## Chicago Number Theory Day University of Illinois at Chicago May 23, 2015

Organizer:

Nathan Jones (UIC)

## Schedule of Talks

The talks will take place in SEO 636 in the SEO building of UIC campus. To get to campus, take the Blue Line train to UIC-Halsted, or walk to the intersection of Harrison and Halsted. Here are more detailed directions.

 Saturday, May 23, 2015 8:30 am - 9:00 am COFFEE 9 am - 9:50 am Roman Holowinsky (The Ohio State) Variations of the delta-method in application to shifted convolution sums and subconvexity 10 am - 10:25 am Yueke Hu (Wisconsin) Triple product formula and applications to subconvexity bound and mass equidistributions 10:30 am - 10:45 am BREAK 10:45 am - 11:10 am Hongbo Yin (Wisconsin) A kind of non-congruence subgroups and their modular curves 11:15 am - 12:05 pm Kevin Ford (UIUC) New bounds for large gaps between consecutive primes 12:15 pm - 2:00 pm LUNCH 2:00 pm - 2:50 pm Wei Ho (Michigan) Distributions of ranks of elliptic curves 3:00 pm - 3:25 pm Gene Kopp (Michigan) A Diophantine Problem from Fluid Dynamics: Resonances of Rossby Waves 3:30 pm - 3:45 pm BREAK 3:45 pm - 4:10 pm Ayla Gafni (Penn State) Power Partitions 4:15 pm - 4:45 pm Victor Guo (Missouri) Squarefree Numbers of the Form $\left\lfloor p^c \right\rfloor$

ABSTRACTS

• Roman Holowinsky, Variations of the delta-method in application to shifted convolution sums and subconvexity

We will present simple "conductor lowering" techniques in the delta-method which have proven useful in various shifted convolution sum and subconvexity problems. Some of the recent works of Ritabrata Munshi, Zhi Qi, and others will be highlighted.

• Yueke Hu, Triple product formula and applications to subconvexity bound and mass equidistributions

Triple product formula relates a global period integral with the special value of triple product L-function and local integrals at ramified places. Before my work, not much was known for the local integrals with ramifications. But they are important for many applications. I will talk about my work on explicit computation and upper bound of the local integrals for very general ramifications. Such results can be applied to prove the subconvexity bound of triple product L-function in level aspect, and mass equidistribution on modular curve of level N.

• Hongbo Yin, A kind of non-congruence subgroups and their modular curves

I will introduce a new kind of non-congruence subgroup. The definition of this new kind of group is similar to the Fermat groups, but the property is different in some sense. I will mainly talk about properties of their modular curves.

• Kevin Ford, New bounds for large gaps between consecutive primes

We describe recent improvements to lower bounds on the largest gap between consecutive primes less than x. This is joint work with Ben Green, Sergei Konyagin, James Maynard and Terence Tao.

• Wei Ho, Distributions of ranks of elliptic curves

In the last five years, there has been significant theoretical progress on understanding the average rank of all elliptic curves over Q, ordered by height, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as discuss generalizations in many directions (e.g., to other families of elliptic curves, higher genus curves, and higher-dimensional varieties) and some corollaries of these types of theorems. We will also describe recently collected data on ranks and Selmer groups of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).

• Gene Kopp, A Diophantine Problem from Fluid Dynamics: Resonances of Rossby Waves

Rossby waves are large-scale atmospheric waves that influence weather patterns, described by the Charney-Hasegawa-Mima partial differential equation. Resonant triads of Rossby waves occur at integer solutions to a particular degree five homogeneous equation in four variables. We show that the hypersurface X defined by this equation is birational to P^2, giving a rational parametrization of all resonant triads. Moreover, we find infinitely many reduced resonances for each nontrivial ratio of zonal and meridional wavenumbers, by regarding X as a family of elliptic curves (over P^1 minus five points) and showing that the Mordell-Weil group is generically rank 1 (but can have larger rank). Finally, we discuss physically important and unresolved questions about the asymptotic behavior of the wavenumber set and the efficient computation of all resonances up to a fixed bound.

• Ayla Gafni, Power Partitions

In 1918, Hardy and Ramanujan published a seminal paper which included an asymptotic formula for the partition function. In their paper, they also state without proof an asymptotic equivalence for the number of partitions of a number into $k$-th powers. In 1934, E.~Maitland Wright [Acta Mathematica, 63 (1934) 143--191] gives a very precise asymptotic formula for this restricted partition function, but his argument is quite long and difficult. In this talk, I will present an asymptotic formula for the number of partitions into $k$-th powers using a relatively simple method, while maintaining a decent error term.

• Victor Guo, Squarefree Numbers of the Form $\left\lfloor p^c \right\rfloor$

For fixed $c \in (1, 149/87)$, we derive an asymptotic formula for the number of primes $p \le x$ such that $\left\lfloor p^c \right\rfloor$ is squarefree, where $\left\lfloor . \right\rfloor$ is the floor function.