I am familiar with a lot of modern research in algebraic geometry (over complex or real numbers) where only very classical algebraic geometry is used (pre Grothendieck). It is both "pure" and "applied". (S. Abhyankar worked in the same department where I am:-)

But of course I cannot EXCLUDE that some modern algebraic geometry is useful in some applied questions, even very applied ones like control theory.

Usually this is a question of training of the writer, and his/her intended audience. In most cases, modern algebraic geometry (Hartshorne-like) can be translated into completely classical terms. So in many cases this is simply a choice of language and thus depends on the author's preferences. Of course, there is a problem that it is hard for people without this modern training to understand the papers written in the modern language. But there are several areas of algebraic geometry (both pure and applied) where the classical language still dominates.

As an example, I can mention this book

MR1117227 Malgrange, B. Équations différentielles à coefficients polynomiaux. Birkhäuser Boston, Inc., Boston, MA, 1991.

which most people with classical training (in differential equations) cannot read.

I am familiar with a lot of modern research in algebraic geometry (over complex or real numbers) where only very classical algebraic geometry is used (pre Grothendieck). It is both "pure" and "applied". (S. Abhyankar worked in the same department where I am:-)

But of course I cannot EXCLUDE that some modern algebraic geometry is useful in some applied questions, even very applied ones like control theory.

Usually this is a question of training of the writer, and his/her intended audience. In most cases, modern algebraic geometry (Hartshorne-like) can be translated into completely classical terms. So in many cases this is simply a choice of language and thus depends on the author's preferences. Of course, there is a problem that it is hard for people without this modern training to understand the papers written in the modern language. But there are several areas of algebraic geometry (both pure and applied) where the classical language still dominates.

As an example, where modern language is used in "applications" (to differential equations) I can mention this book:

MR1117227 Malgrange, B. Équations différentielles à coefficients polynomiaux. Birkhäuser Boston, Inc., Boston, MA, 1991.

which most people with classical training in differential equations cannot read.

I am familiar with a lot of modern research in algebraic geometry (over complex or real numbers) where only very classical algebraic geometry is used (pre Grothendieck). It is both "pure" and "applied". (S. Abhyankar worked in the same department where I am:-)

But of course I cannot EXCLUDE that some modern algebraic geometry is useful in some applied questions, even very applied ones like control theory.

Usually this is a question of training of the writer, and his/her intended audience. In most cases, modern algebraic geometry (Hartshorne-like) can be translated into completely classical terms. So in many cases this is simply a choice of language and thus depends on the author's preferences. Of course, there is a problem that it is hard for people without this modern training to understand the papers written in the modern language. But there are several areas of algebraic geometry (both pure and applied) where the classical language still dominates.

I am familiar with a lot of modern research in algebraic geometry (over complex or real numbers) where only very classical algebraic geometry is used (pre Grothendieck). It is both "pure" and "applied". (S. Abhyankar worked in the same department where I am:-)

But of course I cannot EXCLUDE that some modern algebraic geometry is useful in some applied questions, even very applied ones like control theory.

Usually this is a question of training of the writer, and his/her intended audience. In most cases, modern algebraic geometry (Hartshorne-like) can be translated into completely classical terms. So in many cases this is simply a choice of language and thus depends on the author's preferences. Of course, there is a problem that it is hard for people without this modern training to understand the papers written in the modern language. But there are several areas of algebraic geometry (both pure and applied) where the classical language still dominates.

As an example, I can mention this book

MR1117227 Malgrange, B. Équations différentielles à coefficients polynomiaux. Birkhäuser Boston, Inc., Boston, MA, 1991.

which most people with classical training (in differential equations) cannot read.