## Ninth Bucharest Number Theory Days |

Home | Schedule & Abstracts | Registration |

Chicago time | June 1st | June 2nd | June 3rd | Bucharest time |
---|---|---|---|---|

8:00-8:40 | Berndt | Shparlinski | Kim | 16:00-16:40 |

8:50-9:30 | Murty | Xiong | Dixit | 16:50-17:30 |

9:40-10:20 | Rudnick | Yee | Matomäki | 17:40-18:20 |

10:30-11:10 | Buium | Diaconu | Marklof | 18:30-19:10 |

11:10-11:40 | Break | Break | Break | 19:10-19:40 |

11:40-12:20 | Sarnak | Ford | Kontorovich | 19:40-20:20 |

12:30-13:10 | Granville | Soundararajan | Athreya | 20:30-21:10 |

13:20-14:00 | Radziwill | Ioviță | Luca | 21:20-22:00 |

14:10-14:50 | Remarks/speeches | Popescu | Cobeli | 22:10-22:50 |

**Jayadev Athreya,** *Counting Tripods*

Motivated by some problems from theoretical physics, we describe a toy model problem of counting isometric immersions of certain graphs
in the flat torus and related lattice point counting problems. This is joint work with David Aulicino and Harry Richman.
There will be lots of pictures, and hopefully, some discussion of the motivations.

**Bruce Berndt,**
*Balanced Derivatives, Identities, and Bounds for Trigonometric Sums and Bessel Series*

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively,
with the Gauss circle problem and the Dirichlet divisor problem, in a paper published in
J. Reine Angew. Math. in 2013, Zaharescu, Sun Kim, and the speaker derived representations
for certain sums of products of trigonometric functions as double series of Bessel functions.
With Martino Fassina, in a recently submitted paper, we returned to the ideas from our Crelle
paper. The aforementioned series are generalized by introducing the novel notion of balanced
derivatives, leading to further theorems. The regions of convergence in the unbalanced case
are different from those in the balanced case. If \(x\) denotes the number of products of
the trigonometric functions appearing in our sums, in addition to finding more efficient
proofs of the identities mentioned above, we establish theorems and conjectures for upper and
lower bounds for the trig sums as \(x\to\infty\).

**Alexandru Buium,** *Arithmetic PDEs*

Some of the classical ODEs (Riccati, Weierstrass, Painleve', etc.)
have arithmetic analogues in which derivatives of functions are replaced by Fermat quotients of numbers.
These lead to Diophantine applications (related to the Manin-Mumford conjecture, Heegner points, etc.)
Recently, in joint work with L. Miller, arithmetic analogues of some classical PDEs were introduced.
An application to a new type of reciprocity law will be presented.

**Cristian Cobeli,** *TBA*

**Atul Dixit,** *TBA*

**Adrian Diaconu,** *TBA*

**Kevin Ford,** *TBA*

**Andrew Granville,** *TBA*

**Adrian Ioviță,** *TBA*

**Sun Kim,** *TBA*

**Alex Kontorovich,** *TBA*

**Florian Luca,** *TBA*

**Jens Marklof,** *TBA*

**Kaisa Matomäki,** *TBA*

**Ram Murty,** *TBA*

**Cristian Popescu,** *TBA*

**Maksym Radziwill,** *TBA*

**Zeev Rudnick,** *TBA*

**Peter Sarnak,** *TBA*

**Igor Shparlinski,** *Weyl Sums: Large, Small and Typical*

While Vinogradov's Mean Value Theorem, in the form given by J. Bourgain, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019), gives an essentially optimal result on the power moments of the Weyl sums
\[
S(u;N) =\sum_{1\le n \le N} \exp(2 \pi i (u_1n+\ldots+u_dn^d))
\]
where \(u = (u_1,\ldots,u_d) \in [0,1)^d,\) very little is known about the distribution, or even existence, of \(u \in [0,1)^d\),
for which these sums are very large, or small, or close to their average value \(N^{1/2}\). In this talk, we describe recent progress towards these and some related questions.

In particular, we give new results on the spectrum of possible sizes of Weyl sums. We also present some new bounds on `slices' of \(S(u;N)\),
that is, bounds which hold for almost all \((u_i)_{i\in I}\) and all \((u_j)_{j\in J}\), where \(I \cup J\) is a partition of \({1,\ldots,,d}\),
which improve a previous result of T. Wooley (2015).

Finally, we discuss lower bounds on slices of Weyl sums, generalising that of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020).

Joint work with Julia Brandes, Changhao Chen, Bryce Kerr, James Maynard.

**Kannan Soundararajan,** *TBA*

**Maosheng Xiong,** *TBA*

**Ae Ja Yee,** *TBA*