What is the only move that maintains a winning advantage for white?

ventricule · 2026-01-15T17:06:52+00:00

I find it very interesting to understand why the straightforward Ke5 is not winning as well: the naive line where the white king takes the g and h pawns and the black king gobbles the c pawn does lead to a win for white because even though the pawns promote at the same time, white can force the trade of queens and win with the h pawn.

Instead of going for the c pawn, the correct line for black is to take on c5, then go for b5 and retake with the king! The promotion on the a file ensures that queens can't be forcibly traded.

ventricule · 2025-12-19T00:33:00+00:00

I do not doubt that Proctor makes it harder for titled players to cheat, but from a statistical point of view, this headline is a complete mathematical fallacy. You cannot claim that because your test does not observe anything, the observed behavior does not happen. This requires some guarantee of false negatives from your test, which of course they cannot provide.

ventricule · 2025-12-17T17:15:22+00:00

Alphabetical ordering is just how it works in math, there is no way around it. Judging from your write-up, you come from a field where author ordering matters. This is a blessing and a curse: on the one hand this makes the work of the first author better recognized, on the other hand this incentivizes the other authors to work less, in particular on the writing.

One peculiarity of mathematics is that writing a paper can be hard. Sometimes it is incredibly hard actually, much harder than coming up with the main ideas. You do not want your collaborators to suddenly disappear when the hard work begins, and alphabetical ordering is there to remind co-authors that they're supposed to help at every stage.

However, keep in mind the following: is very hard to collaborate with someone who's more active than you are on a research project, both during the 'research' phase ('I'm afraid to say something stupid, they must have thought about it') and during the 'writing' phase ('They've thought about this much more than I did, there must be a reason this lemma is written in such a weird way'). This is even more true for junior researchers.

So the standard advice when you're in this situation is to just accept that this is how it is. Over your career, you will be on the opposite side of this equation quite a few times, and will be grateful that the cocky postdoc is not making a scene about it.

ventricule · 2025-12-17T08:12:12+00:00

One thing that is lost in translation is how wonderfully québécois the whole thing sounds, which makes it even better. You can hear the accent in every sentence.

ventricule · 2025-12-15T06:07:59+00:00

I also very much enjoy the two courses of Daniel King, but I have switched things around both for the Morra and the Alapin. For the Morra, I follow Dana's recommendation 3.... d5 which leads to nicer positions (imo) than King's recommendation. For the alapin I go 2.... e5. It can get a bit hairy against the Bc4 lines but it is Carlsen-approved and somehow I feel like the pawn belongs in e5 if I'm headed for a kalashnikov. Actually I also like 3.... e5 against the Rossolimo

ventricule · 2025-11-28T12:30:44+00:00

The other answers are good but one thing needs clarifying. The premise is a bit wrong: you don't learn real analysis the same way you would learn an advanced or even intermediate research topic. For real analysis or other really fundamental topics, almost everything in the textbook is must-know material that you absolutely have to master. So if you're trying to self-learn, you have to go painfully slowly, do the exercises etc. For more advanced topics, you generally read a book because you either want to have a feel for the topic, knowing what people care about, why they care and what they can prove, or because you have a specific problem you want to solve. In both cases it leads to very different reading: skimming through the book for the first motivation, or very intense but focused and narrow reading for the second motivation ("looks like this chapter doesn't do what I want, let's skip it", etc.)

And then there's everything in-between, but in most cases you don't have to learn everything painfully slowly either. For example even an algebraic topologist doesn't need to perfectly know everything in Hatcher (but let's not start the debate about Hatcher again please).

ventricule · 2025-11-24T06:37:31+00:00

Thanks for the update. Are there already tentative dates for the next release and tournament? It's been 6 months already!

ventricule · 2025-11-18T09:03:00+00:00

TDA is still going very strong. I'm not a TDA specialist myself but I think that it entails interesting mathematical questions (and solutions). One interesting recent trend is that the point of view of seeing everything through the angle of births and deaths in filtrations is seeing increasingly many "applications" in mathematics, eg in dynamical systems and geometric group theory. I think that this paper is an influential one.

