Dear all,
I'm applying to the Fulbright Visiting Research Student Scholarship and they asked me to write two essays. The first one is about my research objective and it's thought to be red by both experts and non-experts. Although some parts may be a little too specialistic I would like to have some comments and some grammatical help. Here is the guidelines they sent me:
- Clearly identify the area(s) within your field of study in which you want to specialize or concentrate.
- Write the document in a narrative way
- Describe your future academic and professional objectives when you return back in your country
- Specify the US university where you want to pursue your objectives and explain why you choose it
And here is my essay. Note that the citations of books and articles will be more detailed in my final version.
"Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry" said Serge Lang when he was investigating the deep relation between Diophantine equations and Algebraic Geometry (which has led him to coin the phrase "Diophantine Geometry"). In the last century Grothendieck's revolution in the field of Algebraic Geometry has thrown light on this profound connection "in such a way that one might plausibly argue that (...) the theory of Diophantine equations is simply the special case of Algebraic Geometry" ([6]).
From this point of view a broad knowledge about geometric aspects of arithmetic problems and in the geometric techniques (that can be applied in their study) is a naturally prerequisite for expanding ideas and researches in the field of Diophantine Geometry.
The purpose of this project is to study some of the recent trends in Algebraic Geometry which have got wide outcomes on diophantine problems and may very likely be applied in future research on this topics. In particular I would like to concentrate on two areas of geometric research, namely Logarithmic Algebraic Geometry and the Theory of Moduli of curves; both of them have common thread in the theory of Gromov-Witten Invariants. Let me briefly analyze more in detail the importance and the future consequences that both of this areas can have in my research.
Logarithmic Algebraic Geometry, or Log-Geometry is a theme of special interest, especially in this recent years; this field of research can have a deal with the case of open subvarieties of a projective variety and seems particularly good fit to study surfaces of log-general type where the affine surface is seen as the complement, in a suitable projective embedding, of a divisor "at the infinity" in a projective surface. Thus the problem of imposing contact conditions on curves over such surfaces can be treated in such a contest and applied to the study of "boundary" situation in Vojta's conjecture, such as the complement of a plane cubic in P2. At the same time, recent developments in Gromov-Witten Invariants in the Logarithmic case can possibly be applied to the study of Vojta's conjecture over function field, namely (at least in the enumerative case) some of these invariants are expected to count curves with particular tangency conditions to the divisor (see for example [4]).
Another question in which geometric settings can be applied with success to diophantine problems is the Theory of Moduli of curves. This wide theory is continuously showing all its strength and it has been used a lot in the study of Arithmetic Geometry. In detail the construction of Moduli space of curves of fixed genus (at least with some regularity conditions such as stability) is a natural space where investigate Vojta's conjecture over function field: in fact the goal of limiting the degree of the image of non constant morphisms from an open affine subset of a nonsingular projective curve to a surface of log-general type regarding only the genus and the cardinality of the complement of the affine curve is naturally lifted to the study of the moduli of (stable) maps. Moreover such constructions have just shown a lot of impressive results like [5] or [2].
In my humble opinion Brown University is the best option to grow up inside the above specified argument. There might be several explainations for supporting my idea, but just citing the names of Abramovich or Sirverman (among many others) it is enough to get how much prestigious Brown is (another example may be seen on the wide range of results the Mathamatical Department achieved matching the geometric and arithmetic tools: e.g. [1], [3], [8] and [9]). Furthermore there are also personal motivations why Brown represent my first choice.
First, there are plenty of Ph.D. students whom I can have a deal with and talk about math issues. At university of Udine I am the only-one who is working on the arithmetic and geometry field of study, despite our istitution has one of the major expert involved in it. Hence, I believe that being a student at Brown will allow myself to obtain further background knowledge. Second I would be in a highly ranked University where I can learn different approaches to math research: in particular the focus on the applicative aspect of the above specified theoretical problems. All the American University System owns positive feedback from over the world but I think that at Brown the math research fulfill the industrial and applicative world in a particularly fruitful way (as an example prof. Silverman is cofounder of the NTRU Cyroptosystem).
Concluding and talking about long term point of view, I would like to come back to my old department so that I could bring any further knowledge and technique I will have faced studying, training and working at Brown and use it in my future research career. At the same time I could apply this skills in all the academic and professional challenges that will occur in my future.
Thank you very much for your help. I really appreciate it.
