Posts Categorized: geometry

  • absolute, geometry, number theory

    How to dismantle scheme theory?

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    In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ is not suited to tackle problems such as the ABC-conjecture. The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2-dimensional objects’ (and clearly we use both ‘dimensions’ in the theory… Read more »

  • games, geometry

    The geometry of football

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    Soon, we will be teaching computational geometry courses to football commentators. If a player is going to be substituted we’ll hear sentences like: “no surprise he’s being replaced, his Voronoi cell has been shrinking since the beginning of the second half!” David Sumpter, the author of Soccermatics: Mathematical Adventures in the Beautiful Game, wrote a… Read more »

  • geometry, stories

    The subway singularity

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    The Boston subway is a complex system, spreading out from a focus at Park Street. On March 3rd, the Boylston shuttle went into service, tying together the seven principal lines, on four different levels. A day later, train 86 went missing on the Cambridge-Dorchester line. The Harvard algebraist R. Tupelo suggested the train might have… Read more »

  • geometry

    Forgetting can’t be that hard, can it?

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    Geometers will tell you there are two ways to introduce affine schemes. You can use structure sheaves. That is, compute all prime ideals of your ring and turn them into a space. Then, put a sheaf of rings on this space by localisation. You’ll get your ring back taking global sections. Or, you might try… Read more »

  • geometry, personal, rants, representations

    Stirring a cup of coffee

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    Please allow for a couple of end-of-semester bluesy ramblings. I just finished grading the final test of the last of five courses I lectured this semester. Most of them went, I believe, rather well. As always, it was fun to teach an introductory group theory course to second year physics students. Personally, I did enjoy… Read more »

  • geometry, stories

    Where are Grothendieck’s writings? (2)

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    A couple of days ago, there was yet another article by Philippe Douroux on Grothendieck’s Lasserre writings “Inestimables mathématiques, avez-vous donc un prix?” in the French newspaper Liberation. Not that there is much news to report. I’ve posted on this before: Grothendieck’s gribouillis, Grothendieck’s gribouillis (2), and more recently Where are Grothendieck’s writings? In that… Read more »

  • books, geometry

    how much to spend on (cat)books?

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    My favourite tags on MathOverflow are big-lists, big-picture, soft-question, reference-request and the like. Often, answers to such tagged questions contain sound reading advice, style: “road-map to important result/theory X”. Two more K to go, so let’s spend some more money. [section_title text=”Category theory”] [full_width_image] [/full_width_image] One of the problems with my master course on algebraic… Read more »

  • books, geometry, noncommutative, representations

    let’s spend 3K on (math)books

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    Santa gave me 3000 Euros to spend on books. One downside: I have to give him my wish-list before monday. So, I’d better get started. Clearly, any further suggestions you might have will be much appreciated, either in the comments below or more directly via email. Today I’ll focus on my own interests: algebraic geometry,… Read more »

  • books, geometry, personal, stories

    NaNoWriMo (3)

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    In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote: “Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can… Read more »

  • books, geometry, personal, stories

    NaNoWriMo (2)

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    Two more days to go in the NaNoWriMo 2016 challenge. Alas, it was clear from the outset that I would fail, bad. I didn’t have a sound battle plan. Hell, I didn’t even have a a clue which book to write… But then, I may treat myself to a SloWriMo over the Christmas break. For… Read more »

  • 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 »

  • geometry, stories

    le lemme de la Gare du Nord

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    Theorems have the tendency to pop into existence when you least expect them: taking a bath, during your sleep, dreaming away during a dull lecture, waiting for an airplane, bicycling, whatever. One of the most famous (and useful) lemmas was dreamed up in the Parisian Gare du Nord station, during a conversation between Saunders Mac… Read more »

  • geometry

    Car crashes in scheme theory

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    What do you get when two cars crash head on at full speed? A heap of twisted metal. What do you get when two tiny cars crash head on at full speed? A smaller heap of twisted metal. In the limit, what do you get when two point cars crash head on at full speed?… Read more »

  • geometry

    Toposes alive and kicking at IHES

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    After 50 years, vivid interest in topos theory seems to have returned to one of the most prestigious research institutes, the IHES. Last november, there was the meeting Topos a l’IHES. At the meeting, Celine Loozen filmed a documentary which is supposed to have as its title “Unifying Worlds”. Its very classy trailer is now… Read more »

  • absolute, geometry, number theory

    The group algebra of all algebraic numbers

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    Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field $\mathbb{Q}_{cyc}$. This gives, at least in theory, a handle on… Read more »

  • absolute, geometry, number theory

    Topology and the symmetries of roots

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    We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. In the title picture the roots of polynomials of degree $\leq 4$ with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots… 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 »

  • absolute, geometry, number theory, stories

    The Log Lady and the Frobenioid of $\mathbb{Z}$

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    “Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.” “All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can… Read more »

  • absolute, geometry

    Two lecture series on absolute geometry

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    Absolute geometry is the attempt to develop algebraic geometry over the elusive field with one element $\mathbb{F}_1$. The idea being that the set of all prime numbers is just too large for $\mathbf{Spec}(\mathbb{Z})$ to be a terminal object (as it is in the category of schemes). So, one wants to view $\mathbf{Spec}(\mathbb{Z})$ as a geometric… Read more »

  • geometry, stories

    Children have always loved colimits

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    If Chad Orzel is able to teach quantum theory to his dog, surely it must be possible to explain schemes, stacks, toposes and motives to hipsters? Perhaps an idea for a series of posts? It’s early days yet. So far, I’ve only added the tag sga4hipsters (pun intended) and googled around for ‘real-life’ applications of… Read more »

  • geometry, stories

    Can one explain schemes to hipsters?

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    Nature (the journal) asked David Mumford and John Tate (of Fields and Abel fame) to write an obituary for Alexander Grothendieck. Probably, it was their first experience ever to get a paper… rejected! What was their plan? How did they carry it out? What went wrong? And, can we learn from this? the plan Mumford… Read more »

  • geometry, stories, web

    Grothendieck’s gribouillis

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    A math-story well worth following in 2015. What will happen to Grothendieck’s unpublished notes, or as he preferred, his “gribouillis” (scribbles)? Here’s the little I know about this: 1. The Mormoiron scribbles During the 80ties Grothendieck lived in ‘Les Aumettes’ in Mormoiron In 1991, just before he moved to the Pyrenees he burned almost all… Read more »

  • geometry, stories

    Grothendieck’s Café

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    “A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”. At the time only an undefined idea of “schéma” was current in Paris, meaning more or less whatever would improve on Weil’s foundations.” (McLarty in The Rising Sea) Finding that particular café in Paris, presumably in the… Read more »

  • geometry, stories

    the birthday of Grothendieck topologies

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    This is the story of the day the notion of ‘neighbourhood’ changed forever (at least in the geometric sense). For ages a neighbourhood of a point was understood to be an open set of the topology containing that point. But on that day, it was demonstrated that the topology of choice of algebraic geometry, the… 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 »