NTSGrad Fall 2015/Abstracts

Sep 02

Lalit Jain

Monodromy computations in topology and number theory

The monodromy of a family of varieties is a measure of how homology classes vary. Surprisingly, many familiar ideas in number theory, such as Galois representations and Cohen-Lenstra heuristics, are closely linked to monodromy of specific families. In this talk I will define monodromy, explain some number theoretic applications, and describe original work of computing monodromy for moduli spaces of covers of the projective line (Hurwitz spaces). This work generalizes previous results of Achter-Pries, Yu and Hall on hyperelliptic families. Only basic knowledge of algebraic topology and number theory is required.

Sep 09

Megan Maguire

Infintely many supersingular primes for every elliptic curve over the rationals.

In his 1987 Inventiones paper, Dr. Noam Elkies proved that every elliptic curve over [math]\displaystyle{ \mathbb{Q} }[/math] has infinitely many supersingular primes. We shall discuss some of the mathematics needed to prove this result and give a proof.

Sep 16

Silas Johnson

Alternate Discriminants and Mass Formulas for Number Fields

Kedlaya and Wood have explored alternate invariants for number fields, with the idea of replacing the discriminant in standard field-counting questions with one of these alternate invariants. We further explore the space of “reasonable” invariants, expanding on Kedlaya and Wood’s definition. We also discuss a theorem on mass formulas for these invariants.

Sep 23

Daniel Hast

Moments of prime polynomials in short intervals.

How many prime numbers occur in a typical "short interval" of fixed width, and how are primes distributed among such intervals? We examine the analogue of this problem for polynomials over a finite field. Our approach is geometric: we interpret each moment of the distribution in terms of counting certain points on an algebraic variety, and we use an algebraic analogue of the Lefschetz fixed-point theorem to compute the leading terms of this "twisted" point-count. As a motivating example, we realize the "prime polynomial theorem" as a geometric statement.

Sep 30

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Oct 07

Will Cocke

The Trouble with Sharblies

The Sharbly complex provides a generalization of modular symbols and can be used to compute the Hecke eigenvalues on arithmetic cohomology. Such eigenvalues provide useful information pertaining to generalizations of Serre's conjecture. I will introduce the Sharbly complex and examine the necessary reduction techniques needed to compute the Hecke action. A friendly introduction to a new and emerging tool in computational number theory.

Oct 14

Will Cocke

The Trouble with Sharblies

The Sharbly complex provides a generalization of modular symbols and can be used to compute the Hecke eigenvalues on arithmetic cohomology. Such eigenvalues provide useful information pertaining to generalizations of Serre's conjecture. I will introduce the Sharbly complex and examine the necessary reduction techniques needed to compute the Hecke action. A friendly introduction to a new and emerging tool in computational number theory.

Oct 21

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Oct 28

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Nov 04

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Nov 11

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Nov 18

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Nov 25

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Dec 02

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Dec 09

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Organizer contact information

Sean Rostami (srostami@math.wisc.edu)

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