I knew Maryam was incredible the first time I heard her give a talk at Stanford. To my fellow cancer researchers out there: this is about as good motivation as we'll ever have to keep plugging along.
So I am trying to undesrstand what her work was about. It seems fields medal was for "the dynamics and geometry of Riemann surfaces and their moduli spaces" (my emphasis).
What is dynamics to a geometer? Clearly it is important -- because it is important in physics. But I want to understand what mathematicians are getting excited about.
We can talk about a family of geometrical objects foo(t) where the parameter t is a single real value, which stands in for time. But -- other than its relation to the physical world -- why is that $t$ parametrisation important? Why not complex parameters, or multiple parameters, or something completely different?
My memory may be a bit fuzzy on this. But I believe one way the term "dynamics" is used in her work is in reference to the space of Riemann surfaces (or equivalently, the space of hyperbolic geometries on a surface). This space is called moduli space. Certain topological operations on the surface (e.g. twisting a "handle") give rise to a corresponding transformation on moduli space. So one of the things mathematicians in this area can study is the dynamics of such operations on moduli space. This means, for example,
studying the behavior of these operations on moduli space, in the limit.
Also, to a mathematician, "dynamics" can often simply mean studying the behavior of iterations of a single function from a space to itself. So the parameter "t" in this case is over the positive integers (or all integers if the function has an inverse).
Not gonna describe what dynamics is because that's a broad question best answered by some googling. Will comment on the more restricted question "why do we like 1-parameter families".
1-parameter families of objects usually arise from differential equations in geometric settings. Some diffrential geometric examples that one should be able to appreciate without needing motivation from physics:
- Integral curves and 1-parameter flows generated by integrable vector fields. A important special case in dynamics is Hamiltonian diffeomorphisms on symplectic manifolds.
- Ricci flow in Riemannian manifolds
- Geodesics and Jacobi fields on Riemannian manifolds
One plausible reason for why we like 1-parameter families is that they're just easier, than, say, 2-parameter families. This is true in more than one way
- 1-parameter families tend to arise from easier differential equations. For example, compare a geodesic problem to a minimal surface problem. Conversely it's easier to write down natural PDE systems when you're only trying to evolve 1 time variable.
- The domain is easier to understand. For simplicity let's say we're parametrizing things continuously with a connected, compact domain. Topologically there's just two such 1-manifolds, the circle and the closed interval. When we consider a family of object parametrized by, say, 2 parameters, the domain is a surface (which we at least know a classification). If you then consider 3 parameters, the domain is a 3-manifold (which we don't know how to classify).
As an example, a simple "family of geometric objects" you can parametrize is just a family of points on a manifold. In dimension 1, this amounts to some classification of curves. Things that come from this are, for example, the fundamental group, rank 1 (say, singular) homology, morse-smale theory which counts curves between critical points, geodesics which are solutions to some length functional. All these things are fairly easy to understand given some standard education in differential/algebraic topology.
Now if you consider the same problem in dimension 2, the first step is to understand the ways you can embed your domain into your target manifold. This is more or less a starting point for the theory of moduli spaces which is already non-trivial even in the genus 0 case. For example, under certain technical assumptions this can be done with Gromov-Witten theory. These are much more sophisticated issues.
I'm starting to think this black bar was a bad idea. All it gives is fruitless discussions like this one on the merit of a person to be awarded the black bar upon their death, like if it was a trophy or a medal of accomplishment.
I think the first implementation of the black bar was for Steve Jobs. Well, I'm not a fan of Steve Jobs, and I think that individuals like Maryam Mirzakhani are far more important in the course of humanity than Steve Jobs ever was. But I don't question the decision to put the black bar for Jobs: it corresponded to an emotion, felt by a lot of people here and more importantly by the owners of HN. It was their way to express their emotion, other people wrote blog posts, tweets or Facebook status. But it was all a question of emotion, and the merit of the person was secondary. The black bar is like the black clothes of people at burials or the black veil of the widow: it's there to express sorrow, not to mark how much the person was a great scientist or entrepreneur. And who are we to judge emotions or impose them to other?
