Posts Categorized: math

  • geometry, math

    Grothendieck topologies as functors to Top

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    Either this is horribly wrong, or it must be well-known. So I guess I’m asking for either a rebuttal or a reference. Take a ‘smallish’ category $\mathbf{C}$. By this I mean that for every object $C$ the collection of all maps ending in $C$ must be a set. On this set, let’s call it $y(C)$… Read more »

  • books, france, math, personal, stories

    NaNoWriMo (1)

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    Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution. When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked (PS) And Lieven, what are your resolutions for 2016? she didn’t really expect an answer (for decades my generic reply has… Read more »

  • math

    from chocolate bars to constructivism

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    A fun way to teach first year students the different methods of proof is to play a game with chocolate bars, Chomp. The players take turns to choose one chocolate block and “eat it”, together with all other blocks that are below it and to its right. There is a catch: the top left block… Read more »

  • france, geometry, math, stories

    Grothendieck’s gribouillis (2)

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    We left the story of Grothendieck’s Lasserre notes early 2015, uncertain whether they would ever be made public. Some things have happened since. Georges Maltsiniotis gave a talk at the Gothendieck conference in Montpellier in june 2015 having as title “Grothendieck’s manuscripts in Lasserre”, raising perhaps even more questions. Philippe Douroux, a journalist at the… Read more »

  • france, math, stories

    Map of the Parisian mathematical scene 1933-39

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    . Michele Audin has written a book on the history of the Julia seminar (hat tip +Chandan Dalawat via Google+). The “Julia Seminar” was organised between 1933 and 1939, on monday afternoons, in the Darboux lecture hall of the Institut Henri Poincare. After good German tradition, the talks were followed by tea, “aimablement servi par… Read more »

  • math, number theory

    $\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$

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    Note to self: check Jack Morava’s arXiv notes on a more regular basis! It started with the G+-post below by +David Roberts: Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately. A couple of years ago I had a brief email exchange with him on the Habiro… Read more »

  • math, noncommutative

    A noncommutative moduli space

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    Supernatural numbers also appear in noncommutative geometry via James Glimm’s characterisation of a class of simple $C^*$-algebras, the UHF-algebras. A uniformly hyperfine (or, UHF) algebra $A$ is a $C^*$-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras $M_{c_1}(\mathbb{C}) \subset M_{c_2}(\mathbb{C}) \subset … \quad… Read more »

  • math, noncommutative, representations

    Quiver Grassmannians can be anything

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    A standard Grassmannian $Gr(m,V)$ is the manifold having as its points all possible $m$-dimensional subspaces of a given vectorspace $V$. As an example, $Gr(1,V)$ is the set of lines through the origin in $V$ and therefore is the projective space $\mathbb{P}(V)$. Grassmannians are among the nicest projective varieties, they are smooth and allow a cell… Read more »

  • math, stories

    The empty set according to bourbaki

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    The footnote on page E. II.6 in Bourbaki’s 1970 edition of “Theorie des ensembles” reads If this is completely obvious to you, stop reading now and start getting a life. For the rest of us, it took me quite some time before i was able to parse this formula, and when i finally did, it… Read more »

  • math, number theory

    Farey symbols in SAGE 5.0

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    The sporadic second Janko group $J_2$ is generated by an element of order two and one of order three and hence is a quotient of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$. This Janko group has a 100-dimensional permutation representation and hence there is an index 100 subgroup $G$ of the modular group such… Read more »

  • math, number theory

    Aaron Siegel on transfinite number hacking

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    One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication. Some time ago we did run a couple of posts on this. In… Read more »

  • absolute, math, number theory

    Quiver Grassmannians and $\mathbb{F}_1$-geometry

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    Reineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects… Read more »

  • math, noncommutative

    noncommutative geometry at the Lorentz center

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    This week i was at the conference Noncommutative Algebraic Geometry and its Applications to Physics at the Lorentz center in Leiden. It was refreshing to go to a conference where i knew only a handful of people beforehand and where everything was organized to Oberwolfach perfection. Perhaps i’ll post someday on some of the (to… Read more »

  • absolute, math, noncommutative

    Manin’s three-space-2000

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    Almost three decades ago, Yuri Manin submitted the paper “New dimensions in geometry” to the 25th Arbeitstagung, Bonn 1984. It is published in its proceedings, Springer Lecture Notes in Mathematics 1111, 59-101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns. In the introduction Manin… Read more »

  • groups, math

    Monsters and Moonshine : a booklet

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    I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download. The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game which uses the sporadic Mathieu… Read more »

  • math

    The martial art of giving talks

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    Last fall, Matilde Marcolli gave a course at CalTech entitled Oral Presentation: The (Martial) Art of Giving Talks. The purpose of this course was to teach students “how to effectively communicate their work in seminars and conferences and how to defend it from criticism from the audience”. The lecture notes contain basic information on the… Read more »

  • absolute, math, number theory

    Lambda-rings for formula-phobics

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    In 1956, Alexander Grothendieck (middle) introduced $\lambda $-rings in an algebraic-geometric context to be commutative rings A equipped with a bunch of operations $\lambda^i $ (for all numbers $i \in \mathbb{N}_+ $) satisfying a list of rather obscure identities. From the easier ones, such as $\lambda^0(x)=1, \lambda^1(x)=x, \lambda^n(x+y) = \sum_i \lambda^i(x) \lambda^{n-i}(y) $ to those… Read more »

  • geometry, math

    Grothendieck’s functor of points

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    A comment-thread well worth following while on vacation was Algebraic Geometry without Prime Ideals at the Secret Blogging Seminar. Peter Woit became lyric about it : My nomination for the all-time highest quality discussion ever held in a blog comment section goes to the comments on this posting at Secret Blogging Seminar, where several of… Read more »

  • groups, math

    Arnold’s trinities version 2.0

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    Arnold has written a follow-up to the paper mentioned last time called “Polymathematics : is mathematics a single science or a set of arts?” (or here for a (huge) PDF-conversion). On page 8 of that paper is a nice summary of his 25 trinities : I learned of this newer paper from a comment by… Read more »

  • geometry, groups, math, number theory

    Arnold’s trinities

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    Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic… David Corfield of the n-category cafe and a dialogue on… Read more »