BC-MIT Number Theory Seminar

Time: 3:30 - 4:30pm

Speaker: Vesselin Dimitrov (Caltech)

Title: Arithmetic Gevrey series and the nearest integer to $\log{n}$

Abstract:

I will report on the continued joint project with Calegari and Tang on arithmetic holonomy bounds, this time in the broader context of Gevrey series.

We give an extension of Andre's algebraicity criterion to the arithmetic Gevrey class, which turns out to contain the Hermite-Lindemann theorem directly as a special case. Another application solves the following problem first raised by Mahler: We prove that for all integers $n > 1$, the nearest integer distance to $\log{n}$ is bounded below by some universal polynomial in $1/n$.

In the language of the theory of linear forms in logarithms, this supplies precisely the Waldschmidt parameter feature (which has only been proved for homogeneous linear forms with rational coefficients) in the inhomogeneous case of linear forms in logarithms on $\mathbb{G}a \times \mathbb{G}m$.

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Time: 5:00 - 6:00pm

Speaker: Hélène Esnault (FU Berlin)

Title: Arithmetic of complex local systems

Abstract:

Over the last few years I studied with Michael Groechenig, Moritz Kerz and in part with Johan de Jong properties of complex local systems on smooth quasi-projective varieties which are expected by Simpson's general conjectures to be motivic.

I'll try to give an overview of some of the problems which are currently on my mind. Everybody is welcome. Please feel free to interrupt me for more explanation and details.