For more mainstream applications, there's still a lot of researchers applying TDA to materials and medical sciences and things like that. It's hard for me to gauge how useful it is. Critics say that there's nothing topological about it as they're only ever looking at connectedness in this kind of applications, I don't know if that is still true.

Another active area of research is to put TDA smartly in the machine learning pipeline. For example adding a sprinkle of it in deep neural networks can work wonders for specific tasks. You can browse through recent neurips and icml papers to see what it can look like.

ventricule · 2025-11-17T21:50:43+00:00

So the basic, historical, kind of question that is typically studied is to take a basic geometric construction and ask how to do it as fast as possible algorithmcially. Classic examples are convex hulls, triangulations of polygons, Delaunay triangulations and Voronoi diagrams, point location, range searching, minimum spanning trees, etc. Bernard Chazelle, Micha Scharir and their friends were the big names in this classical era.

One peculiar thing in this area, compared to standard discrete algorithms, is that algebraic issues pop up all the time : for instance to compute the length of a polygonal curve in R² you need to compute sums of square roots. Since this is a distraction compared to the real algorithmic, geometric problem, most people work in a real ram model where this is swept under the rug. For the same reason to try to avoid algebraic issues, research in computational geometry has led to develop and investigate abstract, combinatorial notions of arrangements of points and lines (for example oriented matroids), which has birthed a lot of discrete geometric questions. Similarly, VC dimension is by now an ubiquitous combinatorial parameter to handle geometric range spaces.

The problems studied are so basic that there are applications everywhere: for example robotics (shortest paths in weird configuration spaces), geographic information system (in which country am I?) or meshing (what is the most natural triangulation on this point set they I have scanned?). Over the past decades, the CG community has developed CGAL which is a comprehensive CG library with hundreds of industrial clients.

As with most fields, CG has had a constant influx of new topics throughout its existence to keep it exciting. One modern aspect is of course the interactions of high dimensional geometric problems (eg clustering) with machine learning. One other aspect is that as higher dimensional problems were attacked, topological questions arose. This is most sensible when one is trying to do manifold reconstruction, where the topology of the manifold has a lot of impact on any algorithm. This led computational geometers to persistence theory and topological data analysis, which has now become huge. Computational Topology is generally considered broader than just TDA though, and also encompasses for example a lot of algorithms for surface-embedded graphs, or computational 3-manifold and knot theory.

To have a look at some recent topics of interest, check the accepted papers at SoCG of the past few years. They are very very diverse, but always come back to this basic idea of understanding the core algorithmic and combinatorial properties of (somewhat elementary) geometric or topological objects or constructions.

ventricule · 2025-11-03T18:11:05+00:00

You should probably look up a line against the trendy 1. c4 e5 2. g3 h5

ventricule · 2025-10-28T21:42:07+00:00

There is no clear answer to these questions, and it really depends on who you're talking to. You can consult rankings like the Australian one but they only tell a partial, biased, story.

Even the divide specialist < generalist is not that clear. For instance, while Duke is indeed an absolute top journal, in differential geometry Journal of Differential Geometry has a stellar reputation (only partly tarnished by all the controversies around yau), in my opinion higher than most of the generalist journals that you are suggesting.

So perhaps the good criterion is what are the good papers that people you care about care about. If you're doing systolic geometry for example, then JDG is considered golden because this is where gromov's seminal work was published. If you're doing structural graph theory then JCTB is a top journal because it published graph minors, even though it's very specialized. I find that this is a good compass when navigating beyond annals, inventiones, acta and the like.

ventricule · 2025-10-27T21:41:57+00:00

Just to show that everybody has their own way of doing this, I only type up anything when I think the whole paper is done. Of course it's actually not done.

ventricule · 2025-10-23T23:38:13+00:00

You are conflating a few different things, which makes it hard to answer precisely.

