic geometry closely. Then if time permits, I will give a brief introduction to trace formula approach, which is the most powerful and promising tools in this field.


Jan 28

Asvin Gothandaraman
Modular forms and class groups

In preparation for Thursday's talk, I will review some concepts from Galois Cohomology. I will also give an introduction to the Herbrand-Ribet theorem.


Feb 4

Johnnie Han
ABC's of Shimura Varieties

I'll present some of the formalization of Shimura varieties, with a strong emphasis on examples so that we can all get a small foothold whenever someone says the term Shimura variety.


Feb 11

Will Hardt and John Yin
Primality Tests Arising From Counting Points on Elliptic Curves Over Finite Fields

We will give some background on counting the rational points on an elliptic curve over a finite field. Then we will apply this theory to a couple of specific elliptic curves and explain how it results in (impractical) primality tests.


Feb 25

Ivan Aidun
Golomb Topologies and Infinitely Many Irreducibles

In 1955, Furstenberg gave a proof of the infinitude of primes by imposing a topology on Z. Under this topology, all open sets are infinite, but if you assume only finitely many primes then {1} is open. A new, similar, topology was introduced by Golomb in 1959, which turned Z^+ into a connected Hausdorff space. A general Golomb topology on a domain R was introduced by Knopfmacher and Porubský in 1997. I will draw on a paper of Clark, Lebowitz-Lockard, and Pollack, and discuss interesting properties of these topologies, and how they relate to properties of the domain R.


Mar 3

Soumya Sankar
Perspectives on Rational Points

This talk is going to be an all-you-can-eat smorgasbord of techniques for finding rational points on curves.


Mar 24

Brandon Boggess
Squares in Arithmetic Progressions

We will see how results about rational points on curves can say something about integers in arithmetic progressions.

Slides


Mar 31

Hyun Jong Kim
A1 Homotopy Degree

Morel and Voevodsky's A1 Homotopy Theory (c. 1998-1999) develops a homotopy theory for algebraic geometry that is analogous to the more familiar homotopy theory from algebraic topology - here, the unit interval [0,1] is replaced with the affine line A1. One concept that emerges from Morel and Voevodsky's theory is A1 Homotopy Degree, which can be defined for maps from the A1 homotopy sphere to itself. I will focus on how to compute the A1 Homotopy Degrees of such maps in a nice case, which is tractable due to work by Kass and Wickelgren (2016).


Apr 7

Peter Wei
An introduction to l-adic Galois representations

The goal of this talk is to introduce l-aidc Galois representations in various aspects, mostly related to the ones appearing in geometry through l-aidc étale cohomology of varieties. I will focus on Galois representations in two cases, finite fields and local fields.