Mathematics Department - Number Theory Learning Seminar - Spring 2017

Number Theory Learning Seminar - Spring 2017



Organizer(s)

Matthew C Welsh

Archive



Upcoming Talks


Wednesday, April 26th

Matthew Welsh, Rutgers University

"Equidistribution of 3-Torsion Ideals"

Time: 3:15 PM
Location: Hill 525
Abstract: For an imaginary quadratic field, the Cohen-Lenstra heuristics predict asymptotics for the average size of the k-torsion of the class group. Apart from what is given by genus theory, only the asymptotics for k=3 is known, proven by Davenport and Heilbronn. I will discuss a new method of Hough that recovers the Davenport-Heilbronn theorem and a lower order term in the asymptotic by proving that the 3-torsion ideals are equidistributed in H/SL_2(Z).





Past Talks


Wednesday, April 19th

Ed Karasiewicz, Rutgers University

"A Kronecker Limit Formula for Totally Real Cubic Fields and Schubert Eisenstein Series"

Time: 3:15 PM
Location: Hill 525
Abstract: In 1984 Bump and Goldfeld proved a Kronecker Limit Formula for totally real cubic fields using Automorphic forms on GL(3). More recently Bump and Choie introduced the notation of Schubert Eisenstein series, described a connection to the previous work of Bump and Goldfeld. We will discuss both works and describe this connection.


Wednesday, April 12th

George Hauser, Rutgers University

"The Markov Equation mod p"

Time: 3:15 PM
Location: Hill 525
Abstract: The markov equation is the equation x^2+y^2+z^2=3xyz. All of its integer solutions are obtained from the solution (1,1,1) by repeated application of "Vieta moves" and permutations of the coordinates. However, it is an open problem whether every solution in Z/pZ comes from a solution in Z. In this we will discuss some basic properties of the Markov equation mod p, and some recent results of Bourgain, Gamburd, and Sarnak in the direction of proving this conjecture.


Wednesday, April 5th

Surya Teja Gavva, Rutgers University

"Vinogradov's three primes with almost twin primes"

Time: 3:15 PM
Location: Hill 525
Abstract: One strategy to produce prime solutions to equations is to produce a dense model of primes (Transference Technique) and then show solutions exist in the dense model by results like Szemeredi's theorem. But in the case of Vinogradov's equation N=x_1 +x_2+x_3, the dense sets needn't have solutions. I'll try to discuss a paper of Matomaki, Shao where they further analyse the dense model to produce solutions in almost twin primes. I'll first start by discussing Transference principle, Restriction theorem and if time permits sketch the proof of Matomaki. Shao's result. 


Wednesday, March 29th

Katie McKeon, Rutgers University

"Hausdorff Dimension for part of the Markoff Spectrum"

Time: 3:15 PM
Location: Hill 525
Abstract: Last week, Professor Baragar mentioned a part of the Markoff spectrum where little is currently known. I'll discuss a recent work by Matheus and Moreira which sheds more light on this topic. The paper can be found here: https://arxiv.org/pdf/1703.04302.pdf


Wednesday, March 22nd

Arthur Baragar , UNLV

"Circles"

Time: 3:15 PM
Location: Hill 525
Abstract: A quick journey through a number of interrelated results, touching on continued fractions, the Apollonian packing, the Markoff equation, Tchebechev polynomials, descent, heights, K3 surfaces, elliptic curves, Bezout's theorem, and intersections.


Wednesday, March 1st

Matthew Welsh, Rutgers University

"L-functions for a Family of K3 Surfaces"

Time: 3:15 PM
Location: Hill 525
Abstract: I will attempt to explain how to compute the L-functions of an infinite family of K3 surfaces in terms of those of a family of elliptic curves. The calculation will show modularity for a few of these L-functions.


Wednesday, February 22nd

Surya Teja Gavva, Rutgers University

"Trilinear forms in Kloosterman fractions."

Time: 3:15 PM
Location: Hill 525
Abstract: I will talk about the paper of S.Bettin and V. Chandee where they obtain bounds on trilinear forms in Kloosterman fractions, improving the work of Duke, Friedlander, Iwaniec [DFI97


Wednesday, February 15th

Katie McKeon, Rutgers University

" Dirichlet Series and Kakeya Sets"

Time: 3:15 PM
Location: Hill 525
Abstract: We will perform some calculations and elaborate on a few comments made by Jean Bourgain in the essay 'Harmonic Analysis and Combinatorics: how much may they contribute to each other?' In particular, we will see how Montgomery's large values conjecture relates to a conjecture about Kakeya sets.


Wednesday, February 1st

Surya Teja Gavva, Rutgers University

"Twisted Second moment of the Riemann zeta function."

Time: 3:15 PM
Location: Hill 525
Abstract: The asymptotics for mean-square of the product of the Riemann zeta-function with an arbitrary Dirichlet polynomial have been consistently used to understand the distribution of values, zeroes, upper and lower bounds for sizes of the zeta function. I'll try to talk about a recent result of Bettin, Chandee, Radziwill where they go beyond theta=1/2, i.e., length of the Dirichlet polynomial is T^{theta}, theta > 1/2. Their main tool is an estimate for trilinear forms in Kloosterman fractions.


Wednesday, January 25th

Claire Burrin, Rutgers University

"Effective lattice-point counting, after Gorodnik and Nevo"

Time: 3:15 PM
Location: Hill 525
Abstract: To conclude my review of dynamical counting methods, I want to discuss the approach to lattice point counting problems of Gorodnik and Nevo.


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