Difference between revisions of "NTSGrad Spring 2020/Abstracts"
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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. | 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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+ | | 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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Revision as of 13:34, 23 February 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. |