NTSGrad Fall 2015/Abstracts
Sep 02
| Lalit Jain |
| Monodromy computations in topology and number theory |
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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 |
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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 |
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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 |
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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 |
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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
| Brandon Alberts |
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Oct 21
| Yueke Hu |
| Mass equidistribution on modular curve of level N |
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It was shown in previous works that the measure associated to holomorphic newforms of weight k and level q will tend weakly to the Haar measure on modular curve of level 1, as qk goes to infinity. In this talk I will show that this phenomenon is also true on modular curves of general level N. |
Oct 28
| David Bruce |
| Intro to Complex Dynamics |
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Given a polynomial f(z) with complex coefficients, we can ask for which complex numbers p is the set {f(p), f(f(p)), f(f(f(p))),...} bounded, that is to ask which complex numbers have bounded forward orbit under f(z)? Alternatively we can turn the question around and ask for a fixed complex number p, for which (complex) polynomials is the forward orbit of p bounded? Finite? Periodic? These questions give the interesting fractal pictures many of you have probably seen. Amazingly many of the tools needed to approach these questions, arose well before computers allowed us to generate images like the one above. In this talk we will explore some of the basic tools and results of complex dynamics paying particular attention relations to number theory. The goal being to present some of the background material need for Laura DeMarco’s talk later in the week. (Also getting to see a really cool area of mathematics!) |
Nov 04
| Vlad Matei |
| Modular forms for definite quaternion algebras |
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The Jacquet-Langlands theorem states that given two quaternion algebras, then certain automorphic forms for one of them are in canonical bijection with certain automorphic forms for the other. This seems far too general and also a bit vague.So if one translates the statement of the JL theorem down a bit, we should have that certain classical modular forms should be related to certain "modular forms" on other quaternion algebras. We will define modular forms for quaternion algebras, and we will see that for definite quaternion algebras they are very concrete algebraic objects. |
Nov 11
| Ryan Julian |
| What is a K3 surface, and why are K1 and K2 surfaces only studied by mountain climbers? |
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In preparation for Thursday's talk on the Shafarevich conjecture for K3 surfaces, I will attempt to build up enough of the definitions and background theory of differential geometry to define what a K3 surface is. In particular, I hope to explain how K3 surfaces fit into a larger classification of algebraic surfaces, allowing us to prove theorems in a more restricted setting before tackling surfaces of general type. Time permitting, I might even give a couple examples of K3 surfaces. If we're really lucky, I might even have time to explain the hilarious semi-joke in the title above. |
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
Megan Maguire (mmaguire2@math.wisc.edu)
Ryan Julian (mrjulian@math.wisc.edu)
Sean Rostami (srostami@math.wisc.edu)
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