Hodge theory is both beautiful and extremely powerful; it was used in ingenious ways by another prominent mathematician, the Fields medalist Maryam Mirzakhani, to study the deformations of flat surfaces. These smooth surfaces, not obviously connected to algebraic geometry, were found woven into a tapestry of rich algebraic structures by work of Eskin, Mirzakhani and Wright.
The Hodge conjecture has a similar philosophy: you define certain data in terms of "harmonic analysis" on the geometric object, and the conjecture says that they all correspond to the algebraic data out of which the object can be rebuilt.
Context: I don't know much about Mirzakhani's work, but I'm a PhD student in algebraic geometry and I studied Hodge theory and complex manifolds.
> These smooth surfaces, not obviously connected to algebraic geometry
According to Wikipedia:
> Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".
But Riemann surfaces are the simplest complex manifolds, and the compact ones are algebraic curves. I don't see how the connection to algebraic geometry is "not obvious". Is there something deeper going on? I would like to learn a bit more :)
I wish I'd read "Is Maths Real" first before commenting here! Oh well...anyone with more insights? Sometimes I feel once the principles/axioms are laid out, a giant puzzle is untangled within that logical system, and I wonder how much is really practical math vs academic sandbox?
E.g. this quote "You need a new idea, a good definition, a statement that you think you’ll be able to exploit. Only then can your work start."
Why is such an idea necessary in the first place, is the context missing here?
"By learning about the objects, by manipulating them and using them in mathematical arguments, they ultimately become your friend."
Yeah. She was awarded the Crafoord prize this year, a distinction similar to the Nobel prize in other disciplines.
(Other mathematicians who were awarded the prize include Grothendieck, Deligne, Connes, Tao, Bombieri...).
The title is also weird. It's about her view on her (personal) ideas about maths, how she sees it, how it works for her. There are (almost) no statements about creativity, art, and language in general.
Yeah and at 19, she got into the top math school in France where less than 30 students are admitted each year.
In the article she says she didn’t feel like she’d invent math herself until later though. And actually it happens a lot in France. People study maths, cause that’s what you do when you’re good in school. A good share of French “important people” studied fairly advanced maths.
> Voisin sometimes relaxes by painting both original works (shown behind her) and copies of famous portraits, like the Amedeo Modigliani at bottom left.
She has colour co-ordinated pictures, with a material backdrop. This is then - badly - co-ordinated with the couch and its colours and patterns.
I'm genuinely surprised that this person's art choices and arrangements are so bad. She is meant to be a top tier mathematician + into art - so when these combine in her own life, can this really be the result?
I'm not critiquing her interior decor for the sake of being mean - I just find the example of art + reasoning that we see in the pic to be so inconsistent with the person that is being portrayed. Can this be how a top mathematician arranges things they love?
That their work must be dumbed down to fit into a coffee table book or match the colour coordination of some interior design?
Art appreciation has many dimensions, and arguably the most meaningful ones are the most personal and authentic, and do not require justification to other people.
It seems entirely plausible that there wasn’t a good setting just emphasizing her and all her art for an article photo, and that paintings were temporarily moved to create one - in which case what you’re picking up on would represent the photographer and a compromise with expediency rather than the mathematician.
[+] [-] ykonstant|2 years ago|reply
The Hodge conjecture has a similar philosophy: you define certain data in terms of "harmonic analysis" on the geometric object, and the conjecture says that they all correspond to the algebraic data out of which the object can be rebuilt.
[+] [-] red_trumpet|2 years ago|reply
> These smooth surfaces, not obviously connected to algebraic geometry
According to Wikipedia:
> Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".
But Riemann surfaces are the simplest complex manifolds, and the compact ones are algebraic curves. I don't see how the connection to algebraic geometry is "not obvious". Is there something deeper going on? I would like to learn a bit more :)
[+] [-] megamix|2 years ago|reply
E.g. this quote "You need a new idea, a good definition, a statement that you think you’ll be able to exploit. Only then can your work start."
Why is such an idea necessary in the first place, is the context missing here?
"By learning about the objects, by manipulating them and using them in mathematical arguments, they ultimately become your friend."
[+] [-] empath-nirvana|2 years ago|reply
[+] [-] yau8edq12i|2 years ago|reply
[+] [-] fermigier|2 years ago|reply
[+] [-] tgv|2 years ago|reply
[+] [-] Warwolt|2 years ago|reply
[+] [-] Legend2440|2 years ago|reply
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[+] [-] d--b|2 years ago|reply
In the article she says she didn’t feel like she’d invent math herself until later though. And actually it happens a lot in France. People study maths, cause that’s what you do when you’re good in school. A good share of French “important people” studied fairly advanced maths.
[+] [-] Cacti|2 years ago|reply
[+] [-] unknown|2 years ago|reply
[deleted]
[+] [-] verisimi|2 years ago|reply
And yet, we see this pic:
https://d2r55xnwy6nx47.cloudfront.net/uploads/2024/03/Claire...
with this caption:
> Voisin sometimes relaxes by painting both original works (shown behind her) and copies of famous portraits, like the Amedeo Modigliani at bottom left.
She has colour co-ordinated pictures, with a material backdrop. This is then - badly - co-ordinated with the couch and its colours and patterns.
I'm genuinely surprised that this person's art choices and arrangements are so bad. She is meant to be a top tier mathematician + into art - so when these combine in her own life, can this really be the result?
I'm not critiquing her interior decor for the sake of being mean - I just find the example of art + reasoning that we see in the pic to be so inconsistent with the person that is being portrayed. Can this be how a top mathematician arranges things they love?
[+] [-] pnut|2 years ago|reply
That their work must be dumbed down to fit into a coffee table book or match the colour coordination of some interior design?
Art appreciation has many dimensions, and arguably the most meaningful ones are the most personal and authentic, and do not require justification to other people.
[+] [-] Gregaros|2 years ago|reply
[+] [-] lupire|2 years ago|reply