Here are some reflections based on my experience of teaching Ph.D. students, advanced undergraduates (primarily in the MASS program), supervising post-docs, as well as giving numerous research seminar talks, colloquia and other mathematical presentations. I present my thoughts in increasing order of importance of their subject for me.

Teaching a lecture. Delivering a live mathematical lecture is a sophisticated skill whose appreciation is somewhat limited since it requires various degrees of preparation. A single lecture, whether a research seminar, a colloquium for a non-specialized but mathematically advanced audience, or a popular lecture for students or even for general public, must follow certain general principles which are different from those for lectures delivered as parts of a course. Here are some of these:

1. Be aware of your audience's knowledge. Sometimes a single key term used without an explanation loses many people in the audience irretrievably. If I am in the audience and suspect this is happening, I am not afraid to interrupt the speaker and ask a ``naive question''. I also appreciate when people do that in my lectures and try not only to answer but to follow up.

2. Keep listeners motivated. They should feel that what is being discussed is interesting. This helps to keep their attention.

3. Remember that the rate of absorption of new material is finite. Even locally well-presented material may at the end be lost if too much is packed in too little time.

4. Adjust the tempo of your presentation continuously. It is better to omit less important topics in the middle than to cut presentation abruptly and miss a punch line when time expires.

5. If the level of your audience is uneven, try to keep several levels of presentation simultaneously. I remember a beautiful MASS colloquium (with advanced undergraduates from our MASS program as the core and highly appreciative audience) being commended by my young colleague, one of the most brilliant geometers in the world: ''Now I understand what Witten really meant''. (Ed Witten is a great physicist extremely influential and popular in the mathematical community for his unorthodox but incredibly fruitful ideas).

And, finally, my personal rule which I do not recommend to everyone but which to a large extent helps to implement the previous ones:

Do not prepare a lecture (especially a completely new one) too meticulously. Have a good idea of what you want to say and how, but be ready to skip topics of less import, to change course if you feel the audience is unprepared or being lost.

Teaching a course. Teaching a graduate or an advanced undergraduate course, i.e. a course addressed to an audience of future professionals or at least serious learners who plan to use the subject in their future professional work, has a double purpose.

On the one hand, there is a certain body of advanced material (notions, proofs, etc.) to be presented and learned by the members of the audience. At the plain level this means that the students should be able to reproduce the proofs and apply results in certain situations (solving problems). On the other hand, such a course is an opportunity to teach students certain ways of thinking in mathematics where specific material serves a purpose somewhat akin to the subject matter for a figurative painter.

These two tasks are not necessarily in tune. The real challenge here is to achieve a proper synthesis. Sometimes a very clear and polished presentation hides the well-springs of the argument and does not teach students how the facts can be FOUND not just LEARNED. Such a presentation may be replaceable by an equally well written book, sometimes authored by the instructor. On the other hand, a muddled presentation which for the teacher may seem to be a road toward the discovery of truth usually distracts and confuses students who do not yet see the ``big picture''.

My favored approach (which I do not recommend as superior to others or suitable for everyone) is to emphasize the ``big picture'' upfront, often skipping even essential details at first and bringing the students' attention to the structural and logical elements which will play key roles in the development. After that the picture gets filled in, sometimes straightforwardly using the most elegant or most useful available arguments (I like sometimes while preparing a lecture to figure out proofs of even standard results rather than look them up in a book), sometimes with a bit of the Socratic method with students reaching certain conclusions themselves. I am not a great fan of repeating well-presented proofs from books, even excellent ones. Given a choice I would rather spend time on key motivations and applications and let students read the technical arguments.

While elegance and formal perfection of presentation (even in a Socratic form) are desirable and have independent value as tools conveying the beauty of mathematics to students, the role of these elements in a course is less crucial than in a single lecture. I remember a younger colleague who used to be a Ph.D. student at Princeton recollecting the lectures of a great mathematician not renown for the clarity and accessibility of his presentation. Conceding that the presentation was often muddled and confused while professor was still trying to come to grips with recently discovered or digested material, he said: ``It was a superior learning experience: we watched a great mind at work''. I do not recommend to any instructor to strive consciously for this kind of recognition. However, it is true that in a course even time wasted on an erroneous argument or a struggle with an overlooked technical difficulty may be redeemed when instructor explains not only how to correct an error or overcome a difficulty, but what led him into trouble and how he found a way out.

To summarize, a well-taught course results not just in students learning specific material but in advancing their understanding of mathematical thinking and their ability to practice this thinking.

Bringing up a Ph.D. student. The task of supervising a Ph.D. student in mathematics in the contemporary environment of a US research university amounts to transforming a person with a certain level of aptitude and general knowledge in mathematics into a professional research mathematician able to work successfully and with a quickly increasing degree of independence in academia or elsewhere.

