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1、Advice to the beginnerA. ConnesColl ege de France, Institut des HautesEtudesScientifiques and Vanderbilt University.Mathematics is the backbone of modern scienceand a remarkably efficient source of new concepts and tools to understand the “reality” in which we participate. The new concepts themselve
2、s are the result of a long process of “distillation” in the alem- bic of the human thought. I was asked to write some advice for young math- ematicians.The first observation is that each mathematician is a special case, and in general mathematicians tend to behave like “fermions” i.e. avoid working
3、in areas which are too trendy whereas physicists behave a lot more like “bosons” which coalesce in large packs and are often “over- selling” their doings, an attitude which mathe- maticians despise.It might be tempting at first to view mathemat- ics as the union of separate parts such as Geome- try,
4、 Algebra, Analysis, Number theory etc. wherethe first is dominated by the understanding of the concept of “space”, the second by the art of ma- nipulating “symbols”, the next by the access to“infinity” and the “continuum” etc. This however does not do justice to one of the most essential features of
5、 the mathematical world, namely that it is virtually impossible to isolate any of the above parts from the others without depriv- ing them from their essence. In that way the cor- pus of mathematics does resemble a biological en- tity which can only survive as a whole and would perish if separated i
6、nto disjoint pieces.The scientific life of mathematicians can be pic- tured as a trip inside the geography of the “math- ematical reality” which they unveil gradually in their own private mental frame. It often begins by an act of rebellion with re- spect to the existing dogmatic description of that
7、reality that one will find in existing books. The young “to be mathematician” realize in their own mind that their perception of the mathematical world captures some features which do not quitefit with the existing dogma. This first act is often due in most cases to ignorance but it allows one to fr
8、ee oneself from the reverence to authority by relying on ones intuition provided it is backed upby actual proofs. Once mathematicians get to re- ally know, in an original and “personal” manner, a small part of the mathematical world, as esotericas it can look at first1, their trip can really start.I
9、t is of course vital all along not to break the “fil darianne” which allows to constantly keep a fresh eye on whatever one will encounter along the way, and also to go back to the source if one feels lost at times. It is also vital to always keep moving. The riskotherwise is to confine oneself in a
10、relatively small area of extreme technical specialization,thus shrinking ones perception of the mathematical world and of its bewildering diversity.The really fundamental point in that respect is that while so many mathematicians have beenspending their entire scientific life exploring that world th
11、ey all agree on its contours and on its connexity: whatever the origin of ones itinerary, one day or another if one walks long enough, one is bound to reach a well known town i.e.for instance to meet elliptic functions, modular forms, zeta functions. “All roads lead to Rome” and the mathematical wor
12、ld is “connected”. Of course this is not to say that all parts of mathematics look alike and it is worth quoting what Grothendieck says (in “R ecoltes et semailles”) in comparing thelandscape of analysis in which he first worked with that of algebraic geometry in which he spent the rest of his mathe
13、matical life:“Je me rappelle encore de cette impression saisis- sante (toute subjective certes), comme si je quittais des steppes arides et rev eches, pour me retrouver soudain dans une sorte de “pays promis” aux richessesluxuriantes, se multipliant a linfini partout o u il plait a la main de se pos
14、er, pour cueillir ou pour fouiller.”Most mathematicians adopt a pragmatic atti- tude and see themselves as the explorers of this “mathematical world” whose existence they dont have any wish to question, and whose structure they uncover by a mixture of intuition, not so for- eign from “poetical desir
15、e”2, and of a great deal of1my starting point was localization of roots of polyno-mials, but I was fortunately invited at a very early age in a conference in Seattle where I found the roots of all my future work on factors. 2as emphasised by the French poet Paul Valery.12rationality requiring intens
16、e periods of concentra- tion. Each generation builds a “mental picture” of their own understanding of this world and con- structs more and more penetrating mental tools to explore previously hidden aspects of that reality. Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were previously believed to be very far remote from each other in the natural mental picture that a generation had elaborated. At that poi
