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edited Dec 14 2011 at 21:39 |
I would like to know other people answers, so perhaps other people will want my answer also. I can offer my opinion about the area of geometric/algorithmic/asymptotic group theory which I am doing. I should start with saying that my opinion most probably does not coincide with opinions of other people in my area which, I think, is normal. I may also forget something. Also by an achievement I mean a concrete result, not a theory. In my area, the "top" achievements of the last year are (IMHO, in no particular order)
Sela's continued work on Tarski-related problems. His series of 10 papers is now more than 1000 pages long and the latest (recent) results include a solution of an old Malcev's problem about elementary equivalence of free products of groups. That of course assuming it is correct: the solution is being checked. Kharlampovich-Myasnikov's solution of another Malcev's problem, also related to Tarski problems, that proper subgroups of free non-Abelian groups cannot be defined by first order formulas. Dani Wise's work on "cubulating" groups, i.e. embedding groups into Right Angled Artin Groups (again assuming it is correct). In particular his solution of an old problem by Baumslag: all 1-related groups with torsion are residually finite. Also his and Agol's results imply that hyperbolic Haken 3-manifold groups are virtually surface-by-cyclic which is a great result. Igor Mineyev's (very short!) proof of the Strong Hanna Neumann conjecture (more than 40 years old). Grigorchuk-Medynets' example of a finitely generated simple amenable group. Bestvina-Bromberg-Fujiwara's proof that mapping class groups have finite asymptotic dimension.
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edited Dec 14 2011 at 9:47 |
I would like to know other people answers, so perhaps other people will want my answer also. I can offer my opinion about the area of geometric/algorithmic/asymptotic group theory which I am doing. I should start with saying that my opinion most probably does not coincide with opinions of other people in my area which, I think, is normal. I may also forget something. Also by an achievement I mean a concrete result, not a theory. In my area, the "top" achievements of the last year are (IMHO, in no particular order)
Sela's continued work on Tarski-related problems. His series of 10 papers is now more than 1000 pages long and the latest (recent) results include a solution of an old Malcev's problem about elementary equivalence of free products of groups. That of course assuming it is correct: the solution is being checked. Kharlampovich-Myasnikov's solution of another Malcev's problem, also related to Tarski problems, that proper subgroups of free non-Abelian groups cannot be defined by first order formulas. Dani Wise's work on "cubulating" groups, i.e. embedding groups into Right Angled Artin Groups (again assuming it is correct). In particular his solution of an old problem by Baumslag: all 1-related groups with torsion are residually finite. Also his and Agol's results imply that hyperbolic Haken 3-manifold groups are virtually surface-by-cyclic which is a great result. Igor Mineyev's (very short!) proof of the Strong Hanna Neumann conjecture (more than 40 years old). Grigorchuk-Medynets' example of a finitely generated simple amenable group. Besvina-Bromberg-Fujiwara's Bestvina-Bromberg-Fujiwara's proof that mapping class groups have finite asymptotic dimension.
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edited Dec 14 2011 at 6:47 |
I would like to know other people answers, so perhaps other people will want my answer also. I can offer my opinion about the area of geometric/algorithmic/asymptotic group theory which I am doing. I should start with saying that my opinion most probably does not coincide with opinions of other people in my area which, I think, is normal. I may also forget something. Also by an achievement I mean a concrete result, not a theory. In my area, the "top" achievements of the last year are (IMHO, in no particular order)
Sela's continued work on Tarski-related problems. His series of 10 papers is now more than 1000 pages long and the latest (recent) results include a solution of an old Malcev's problem about elementary equivalence of free products of groups. That of course assuming it is correct, of course: the solution is being checked. Kharlampovich-Myasnikov's solution of another Malcev's problem, also related to Tarski problems, that proper subgroups of free non-Abelian groups cannot be defined by first order formulas. Dani Wise's work on "cubulating" groups, i.e. embedding groups into Right Angled Artin Groups (again assuming it is correct). In particular his solution of an old problem by Baumslag: all 1-related groups with torsion are residually finite. Also his and Agol's results imply that hyperbolic Haken 3-manifold groups are virtually surface-by-cyclic which is a great result. Igor Mineyev's (very short!) proof of the Strong Hanna Neumann conjecture (more than 40 years old). Grigorchuk-Medynets' example of a finitely generated simple amenable group. Besvina-Bromberg-Fujiwara's proof that mapping class groups have finite asymptotic dimension.
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edited Dec 14 2011 at 1:56 |
I would like to know other people's answerpeople answers, so perhaps other people will want my answer also. I can offer my opinion about the area of geometric/algorithmic/asymptotic group theory which I am doing. I should start with saying that my opinion most probably does not coincide with opinions of other people in my area which, I think, is normal. I may also forget something. Also by an achievement I mean a concrete result, not a theory. In my area, the "top" achievements of the last year are (IMHO, in no particular order)
Sela's continued work on Tarski-related problems. His series of 10 papers is now more than 1000 pages long and the latest (recent) results include a solution of an old Malcev's problem about elementary equivalence of free products of groups. That of course assuming it is correct, of course: the solution is being checked. Kharlampovich-Myasnikov's solution of another Malcev's problem, also related to Tarski problems, that proper subgroups of free non-Abelian groups cannot be defined by first order formulas. Dani Wise's work on "cubulating" groups, i.e. embedding groups into Right Angled Artin Groups (again assuming it is correct). In particular his solution of an old problem by Baumslag: all 1-related groups with torsion are residually finite. Also his and Agol's results imply that hyperbolic Haken 3-manifold groups are virtually surface-by-cyclic which is a great result. Igor Mineyev's (very short!) proof of the Strong Hanna Neumann conjecture (more than 40 years old). Grigorchuk-Medynets' example of a finitely generated simple amenable group. Besvina-Bromberg-Fujiwara's proof that mapping class groups have finite asymptotic dimension.
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answered Dec 14 2011 at 1:39 |
I would like to know other people's answer, so perhaps other people will want my answer also. I can offer my opinion about the area of geometric/algorithmic/asymptotic group theory which I am doing. I should start with saying that my opinion most probably does not coincide with opinions of other people in my area which, I think, is normal. I may also forget something. Also by an achievement I mean a concrete result, not a theory. In my area, the "top" achievements of the last year are (IMHO, in no particular order)
Sela's continued work on Tarski-related problems. His series of 10 papers is now more than 1000 pages long and the latest (recent) results include a solution of an old Malcev's problem about elementary equivalence of free products of groups. That of course assuming it is correct, of course: the solution is being checked. Kharlampovich-Myasnikov's solution of another Malcev's problem, also related to Tarski problems, that proper subgroups of free non-Abelian groups cannot be defined by first order formulas. Dani Wise's work on "cubulating" groups, i.e. embedding groups into Right Angled Artin Groups (again assuming it is correct). In particular his solution of an old problem by Baumslag: all 1-related groups with torsion are residually finite. Also his and Agol's results imply that hyperbolic Haken 3-manifold groups are virtually surface-by-cyclic which is a great result. Igor Mineyev's (very short!) proof of the Strong Hanna Neumann conjecture (more than 40 years old). Grigorchuk-Medynets' example of a finitely generated simple amenable group. Besvina-Bromberg-Fujiwara's proof that mapping class groups have finite asymptotic dimension.
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