Best Regards
I'm applying to the Fulbright Visiting Research Student Scholarship and they asked me to write two essays. The first one is about my research objective and it's thought to be red by both experts and non-experts. Although some parts may be a little too specialistic I would like to have some comments and some grammatical help. Here is the guidelines they sent me:
- Clearly identify the area(s) within your field of study in which you want to specialize or concentrate.
- Write the document in a narrative way
- Describe your future academic and professional objectives when you return back in your country
- Specify the US university where you want to pursue your objectives and explain why you choose it
And here is my essay. Note that the citations of books and articles will be more detailed in my final version.
"Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry" said Serge Lang when he was investigating the deep relation between Diophantine equations and Algebraic Geometry (which has led him to coin the phrase "Diophantine Geometry"). In the last century Grothendieck's revolution in the field of Algebraic Geometry has thrown light on this profound connection "in such a way that one might plausibly argue that (...) the theory of Diophantine equations is simply the special case of Algebraic Geometry" ([6]).
From this point of view a broad knowledge about geometric aspects of arithmetic problems and in the geometric techniques (that can be applied in their study) is a naturally prerequisite for expanding ideas and researches in the field of Diophantine Geometry.
The purpose of this project is to study some of the recent trends in Algebraic Geometry which have got wide outcomes on diophantine problems and may very likely be applied in future research on this topics. In particular I would like to concentrate on two areas of geometric research, namely Logarithmic Algebraic Geometry and the Theory of Moduli of curves; both of them have common thread in the theory of Gromov-Witten Invariants. Let me briefly analyze more in detail the importance and the future consequences that both of this areas can have in my research.
Logarithmic Algebraic Geometry, or Log-Geometry is a theme of special interest, especially in this recent years; this field of research can have a deal with the case of open subvarieties of a projective variety and seems particularly good fit to study surfaces of log-general type where the affine surface is seen as the complement, in a suitable projective embedding, of a divisor "at the infinity" in a projective surface. Thus the problem of imposing contact conditions on curves over such surfaces can be treated in such a contest and applied to the study of "boundary" situation in Vojta's conjecture, such as the complement of a plane cubic in P2. At the same time, recent developments in Gromov-Witten Invariants in the Logarithmic case can possibly be applied to the study of Vojta's conjecture over function field, namely (at least in the enumerative case) some of these invariants are expected to count curves with particular tangency conditions to the divisor (see for example [4]).
Another question in which geometric settings can be applied with success to diophantine problems is the Theory of Moduli of curves. This wide theory is continuously showing all its strength and it has been used a lot in the study of Arithmetic Geometry. In detail the construction of Moduli space of curves of fixed genus (at least with some regularity conditions such as stability) is a natural space where investigate Vojta's conjecture over function field: in fact the goal of limiting the degree of the image of non constant morphisms from an open affine subset of a nonsingular projective curve to a surface of log-general type regarding only the genus and the cardinality of the complement of the affine curve is naturally lifted to the study of the moduli of (stable) maps. Moreover such constructions have just shown a lot of impressive results like [5] or [2].
In my humble opinion Brown University is the best option to grow up inside the above specified argument. There might be several explainations for supporting my idea, but just citing the names of Abramovich or Sirverman (among many others) it is enough to get how much prestigious Brown is (another example may be seen on the wide range of results the Mathamatical Department achieved matching the geometric and arithmetic tools: e.g. [1], [3], [8] and [9]). Furthermore there are also personal motivations why Brown represent my first choice.
First, there are plenty of Ph.D. students whom I can have a deal with and talk about math issues. At university of Udine I am the only-one who is working on the arithmetic and geometry field of study, despite our istitution has one of the major expert involved in it. Hence, I believe that being a student at Brown will allow myself to obtain further background knowledge. Second I would be in a highly ranked University where I can learn different approaches to math research: in particular the focus on the applicative aspect of the above specified theoretical problems. All the American University System owns positive feedback from over the world but I think that at Brown the math research fulfill the industrial and applicative world in a particularly fruitful way (as an example prof. Silverman is cofounder of the NTRU Cyroptosystem).
Concluding and talking about long term point of view, I would like to come back to my old department so that I could bring any further knowledge and technique I will have faced studying, training and working at Brown and use it in my future research career. At the same time I could apply this skills in all the academic and professional challenges that will occur in my future.
Thank you very much for your help. I really appreciate it.
Best Regards