I don't think there should be a specific criterion. Putting on the black bar now when it was not done spontaneously is like faking a deep sorrow that do not exist.
More importantly, I think, is the connection HN's operators felt to Steve Jobs --- for whatever reason, and don't discount that for some of them that reason could be pretty personal and authentic.
I could not agree more with you about these discussions, which occur at the bottom of every thread about any recently deceased, like a sort of ghoulish, amoral Slashdot "first post" contest. It is not OK to litigate how strangers choose to relate grief, least of all in their own damn house.
Stories and threads on HN are, for the most part, a sort of community property. The bar at the top of HN is not: it belongs to the operators of HN. Litigating how and when they use it to express grief is not an especially polite thing to do.
The easiest way, then, to understand the "black bar" is to think of it as an expression of sorrow that has a direct connection to HN's operators.
If it was a private thing I would be totally with you, but the black bar's entire purpose is to send a public message to the community. It seems natural for the public reading the message to wonder why it was sent (or omitted) on a given occasion. It's not "why aren't you grieving?" but rather "what message are you trying to send us through displaying or omitting it?"
The public message is that "someone who has had a profound impact on YC, or HN and its members has passed away". That impact is up to the operators to decide.
Much the same way a family remembers their loved ones, but doesn't grieve every important person who has passed away irrespective of their stature. Like parent was saying, it's a little discomfiting to see this discussed on threads.
I really think all but a handful of HN readers first learned about Mirzakhani upon her passing. It's a tragic event -- highly so -- but not one of particular significance to this community.
I seem to recall feeling similarly a few times when we did have black bars. Sadly I don't remember who they were (which I suppose vaguely confirms what I'm saying?) but I do remember being surprised that I hadn't heard of them before, and didn't expect many had either.
Isn't it generally reserved for important figures in the history of computing/the tech industry? I don't think it takes away from the magnitude of her accomplishments or the tragedy of her early death to say that she was a giant in her field, but not really in our field.
That's a good point, I don't recall everyone who's had a black bar, though the few I do were in tech. I guess it would be an odd list to keep, but if anyone happens to know of one that exists, I'd be curious to see the list of the past N people we've had black bars for.
Edit: I just did some digging around and it seems they didn't have a bar for Neil Armstrong's death either, so this seems right on the mark so far.
I don't agree with gp's inflammatory tone, but do you not think a Fields medalist is not worth a black bar? I can buy not agreeing with tokenism, but her merit stands on its own imo. We lost a great mind; nobody's saying we should be in sack cloth and ashes but I think it's worth recognizing in some manner.
That said, I understand how HN is not really a Math/CS forum, it's more about industry.
I'm referring to the fact that she was the first female medalist. That's not enough to merit a community that usually doesn't bb mathematicians to make an exception - that would be tokenism.
Please be charitable when commenting here. That means responding to the strongest plausible interpretation of what someone might mean, instead of a weaker one that's more offensive or ridiculous.
Sorry if I came off that way, that's not at all what I meant. My comment was more along the lines of putting a face to statistics; reading about the numbers of people dying is a weaker motivator than hearing the stories behind the lives that are lost to cancer.
Hey! There's no reason why you can pretend it's wrong or offensive that someone has a preference or fondness for one person over others. There's no rule that says people ought to like each other the same, and nothing wrong that they don't. Don't be so narrow minded, and try to make someone's joy or love wrong! You doing that is the wrong thing and must be called out. How dare you, seriously! On the news of someone's death to say that another expressing their sincere sentiment toward that is not good enough because it didn't include a reference to all other suffering. You pretend to be sticking up for suffering, but by trying to rob legitimacy from a subjective appreciation of one instance you rob it from all. Do you not see! Ask not for whom the bell tolls. The whole point of feelings is they are subjective. You have no power to judge these people and their emotions have nothing held over them by your scroogelike pronouncements. Be gone from here abject troll or cleanse your vile mouth before next you dare to speak!
- Random wanderer leaping to the defence of a fellow human.
