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Saharon Shelah (Jerusalem and Piscataway, N J Sh:E70 Plenary speakers answer two questions a complex problem with many parame- open to rigorously explain why this ters and degrees of freedom and allow- crystallization occurs, i.e., demonstrat- ing it to tend to its extreme value. ing mathematically that the arrange- I have been interested in each of ments which have the least energy these questions, but particularly in the are necessarily periodic. To me, this is second and third. For paern forma- a very fascinating question. While it is tion, some specific interests of mine quite simple to pose, we still do not have included finding a mathematical have much of an idea of how to ad- derivation of the appearance of the sur- dress it. prising “cross tie walls” paerns that Some results have been obtained in form in micromagnetics, and to ex- the very particular case of the sphere plain the distribution of vortices along packing problem in two dimensions, triangular “Abrikosov laices” for min- and perturbations of it, in the works of imizers of the Ginzburg–Landau ener- Charles Radin and Florian eil. Prov- gy functional. In the laer case, as in ing the same type of result in higher di- many problems from quantum chem- mension or for more general optimiza- istry, one observes that nature seems tion problem is a major challenge for to prefer regular or periodic arrange- the fundamental understanding of the ments, in the form of crystalline struc- structure of maer. tures. It remains almost completely Sylvia Serfaty Laboratoire Jacques-Louis Lions Universit/’e Paris VI and Courant Institute of Mathematical Sciences [email protected] Saharon Shelah (Jerusalem and Piscataway, N J)* Why are you interested in model theory ply, how to find formulas for ar- (a branch of mathematical logic)? eas of squares, rectangles, trian- gles etc – and the natural sciences , mathemat- looked more aractive. en, enter- I ics looked (to me) like just a ing the ninth grade, Euclidean ge- computational skill – how to multi- ometry captured my heart: from the * We thank J. Baldwin, G. Cherlin, U. Hrushovski, M. Malliaris and J. Vaananen for helpful comments. is is [E70] on the author’s publication list. © Polskie Towarzystwo Matematyczne Sh:E70 Saharon Shelah bare bones of assumptions a magnif- e.c., i.e. elementary classes, explained icent structure is built; an intellectu- below. al endeavour in which it is enough Naturally model theorists start to be right. from the boom: Consider K, an e.c. Undergraduate mathematics was (elementary class), i.e. the class of mod- impressive for me, but algebra consid- els of a first order theory T as ex- erably more so than analysis. Read- plained below. e class K (i.e. T) is ing Galois theory, understanding equa- called categorical in the infinite cardi- tions in general fields, was a gem. nal λ if it has a unique model up to Finding order in what looks like a isomorphism of cardinality (= number chaos, not grinding water but find- of elements) λ. Łoś conjectured that if ing natural definitions and hard the- an e.c. K with countable vocabulary is orems; generality, being able to say categorical in one uncountable cardi- something from very few assump- nal then this holds for every uncount- tions, was impressive. From this per- able cardinal. Aer more than a decade, spective mathematical logic was the Morley proved this, and when I started most general direction, so I took my PhD studies I thought it was won- the trouble to do my MSc thesis in derful (and still think so). mathematical logic; the thesis hap- e point of view explained above pened to be on the model theory of naturally leads to the classification pro- infinitary logics. gram. e basic thesis of the classifi- Model theory seemed the epitome cation program is that reasonable fam- of what I was looking for: rather than ilies of classes of mathematical struc- investigating a specific class like “the tures should have natural dividing class of fields”, the “class of rings with lines. Here a dividing line means a par- no zero divisors” or whatever, we have tition into low, analyzable, tame class- a class of structures, called here mod- es on the one hand, and high, compli- els. For this to be meaningful, we have cated, wild classes on the other. ese to restrict somewhat the class, first by partitions will generate a tameness hi- saying they are all of the same “kind”, erarchy. For each such partition, if the i.e. have the same function symbols class is on the tame side one should (for rings: addition, multiplication; al- have useful structural