List of proposed topics

These are the possible topics that we could study during the 2008/2009 season. See also the List of proposed problems.

Feel free to expand, comment, ask…

Topic 1: Tropical geometry

From wikipedia: "Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry."

For an introduction to the tropical geometry of plane curves see http://www.mathematik.uni-kl.de/~gathmann/pub/trop-0601322.pdf .

  • Sounds interesting, and it seems to be in fashion these days. — David
  • I am not in favour of this topic (though it is a nice one), since too many people jumped on it meanwhile and we should rather learn concepts which we daily need. — hh
  • If we have several shorter topics I vote for tropical geometry, since as a mainstream topic we should at least know a bit about it. i agree that we might not benefit from a full semester course in trometry — C

Topic 2: Clifford algebras, pin and spin

In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry and theoretical physics. Pin and Spin groups are also used in robotics and theoretical physics.
wikipedia entry: http://en.wikipedia.org/wiki/Clifford_algebra

  • This is a topic Markus and I wanted to look at, at some point, because of the applications in particle physics and robotics. — Brian
  • Even though interesting, this topic seems to me too far away from the interests of the majority. - hh

Topic 3: Algebraic geometry in robotics

Several topics could be investigated. For example, cuspidal robots are robots that can change posture without meeting a singularity [Wenger:2007]. The study of flagged redundant manipulators using flagged manifolds [Alberich-Carraminana].

  • would be a good counterweight for too theoretical considerations — C

Topic 4: Lie groups and Lie algebras

General introduction and applications to algebraic geometry, assuming that nothing is known about the topic.

  • maybe we should not go to deep into this theory, just explain what a lie algebra is and an application in algebraic geometry (eg pmz of rational normal curves?). in this case see Topic 13 :)
    • Or maybe we should go deeper into this, it is an important topic in itself that can have many applications and that everyone should probably know about beyond a definition. — David

Topic 5: Sheaf Cohomology and its applications

Continuing from the last semester seminar we could try to understand/learn the algebraic/analytic side of cohomology theory (see eg ueno, hartshorne, griffiths-harris)

  • Continuation of the last seminar has some profits, because cohomology is widely used and everybody should know the fundamentals — but nevertheless it is not my favourite subject. — hh
  • Since we have put so much work into getting a geometric picture of homology and cohomology it seems reasonable to exploit our knowledge and try to understand now the more abstract and more algebraic theory which is widely used in algeom. but i got the impression that the interest for this topic in the last semester was pretty low, strange — C

Topic 6: Singularity Theory

Learn about different approaches to study singularites (algebraic, topological, analytic, …) and its invariants; probably have a look at classification problems (see for e.g. Milnor, Arnold, amm).

  • since "sing theory" has so many flavours we could choose topics suiting many seminar participants — C

Topic 7: Deformation Theory

Learn the basics of analytic and algebraic deformation theory

Topic 8: Moduli Spaces

What are moduli spaces and why are they useful? What is the Hilbert Scheme?

  • i think Topic 9 is prerequisit — niels

Topic 9: Schemes

Sheafves and schemes from the beginning.

  • To begin with, we will learn there how to spell the plural of sheaf. - hh
  • i think Topic 13 is prerequisit — niels
    • it is useful to have some knowledge of basic differential geometry to have more motivation for certain abstract definitions, but topic 13 is not at prerequesit to schemes/sheaves — C
    • Agreed with C — D
    • R. Hartshorne and D. Mumford also agree :), anyway, it is an opinion. — niels

Topic 10: GAGA

This is Serre's famous article from the fifties where he establishes the basic relations between algebraic and analytic geometry (e.g., when is a regular map which is an analytic isomorphism also biregular). Very beautiful material and also part of the "basics". — hh

wikipedia entry: http://en.wikipedia.org/wiki/GAGA#Serre.27s_GAGA

  • beautiful but probably not interesting to a majority — C
  • since part of basics, i would suggest to spend at least some time on it —niels

Topic 11: Atiyah-Singer index theorem

To understand the statement, and maybe for very simple special cases, the proof. Understanding the statement needs some preparation, namely connections and characteristic classes in order to define the topological index. 2-3 talks.

