This is the homepage for the UIUC Number Theory Seminar.
We will meet in Loomis 136.
Seminars from 2012-2022 are archived here.
January 27: Daniel Martin (Clemson University)
Title: Arithmetic on Markoff surfaces
Abstract: The generalized Markoff equation gives rise to a dynamical system via the Markoff group acting on the solution set. This talk considers the resulting dynamics over finite fields, which exhibits strong connections with combinatorial group theory and arithmetic geometry. Specifically, we focus on the orbit structure of the Markoff group action, which McCullough and Wanderley conjecture to be trivial outside of a small number of exceptional orbits. Certain cases of this conjecture have been verified in recent work of Chen and of Bourgain–Gamburd–Sarnak. We will present new techniques that establish the McCullough–Wanderley conjecture in additional settings.
February 3: Dimitris Koukoulopoulos (Université de Montréal)
Title: Erdős's integer dilation approximation problem
Abstract: Let 𝒜 ⊂ ℝ ≥ 1 be a countable set such that limsup x→∞(1/log x)Σ α∈𝒜∩ [1,x](1/α)>0. Erdős conjectured in 1948 that, for every ε>0, there exist infinitely many pairs (α, β)∈𝒜² such that α ≠ β and |nα -β| <ε for some positive integer n. When 𝒜 is a set of integers, the conjecture follows by work of Erdős and Behrend on primitive sets of integers from the 1930s. Moreover, if 𝒜 contains "enough elements" all of whose pairwise ratios are irrational, then Haight proved Erdős's conjecture in 1988. In this talk, I will present recent joint work with Youness Lamzouri and Jared Duker Lichtman that solves the conjecture in full generality. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by Koukoulopoulos-Maynard in the proof of the Duffin-Schaeffer conjecture in Diophantine approximation.
February 10: Anurag Sahay (Purdue University)
Title: Two variations on the theme of divisor correlations
Abstract: The correlations between d(n) and d(n+h), where d(n) is the divisor-counting function and h is a possibly varying non-zero integer are a classical topic in analytic number theory, going back to Ingham. It is intimately related to the fourth moment of the Riemann zeta function. After a quick review of the history of this problem, we will discuss two recent variants that arose in our work.
In the first variant, we will replace the integers with shifted integers n+α, where α is an irrational number. This arose in work joint with Winston Heap on the fourth moment of the Hurwitz zeta function and has connections to Diophantine approximation.
In the second variant, we will replace the integers with the ring of polynomials over a finite field. This was investigated in work joint with Alexandra Florea, Matilde Lalín, and Amita Malik, where we extended the range of uniformity in h for which an asymptotic formula is available in this setting, building on earlier work of Conrey-Florea, Gorodestky, Woo and Yiasemides. The main new input here is a Voronoi summation formula for the divisor function, which appears to be novel in this setting.
February 17: Shifan Zhao (Ohio State University)
Title: TBA
Abstract: TBA
February 24: Sheng-Yang Kevin Ho (UC San Diego)
Title: Ogg's conjectures over function fields
Abstract: In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over ℚ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence of his fundamental study of the arithmetic properties of modular curves and Hecke algebras. In this talk, we review the function field analogues of Ogg's conjectures, their current status, and the methods that have been applied to prove some of these conjectures. The methods are based on the ideas of Mazur and Ogg, but there are interesting differences and technical complications that arise in the function field setting, as well as intriguing possible new directions for generalizations. This is joint work with Cécile Armana and Mihran Papikian.
March 3:
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March 10: Jeremy Rouse (Wake Forest University)
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March 31: Alexandra Florea (UC Irvine)
Title: TBA
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April 7: Katy Woo (Stanford)
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April 14:
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April 21:
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April 28:
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May 5:
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