analyses apply- so the so-called “individual constants” ing to all structures in the class, while 0 and 1, we may have so called predi- if the class is on the wild side one cates, i.e. symbols for relations, but we should have strong evidence of chaot- shall ignore that point; this informa- ic behavior (set theoretic complexity). tion is called the vocabulary). We have ese results should be complemen- to further restrict the classes we con- tary, proving that the dividing lines are sider, and the classical choice in mod- not merely sufficient conditions for be- el theory is to restrict to the so called ing low complexity, or sufficient con- Sh:E70 Plenary speakers answer two questions ditions for being high complexity. is union (i.e. demanding at least one of calls for relevant test questions; we two conditions, logically “or”) and in- expect not to start with a picture of tersection (i.e. “and” ), under comple- the meaning of “analyzable” and look ment and lastly we close under pro- for a general context, as this usual- jections, which means “there is x such ly does not provide evidence for this that…”; but we do not use “there is a being a dividing line. Of course, al- set of elements” or even “there is a fi- though it is hard to refute this thesis nite sequence of elements”.e way we (as you may have chosen the wrong define such a set is called a first order test questions; in fact this is the nature formula, denoted by φ(x0,..., xn 1). If − of a thesis), it may lead us to fruitful n = 0 this will be just true or false or unfruitful directions. e thesis im- in the structure and such formulas are plies the natural expectation that a suc- called sentences. e (complete first cess in developing a worthwhile the- order) theory Th(M) of M is the set ory will lead us also to applications of (first order) sentences it satisfies. in other parts of mathematics, but for An e.c.(=elementary class) is the class me this was neither a prime motiva- ModT of models of T, that is the struc- tion nor a major test, just a welcome tures M (of the relevant kind, vocabu- and not surprising (in principle) side lary) such that Th(M) = T. Natural- benefit and a “proof for the uniniti- ly, N is an elementary extension of M ated”, so we shall not deal with such (and M is an elementary submodel important applications. of N) when for any of those definition, We still have to define what an e.c. on finite sequences from the smaller (elementary class) is. It is a “class of model they agree. ere are many nat- structures satisfying a fixed first order ural classes which are of this form, theory T”. For our purpose, this can be ranging from Abelian groups and alge- explained as follows: given a structure braically closed fields, through random M, we consider subsets of M, sets of graphs to Peano Arithmetic, Set eo- pairs of elements of M, and more gen- ry, and the like. erally sets of n-tuples of elements of A reader may well say that this set- M, which are reasonably definable. By ting is too general, that it is nice to this we mean the following: start with deal with “everything”, but if what we the family of the sets of n-tuples satis- can say is “nothing”, null or just dull, fying an equation (or another atomic then it is not interesting. However, formula if we have also relation sym- this is not the case. e classification bols). ose we call the atomic rela- program has been successfully done tions. But we may also look at the set of for the partition to stable/unstable and parameters for which an equation has further subdivisions have been estab- a solution. More generally, the set of lished on the tame side for the fam- first order definable relations on M is ily of elementary classes. Critical di- the closure of the atomic ones, under viding lines for the taxonomy involve Sh:E70 Saharon Shelah the behavior of the Boolean algebras der can we really dispense with it? of parametrically first order definable See § . sets and relations, i.e.: φ(M, a¯) := Dually, we may feel that as success- ¯b : M satisfies φ(¯b, a¯) . E.g. T = ful as the dividing line stable/unstable { } Th(M), i.e. K = ModT is unstable iff (and finer divisions “below” that) has some first order formula φ(x, y) lin- been, not all unstable classes are com- early orders some infinite set of el- pletely wild, (and what constitutes ements (not necessarily definable it- being complicated, un-analysable de- self!) in some model from K, or sim- pends on your yard-stick). Moreover, ilarly for a set of pairs or, more gen- though many elementary classes are erally, a set of n-tuples.
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