# The next Ŧ₁-seminar (December 21, 2017)

The next Ŧ₁-seminars will take place in December 2017. The speaker will be Lieven Le Bruyn, from University of Antwerp.

WHEN ?: The seminar is scheduled for Thursday December 21, 2017.

WHERE ?: Department of Mathematics, Ghent University, Krijgslaan 281, Building S25, Ghent, Belgium.

SCHEDULE:

13:30 – 14:00: Coffee break (Entrance hall, Building S25)

14:00 – 14:15: Blitzing through Ŧ₁-theory (by K. Thas)
14:20 – 15:10: The shape of all prime numbers, 1 [ABSTRACT] (by L. Le Bruyn)

15:10 – !5:25: Coffee break (Entrance hall, Building S25)

15:25 – 16:15: The shape of all prime numbers, 2 [ABSTRACT] (by L. Le Bruyn)

16:15 – 17:15: Reception (Entrance hall, Building S25)

All lectures will take place in Room “Emmy Noether,” Building S25.

# Inaugural Lectures

The inaugural speaker in “The $\mathbb{F}_1$ Seminars at Ghent University” is Yuri I. Manin (Max Planck Institute for Mathematics, Bonn, GE and Northwestern University, Evanston, US).

WHEN ?: The inaugural seminar is scheduled for Tuesday February 3, 2015.

WHERE ?: Department of Mathematics, Ghent University, Krijgslaan 281, Buildings S22 and S25, Ghent, Belgium.

SCHEDULE:

13:30 – 14:00: Coffee break (Room A, Building S22)

14:00 – 14:25: A casual introduction to $\mathbb{F}_1$-theory (by K. Thas)
14:30 – 15:20: Numbers as functions [ABSTRACT] (by Yu. I. Manin)
15:30 – 16:20: $\mathbb{F}_1$: Mathematical object in search of a definition [ABSTRACT] (by Yu. I. Manin)

All lectures will take place in Room “Emmy Noether,” Building S25.

16:30 – 17:30: Reception (Room A, Building S22)

Professor Yuri I. Manin

# The shape of all prime numbers, 1 and 2 (by L. Le Bruyn)

Our knowledge of the geometric shape of prime numbers changed dramatically over the last century. The most common point of view, dating back to Dedekind and Kronecker, is to picture them just as points on a line. In the early 60ties, Grothendieck and his gang developed étale cohomology, forcing us to view prime numbers as knots in 3-space. In recent years,  Ŧ₁-geometry suggests that the shape of all prime numbers may be infinitely more complex.

-FIRST TALK-

In the first talk I’ll give a low-tech account of this story, focusing on the philosophy and motivation of Ŧ₁-geometry. I will give a new proposal, in the spirit of Jim Borger’s approach via λ-rings, to view additional structure on rings as descent data from the integers ℤ to the field with one element Ŧ₁.

-SECOND TALK-

In the second talk I’ll indicate how this approach fits in nicely with Alexander Smirnov’s proposed attack on the ABC-conjecture via Ŧ₁-geometry and (if time allows) a question posed by Yuri I. Manin in the previous seminar on the motivic interpretation of the local Γ-factor at complex arithmetic infinity.

# Ŧ₁: Mathematical object in search of a definition (by Yu. I. Manin)

In this lecture I will focus on (some of) the recent developments deepening our intuition that roots of unity and more general multiplicative monoids can be considered as “constants” in various kinds of algebraic (and possibly analytic) geometries to which I refer as “geometries in characteristic one.”

# Numbers as functions (by Yu. I. Manin)

Since Kurt Hensel’s introduction of $p$-adic numbers in 1897, it was clear that in a sense they were similar to formal Laurent series of one variable over a field of constants $k$. One essential difference, however, was a different nature of coefficients: the standard choice of, say, $\{0,1,2,\ldots, p-1 \}$ does not look at all like a field or a ring: it is not closed w.r.t. addition and multiplication. A crucial remedy was found by Oswald Teichmueller in the 1930s, when he suggested to use $\{0,$ roots of unity of degree $p \}$ (for odd $p$) as standard coefficients ($\{0,1,-1\}$ for $p=2$). This set is still not a ring, but at least it is a multiplicative monoid.

In this lecture I will explain some of the subsequent developments of this analogy. I will start with an introduction to A. Buium’s theory of “differential equations in the $p$–adic direction” and its interrelations with another active current project “geometries in characteristic one.”