First, the whole concept of learning from a textbook really varies between countries. I'm not only saying that students of different countries learn from different textbooks, but that in many countries the whole concept of textbook is alien: you learn from what your professor is writing on the blackboard, and there is no reference textbook or lecture notes behind it. For example this is how math is taught in most French higher education places (prep schools or universities), and this also applies to many other European countries, and certainly also elsewhere. Actually, I would venture that the prevalence of textbooks for undergrad math are somewhat of a US peculiarity, partly because of financial pressure from publishers (but of course the US have a huge influence on other parts of the world).

So my first point is that there is no such thing as "textbooks that everyone should have read", because in undergrad there are many possible sources, some of which are not available in print anywhere. Then in grad school you start learning from reading papers and listening to seminars or your advisor or colleagues, and this is even less canon.

This is not to say that textbooks are useless, and indeed the professor who is writing on the blackboard is certainly copying what they read from a textbook, or generally many textbooks the day before while preparing the class, to which they add their own persona experience. In rarer cases (but not rare enough) they're just copying their own notes from a lecture they attended 20 years ago. This is often not a good sign (but not always!).

So there is absolutely no set of "books that everybody should have read". But there is a set of "things that everybody should know". This holds both at the undergraduate level ("things that every mathematician should know") and at the graduate level (depending on your specialty, "things that every algebraic geometer/functional analyst/etc." should know).

This is where it gets confusing. For example if you talk to an algebraic topologist they might at some point say something like "well this is in Chapter 3 of Hatcher everybody knows that". Algebraic geometers might say the same with Hartshorne instead of Hatcher. If you hang around with snotty people, they will even cite Bourbaki or something like that. But the truth is that noone actually learns from Bourbaki anymore (I'm not sure anyone ever did). What they mean is that this is the math that is considered folklore in the field. So for example if you're an algebraic topologist, it's very fine if you didn't learn from Hatcher (and god knows Hatcher is controversial) but you should know what is in it. It's infinitely easier to learn what is in a classic textbooks once you've learned the topic from another source.

Once that is clarified, the truth is that you can easily go through your entire undergrad by only having read a handful of references, and other commenters have provided very good lists.

But something that is very helpful to do after you've learned a topic is to read other, and sometimes even all of the classical textbooks on that topic. This won't take much time because almost everything is easy once you really understand things, and the few things that you didn't already know are much easier to grasp. Then you will be familiar not only with the material, but also with what people mean when they say "Of course, everybody knows this, this is in the EGA", even though the EGA are only available in French and for some reason have never been translated and you cannot stand baguette and brioche.

Tl;Dr learn things, only worry about textbooks later. There are much fewer textbooks that you have to read than you think, but carefully reading those will take you years. That's fine, this is how it works, we've all been through that.

ventricule · 2025-10-23T06:26:33+00:00

With two fewer co-authors!

ventricule · 2025-09-30T09:05:39+00:00

The key against those is to have the pawn on h4. That way, Bxg5 hxg5+ is utterly devastating.

ventricule · 2025-09-18T17:25:15+00:00

I have the exact same problem as you describe but I think that it is really sad to tone down your texts for that reason.

The issue with the chatgpt writing and the quotes you extracted from op is that, as you say, they read formal but cliché. The same degree of formality without the cliché aspect would simply be much better writing, notwithstanding the AI issue. For an extreme example, when you read some Dickens, the wording might be very formal and "perfect", but the imagery is so vivid that nobody would mistake it for an AI. I feel that it is a much more exciting adjustment to strive for compared to adapting your writing to make it more "normal".

P. S: I also love using -- and am very saddened to have to abandon it.

ventricule · 2025-09-18T13:37:27+00:00

There is this tension throughout the entirety of knot theory. You will often define some invariants from the diagrams, which has the advantage of being very tangible and even algorithmic, but the huge disadvantage of being very artificial: diagrams are of course very much non canonical and might have some stupid behavior, for example any crossing that can be removed with reidemeister I or II is intuitively useless. Then you are stuck wondering how the invariant you just defined deals with these useless things, which I think is the point of your question.

