Category: noncommutative

  • The $\mathbb{F}_1$ World Seminar

    For some time I knew it was in the making, now they are ready to launch it: The $\mathbb{F}_1$ World Seminar, an online seminar dedicated to the “field with one element”, and its many connections to areas in mathematics such as arithmetic, geometry, representation theory and combinatorics. The organisers are Jaiung Jun, Oliver Lorscheid, Yuri […]

  • Do we need the sphere spectrum?

    Last time I mentioned the talk “From noncommutative geometry to the tropical geometry of the scaling site” by Alain Connes, culminating in the canonical isomorphism (last slide of the talk) Or rather, what is actually proved in his paper with Caterina Consani BC-system, absolute cyclotomy and the quantized calculus (and which they conjectured previously to…

  • Alain Connes on his RH-project

    In recent months, my primary focus was on teaching and family matters, so I make advantage of this Christmas break to catch up with some of the things I’ve missed. Peter Woit’s blog alerted me to the existence of the (virtual) Lake Como-conference, end of september: Unifying themes in Geometry. In Corona times, virtual conferences…

  • the mystery Manin-Marcolli monoid

    A Belyi-extender (or dessinflateur) $\beta$ of degree $d$ is a quotient of two polynomials with rational coefficients \[ \beta(t) = \frac{f(t)}{g(t)} \] with the special properties that for each complex number $c$ the polynomial equation of degree $d$ in $t$ \[ f(t)-c g(t)=0 \] has $d$ distinct solutions, except perhaps for $c=0$ or $c=1$, and,…

  • Monstrous dessins 1

    Dedekind’s Psi-function $\Psi(n)= n \prod_{p |n}(1 + \frac{1}{p})$ pops up in a number of topics: $\Psi(n)$ is the index of the congruence subgroup $\Gamma_0(n)$ in the modular group $\Gamma=PSL_2(\mathbb{Z})$, $\Psi(n)$ is the number of points in the projective line $\mathbb{P}^1(\mathbb{Z}/n\mathbb{Z})$, $\Psi(n)$ is the number of classes of $2$-dimensional lattices $L_{M \frac{g}{h}}$ at hyperdistance $n$ in…

  • the Riemann hypothesis and Psi

    Last time we revisited Robin’s theorem saying that 5040 being the largest counterexample to the bound \[ \frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \] is equivalent to the Riemann hypothesis. There’s an industry of similar results using other arithmetic functions. Today, we’ll focus on Dedekind’s Psi function \[ \Psi(n) = n \prod_{p | n}(1 + \frac{1}{p})…

  • RH and the Ishango bone

    “She simply walked into the pond in Kensington Gardens Sunday morning and drowned herself in three feet of water.” This is the opening sentence of The Ishango Bone, a novel by Paul Hastings Wilson. It (re)tells the story of a young mathematician at Cambridge, Amiele, who (dis)proves the Riemann Hypothesis at the age of 26,…

  • Closing in on Gabriel’s topos?

    ‘Gabriel’s topos’ (see here) is the conjectural, but still elusive topos from which the validity of the Riemann hypothesis would follow. It is the latest attempt in Alain Connes’ 20 year long quest to tackle the RH (before, he tried the tools of noncommutative geometry and later those offered by the field with one element).…

  • Archangel Gabriel will make you a topos

    No kidding, this is the final sentence of Le spectre d’Atacama, the second novel by Alain Connes (written with Danye Chéreau (IRL Mrs. AC) and his former Ph.D. advisor Jacques Dixmier). The book has a promising start. Armand Lafforet (IRL AC) is summoned by his friend Rodrigo to the Chilean observatory Alma in the Altacama…

  • The Langlands program and non-commutative geometry

    The Bulletin of the AMS just made this paper by Julia Mueller available online: “On the genesis of Robert P. Langlands’ conjectures and his letter to Andre Weil” (hat tip +ChandanDalawat and +DavidRoberts on Google+). It recounts the story of the early years of Langlands and the first years of his mathematical career (1960-1966)leading up…