  • characteristic classes can also be used for defining intersection product on cycles. —niels
  • one possible definition for connections uses the lie bracket: see topic 4 —niels
  • We will need to make sure that everybody knows the prerequisites first — David
    • I think Josef wants to withdraw this topic, since it seems a long, stony and not too benefitting way to the the index-theorem — C

Topic 12: Resolution of Singularities in characteristic 0/p

This topic has preparatory character for the upcoming conference on resolution in kyoto, december 08, which will be attended by a good part of the working group. The idea is to have a look at recent ideas developed in the field by kawanoue, matsuki, hironaka, villamayor, … .
Apart from that it is a topic our project leaders are well-known experts for …

  • What is the relation between this and the proposed above "Singularity theory" in terms of their content? — David
    • in short: in singularity theory you can spend 4 lives, in resolution just 2. To be more concrete: With Topic 6 i suggest to learn more about ways to describe singularities (topological, algebraic, analytic, biological,…) and their properties. one way to look at sings is via resolution. in topic 6 i think we should use resolution "just" as a blackbox — which works fine for many many algebraic geometers — and here in topic 12 the resolution process itself is at focus — C

Topic 13: Differential geometry

sub topics:
- manifolds (fibre bundles,tensors, flows, derivative, integration, lie groups+algebras)
- riemannian manifolds (connection, curvature, isometry, killing vectorfields)
- complex manifolds (complex structure, hermitian -, kaehler -, when is a manifold algebraic?)

  • the last question is specific enough to give a reasonable dimensioned topic for the seminar — C

- cohomology (de rham-, dolbeault-, sheaf-)

  • voila! —niels

- hodge theory (understand a bit what Hodge Conjecture is about :))

  • the vienna group got some ideas, better said "impressions", by the katzarkov lecture. i would strongly advise to learn a bit more about cohomology and its value first. but it might be a good extension to more cohomology. of course for those not interested in cohomology this is … — C
  • i think we learned enough about cohomology to start with more concrete applications of it. For me this is more inspiring. If we need something along the way (eg kunneth formula) we can spend time on it anyway. —niels
  • actually i was thinking exactly about applications of the theory as it can be found in hartshorne, griffiths-harris, …; but since formulated as sheaf cohomology we should first revisit known cohomology concepts for sheaves. this is not too much of an effort. — C

- characteristic classes (de rham and dolbeault cohomology, chern class, intersection theory)

  • this topic is quite interesting and will probably take not too much time to understand — C

proposed literature:
- wikipedia
- Nakahara: geometry, topology and physics (though with care and only till chapter 11)
- griffiths and harris: principles of algebraic geometry (assumes that you already understand the material)
- Jaenich: Vector analysis (for integration and de rham cohom. over reals, hodge theorem)
- Lee: Introduction to smooth manifolds
- Hatcher: algebraic topology


comments:

  • i would propose to fix the topic but not the literature. —niels
  • All this content would take us the whole year. If this is broken down into pieces then we can study some individual sub-topics. — David
    • i would propose not to spend too much time on each topic, e.g. the first subtopic manifolds could be done in 1 seminar. We could do some exercises using the concepts as given, but don't discuss too much the definitions together. —niels
      • of the list in the "manifolds" subtopic, I know next to nothing about most of the things mentioned. I may not be the only one. I cannot see how to do that in less than a few seminars. — David
      • i agree to specify smaller, but still vital, topics — C
      • could you specify what you mean with this remark? i guess what you mean is that we should skip some of the concepts, e.g. flow? it seems to me that the topics are specified as small as reasonable possible. —niels
  • griffiths-harris provides a link between differential geometry and algebraic geometry. For example we could choose from the following items to test our knowledge of proposed topics:
    • divisors/sheafs/linebundles/associated map/linear system
      • ok — C
    • relation between ample and metric (ie positivity) (see also Lazardsfeld)
      • hm — C
    • sheaf cohomology
      • yup — C
    • adjunction formula
      • ah, yes that i know — C
    • kodaira vanishing theorem
      • hups — C
    • lefschetz theorem on hyperplane sections
      • umh — C
    • index theorems (p126) —niels
      • gng — C
      • stands for "Go or Not to Go", but also for "Galactica Next Generation", which sounds pretty cool to me! —niels
  • griffiths-harris provides an obstacle to learning algebraic geometry/ diffgeom if you don't have yet a good amount of background knowledge or a very large stomach. i'm not sure that everybody will feel happy with the book — C
          • i am also not so happy with the book, but i think that it gives pointers to interesting topics though. it is the only book i know which takes a more diff. geo. approach to alg. geo. We should find more suitable literature to understand it. —niels

Topic 14: Derived Categories

Definition, Motivation and use for this very abstract but "modern" topic.

Topic 15: Divisors

Learn and understand the basics about Weil- and Cartierdivisors, ample and very ample divisors; connection to invertible sheaves; divisor class group and picard group… and of course the Riemann-Roch Theorem.

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License