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The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

Example. In the early 1980th my friend, then a graduate student, discussed with his adviser possible directions of research. The adviser himself contributed to classical complex functions theory and to holomorphic dynamics. He said that holomorphic dynamics is dead, and recommended to work in classical function theory (the area which was not the most popular at that time either). It is almost exactly at that time (1982) when the revolution in holomorphic dynamics happened, suddenly this become one of the most fashionable area in whole mathematics, and it preserves this status to this day. But in 1960-80 the adviser was almost alone, working in holomorphic dynamics.

The areas which experienced strong revival during my lifetime, after a long oblivi, are besides holomorphic dynamics, hyperbolic geometry, Kleinian groups, analytic theory of differential equations, especially Painleve equations, Schubert calculus, knot theory.

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

Example. In the early 1980th my friend, then a graduate student, discussed with his adviser possible directions of research. The adviser himself contributed to classical complex functions theory and to holomorphic dynamics. He said that holomorphic dynamics is dead, and recommended to work in classical function theory (the area which was not the most popular at that time either). It is almost exactly at that time (1982) when the revolution in holomorphic dynamics happened, suddenly this become one of the most fashionable area in whole mathematics, and it preserves this status to this day. But in 1960-80 the adviser was almost alone, working in holomorphic dynamics.

The areas which experienced strong revival during my lifetime, after a long oblivion, are besides holomorphic dynamics, hyperbolic geometry, Kleinian groups, analytic theory of differential equations, especially Painleve equations, Schubert calculus, knot theory.

Remark. Just for fun: type such terms as "elliptic curve", "automorphic function", "modular form", "enumerative geometry", "Painleve", "Fatou" on Google ngram.

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

Example. In the early 1980th my friend, then a graduate student, discussed with his adviser possible directions of research. The adviser himself contributed to classical complex functions theory and to holomorphic dynamics. He said that holomorphic dynamics is dead, and recommended to work in classical function theory (the area which was not the most popular at that time either). It is almost exactly at that time (1982) when the revolution in holomorphic dynamics happened, suddenly this become one of the most fashionable area in whole mathematics, and it preserves this status to this day. But in 1960-80 the adviser was almost alone, working in holomorphic dynamics.

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

Example. In the early 1980th my friend, then a graduate student, discussed with his adviser possible directions of research. The adviser himself contributed to classical complex functions theory and to holomorphic dynamics. He said that holomorphic dynamics is dead, and recommended to work in classical function theory (the area which was not the most popular at that time either). It is almost exactly at that time (1982) when the revolution in holomorphic dynamics happened, suddenly this become one of the most fashionable area in whole mathematics, and it preserves this status to this day. But in 1960-80 the adviser was almost alone, working in holomorphic dynamics.

The areas which experienced strong revival during my lifetime, after a long oblivi, are besides holomorphic dynamics, hyperbolic geometry, Kleinian groups, analytic theory of differential equations, especially Painleve equations, Schubert calculus, knot theory.

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

Example. In the early 1980th my friend, then a graduate student, discussed with his adviser possible directions of research. The adviser himself contributed to classical complex functions theory and to holomorphic dynamics. He said that holomorphic dynamics is dead, and recommended to work in classical function theory (the area which was not the most popular at that time either). It is almost exactly at that time (1982) when the revolution in holomorphic dynamics happened, suddenly this become one of the most fashionable area in whole mathematics, and it preserves this status to this day. But in 1960-80 the adviser was almost alone, working in holomorphic dynamics.