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Szűcs András topológia jegyzete: letöltés

Juhász András topológia sáv írott jegyzete: [1] [2] [3] [4]

Back to the drawing board

Allen Hatcher's page. Okay, how can I not start by this one? The guy's written darn good books about algebraic topology and he put them up for everybody to use for free! I say, that deserves at least a visit to his homepage (and chances are, you will be staying there for quite a while).

Self-promotion is allegedly an important pastime. So eventually a list of my papers should be linked here as well. If you're still reading and are actually interested, here's a short summary:

• ELTE TTK maths MSci theses have mine, check under year 2004. It's about some silly messing with level sets (point preimages) of a somewhat weird class of mappings. There are some really nice results about them (e.g. this one), and thankfully there are also relatively accessible (if practically unusable) problems that help young people construct their theses.
• Cobordisms of fold maps and maps with prescribed number of cusps, contribution to the work of András Szűcs and Tobias Ekholm. This has been published in Kyushu Journal of Mathematics 61/2 (2007), pp. 395—414, and it's about observing how do the cobordism groups of manifolds or mappings change if we restrict ourselves solely to "nice" maps having only regular and fold points.
• Cobordisms of fold maps of 2k+2-manifolds into R^3k+2, Geometry and topology of caustics, Banach Center Publications vol. 82 (2008), pp. 209—213 (proceedings of the Caustics '06 symposium). Same thing, in another dimension, when the calculations are not quite as simple and easy since the objects to avoid are whole curves of cusps and not just discrete points.
• Calculation of the avoiding ideal for Σ^1,1, Algebraic Topology — Old and New, Banach Center Publications vol.85 (2009), 307—313. Another path for attacking the same problem, and in the complex setting it actually works all the time. In the real setting, well, fold maps is the most complicated case where it still works that I'm aware of.
• Bordism groups of fold maps (joint with András Szűcs), Acta Mathematica Hungarica 126/4 (2010), pp. 334—351. The concept of classifying spaces is what makes singular cobordisms conceptually very nice, one can translate singular cobordisms to homotopies in the classifying space. The one tiny problem with this is that classifying spaces are hard to construct, and after construction we need to calculate their homotopic properties, which is typically hard in itself (just think about the homotopy groups of spheres, and spheres are not complicated to construct). So here we do some calculations in the smallest classifying space that has not been understood by the great figures of the past, who did consider the classifying space for immersions. We can only do homological calculations, but at least those do work and yield some geometric results.
• Fibration of classifying spaces in the cobordism theory of singular maps, Proceedings of the Steklov Institute of Mathematics vol. 267 (2009), pp. 270—277. I am fond of the result of this paper, because it is a geometric proof of a very useful fact that previously only had a quite contrived proof. The statement is that when one constructs classifying spaces for a set of singularities and then adds exactly one other monosingularity, then there is a fibration involving the two classifying spaces and the space that classifies the "new" singular locus as a decorated immersion. The use of this lies in the fact that it allows calculations with homotopy groups, which fibrations handle nicely, of classifying spaces, whose standard construction is a pile of gluings and thus not conductive to homotopic calculations at all. Additionally, making the proof simpler also allows extending it to settings where the original proof didn't work, so good things all around.
• Calculation of the obstruction ideals of Morin maps, Periodica Mathematica Hungarica 63/1 (2011), pp. 89—100. Here we get a set of relations among the characteristic classes of the virtual normal bundle of a map with only some Morin singularities, that is, relatively easily computable obstructions to the existence of a deformation of a given map without sufficiently ugly singularities. This set is actually maximal if we consider not only honest maps, but also fiber-preserving bundle maps. The drawback is that we only look at cohomology with modulo 2 coefficients, and there's the slight aesthetic problem of these obstructions being mostly due to the exclusion of non-Morin singularities. I don't know how to enhance the calculation to address maps that only avoid swallowtails (Σ^1,1,1), for example.
• On bordism and cobordism groups of Morin maps (joint with Endre Szabó and András Szűcs), Journal of Singularities, vol. 1 (2010), pp. 134—145. A pretty technical paper, we get lucky calculating the rational homology of the classifying space of Morin maps, and consequently get results for rational homotopy groups and the rational cobordism groups.
• Proof of a conjecture of V. Nikiforov, Combinatorica 31/6 (2011), pp. 739—754. Nothing to do with topology, although the inspiration did come from Morse theory and I would have liked to push that analogy further than I finally managed to. I heard a friend of mine talk about the conjecture, and under the influence of thinking about dense graph limits I had the idea to work directly on the optimal limit of the question, and there to test optimality locally under reparametrisations. It's slightly annoying that in the end, one still needs to sift through a (luckily small-dimensional) family of candidates, and this causes the method to essentially fail in all the generalisations that I've tried so far.
• Relations among characteristic classes and existence of singular maps (joint with Boldizsár Kalmár), Trans. AMS 364 (2012), pp. 3751—3779. Here we look at negative codimensional maps (so the dimension of the source manifold is larger than the dimension of the target), which are Boldizsár's speciality, not mine. The main idea is to get calculable cohomological obstructions by first eliminating the singularities of the fold or Morin map — by blowing up the source along the singular set and perturbing the result — and then tracking the restrictions obtained from now having a nicer map to work with. There's a surprising amount of concrete calculations that can be done that way, some may be doable in positive codimensional setting as well.
• Large 2-coloured matchings in 3-coloured complete hypergraphs, submitted to Electronic Journal of Combinatorics. This is another paper far from my main competence, I attended a combinatorics workshop, encountered the problem and it stuck in my head. The proof I found is not very enlightening, but I harbour a faint hope that one day similar problems may be attacked with the dense graph limit approach. That would be awesome.
• Singularities and stable homotopy groups of spheres I (joint with Csaba Nagy and András Szűcs) and Singularities and stable homotopy groups of spheres II (joint with András Szűcs) apply the classifying space machinery in a case when the classifying space does not become mind-bogglingly complicated. For that, one needs singularities with few automorphisms, so we start with codimension 1 Morin maps, whose symmetry group is C₂×C₂, require them to be cooriented (reducing the symmetry group to a single C₂), and finally asking for a trivialization of the kernel bundle, eliminating homotopically nontrivial symmetries altogether. From another point of view, these are exactly the projections of immersions into one higher dimension, and this allows us to identify the gluing maps of the classifying space (which describe what the less complicated singular points do around more complicated singular points) with maps in a spectral sequence associated to the filtration of the complex projective space CP^∞ by the CPⁿ, and those latter maps have been studied quite a lot.
• Symmetric Shannon capacity is the independence number minus 1 does what it says on the tin: we consider all collections of unlabeled (hence "symmetric") stones on the vertices of a graph with two collections considered neighbours if one can move all the stones at most to a neighbouring vertex to get from one collection to the other, and determine the asymptotics of the independence number of the resulting graph.

Back to the drawing board

Back to the drawing board

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Travel Cost Calculator for Pardus (unmaintained since forever):

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• Arvo Pärt: De profundis
• Soundgarden: Rowing
• Soundgarden: Fell on black days
• Serj Tankian: Empty walls
• Metallica: Bleeding me
• Die Toten Hosen: Pushed Again
• Tool: Right in two
• Tool: Pneuma
• Tool: 7empest

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