NTSGrad Spring 2020/Abstracts: Difference between revisions
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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. | 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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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar''' | |||
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| bgcolor="#BCD2EE" align="center" | ''Perspectives on Rational Points'' | |||
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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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Revision as of 17:33, 2 March 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 |
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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 |
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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 |
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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. |
Feb 11
| Will Hardt and John Yin |
| 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. |
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 |
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This talk is going to be an all-you-can-eat smorgasbord of techniques for finding rational points on curves. |