Modern mathematical research is still primarily a cottage industry and an enterprise both individualistic and democratic. Researchers exchange ideas widely and either work individually or (currently more often) actively collaborate in teams of two, three, less frequently of four, and very rarely of five or more. There is not much ``vertical'' interaction: collaborators often bring expertise from different areas and insights from different perspectives but conception and execution are not clearly separated between participants in a project. All of these factors as well as the well-known features of the job market in mathematics, put a great premium on creativity and independent thinking.

A necessary prerequisite for developing these traits is a mastery of professional tools, which in the case of a research mathematician includes three principal ingredients:

(i) a fairly deep knowledge of considerable parts of modern mathematics,

(ii) high level of mastery of the various techniques of rigorous proofs, and

(iii) a sufficiently well developed professional intuition which allows to predict heuristically likely outcomes of cumbersome calculations and to chart complicated sequences of technical steps to be carried out rigorously.

How would one approach this task? There is no universal recipe and the whole thing is to a large extent an art form. It also has an element of a gamble about it. I firmly believe and can demonstrate by a number of ``case histories'' of former students, both my own and those who worked with other advisers, that the way a beginning mathematician produces his/her first piece of original research (be that a Ph.D. project or something preceding it) plays a crucial role in the whole research career; it may greatly enhance or dull the natural talent, may stimulate bold, creative but realistic approach to problems, or forever relegate a mathematician to the role of a timid imitator of others, or make one set unrealistic goals and waste creative energy in fruitless attempts to achieve these.

Basically there are two ``pure strategies'' of working with Ph.D. students: at one extreme, to set a well-described goal (solving a well-posed problem or developing an understanding of a certain subject) and to navigate the student toward it, and, on the other, to let the students chose the thesis topic on their own, try to help along the way and act as a sounding board for the students' ideas.

In such a pure form both approaches are rarely successful and even practicable. The problem with the first one is that in mathematical research success which is essentially guaranteed tends to be discounted, so a fixed goal may be either unattainable or (if the adviser has a good blueprint in mind) not that interesting. The second approach sometimes works with really outstanding students but may easily destroy a major but not yet developed talent.

As I said before there is no magical solution and a thoughtful adviser takes into account students' perceived strengths and weaknesses and tries to monitor their progress in order to amplify positive developments (germs of potentially fruitful ideas or constructions, bringing in new viewpoints, expanding into potentially interesting adjacent areas, and so on) and to prevent the students from reaching the point of diminishing returns, as well as the degeneration of their work into uninteresting technicalities. It is the greatest joy for an adviser when suddenly something clicks and the student is transformed from a plain apprentice into a master of his/her own (even if very limited) subject. This means that a mathematician is born. After that point the student usually knows WHAT to do even if she or he does not always see HOW to do it. After that the second mode (trying to give students driven by their own agenda as much expert help as possible and to listen to their ideas) becomes fruitful and natural.

On numerous occasions this happy transformation happened to my Ph.D. students. I consider my contributions in achieving this the most valuable and fruitful part of my teaching experience. In all honesty one must say that it does not always happen this wonderful way; a student may produce a first-rate piece of mathematical research mostly through personal efforts and still retain an attitude of a pupil, not a master. Unfortunately, a Ph.D. produced in this way sometimes remains a mathematician's best lifetime research achievement. However, it often happens that such a student achieves professional maturity and creative independence later as a post-doc or even after that; in fact, some of my experience with post-docs has helped them achieve this creative independence.

But there is still the problem of getting a student started on the hard road to becoming a creative researcher. Several methods help. First, I believe that it is very useful for a student to produce a piece of research before starting to work on the thesis. Working on such a project produces less stress and helps to unleash creativity. Such projects often grow out of topics discussed in more advanced graduate courses or in working seminars. They are not meant to lead to a Ph. D. project but sometimes they do. I also try to expose students beginning their research work to a variety of topics of current research interest and to stimulate them to start thinking in several directions with a hope that it may lead to a fruitful choice of a thesis topic. I am not at all jealous if my official Ph.D. students interact or even work primarily with another faculty member. Conversely, I am happy to discuss (and even suggest if they ask) problems to Ph.D. students working with my colleagues. I think learning in a serious way from more than one mathematician enhances a young mathematician's potential.

Working seminars are very useful both for developing mastery of serious mathematics and for starting participants on new research endeavors. Such a seminar has a format different from usual research seminars (longer hours, usually with a coffee break, sometimes in the evening). It has a diverse group of participants with a core group of graduate students, usually from middle to advanced level, and also post-docs and young faculty. The level of commitment is also higher since participants present papers and other material. Participants do not talk about their recent work unless this is deemed essential for advancing the seminar's program. Interaction between young mathematicians of different status as well as the ``critical mass'' effect are among principal benefits of this format.

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