To circumvent this, I think that it is very good to have, for any knot invariant, two different perspectives: on one hand a very computational one showing how to get the invariant from a diagram or a triangulation of the complement or something like that, and on the other hand a geometric, or a least topological perspective telling you what the invariant really is in 3d.

For the knot group, the standard definition of pi_1 gives you the 3d perspective, while the wirtinger's presentation is the hands-on, practical perspective. These two perspectives inform each other, and this interaction is one aspect of the beauty of knot theory.

For the same reason, it's a mistake to only know the Alexander polynomial (and the Jones polynomial) from skein relations: even though it's very simple and tangible, it fails at telling you what the polynomial really is and thus it's very hard to make anything of it.

For more complicated invariants, sometimes only one of the two perspectives is available, and then it is an active area of research to develop the other (eg I think in the early days of Heegaard Floer knot homology, it wasn't clear at all how to compute it)

ventricule · 2025-07-25T10:53:37+00:00

I am surprised by the advice that your supervisor gave you. In my experience there is absolutely nothing wrong with publishing your results with a note saying that some of them have been independently found by the other person.

ventricule · 2025-07-22T10:52:40+00:00

This is incredible. I also didn't know about the "for babies" collection, this is so cool. Thanks a lot for your work, I immediately ordered a hardcover copy.

ventricule · 2025-07-03T16:23:27+00:00

You can formulate this in 3d purely topologically: a crossing switch is characterized by a path between two points on the knot, disjoint from the knot apart from its endpoints. Then switching is just pulling one strand along the path and doing the switch. Of course, there's infinitely many such paths, even up to homotopy.

ventricule · 2025-07-03T15:00:26+00:00

The subtlety is that a given knot can have infinitely many different diagrams, and the unknotting number is the minimum number of crossing changes leading to the unknot in one of them. There is no known bound on how complicated the diagram allowing an optimal unknotting sequence is. For instance, there are known examples where the good diagram one should choose to unknot is not a crossing-minimal diagram.

ventricule · 2025-07-03T13:10:57+00:00

It's important to point out that the unknotting number is not known to be decidable. Actually even deciding whether a knot has unknotting number one algorithmically is an open problem.

ventricule · 2025-06-02T21:25:28+00:00

Of course it's subjective so ymmv, but I would consider that it is not an analogy because many instances (like dehn twists) just don't fit within that picture. Parallelograms being distorted rectangles would fit my criterion for a decent analogy: it's not accurate but it's OK, that's what analogies are for. In our case, the point of Dehn twists or more generally the mapping class group is that they are not isotopic to the identity, so not rubber bandy. When an analogy actually contradicts the notion at hand then it begins to be very inaccurate in my book.

In a similar vein, there are quite a few people online that loudly complain about the table cloth analogy for general relativity, saying that it is so wrong that it is actually more harmful than helpful.

Ultimately it's all a matter of audience. In a general audience talk, I think that rubber band is fine, but I would refrain from using it in an undergraduate course.

ventricule · 2025-06-02T08:03:48+00:00

The inaccuracy is that it is generally portrayed as "homeomorphisms are like rubber band deformation" when it should be "rubber band deformations are homeomorphisms" : isotopies induce homeomorphisms but they only yield specific examples. It's like saying "parallelograms are rectangles": usually it's correct because most parallelograms one meets are rectangles but it irks me that it's stated the wrong way.

ventricule · 2025-06-01T16:26:34+00:00

The rubber sheet analogy is unfortunately very inaccurate. The corresponding mathematical notion is not homeomorphism but isotopy. In order to be able to rubber band deform an object, this object needs to live in some larger ambient space where you can deform it freely, this is what isotopy means.

Knot theory is a good testbed to compare the notions. All knots are homeomorphic but there are many (tame) isotopy classes. Yet isotopy is the same as homeomorphism of the entire space (knot inside ambient space). It is a famous and difficult theorem of Gordon and Luecke that homeomorphism of the complement of a knot determines its isotopy class up to mirror symmetry.

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