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Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Added by PLC: Yes, this a very nice proof to show to undergraduates, especially those who know about the Chevalley-Warning Theorem. (A little searching on this site and elsewhere will reveal that Chevalley-Warning is one of my very favorite results in undergraduate number theory.) I was extremely taken by Dvir's proof when it came out and wrote up a treatment here. And I agree: if you're trying to convince someone that there is really something to this "polynomial method" business, I think it would be hard to do better than this beautiful result.

Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Added by PLC: Yes, this a very nice proof to show to undergraduates, especially those who know about the Chevalley-Warning Theorem. (A little searching on this site and elsewhere will reveal that Chevalley-Warning is one of my very favorite results in undergraduate number theory.) I was extremely taken by Dvir's proof when it came out and wrote up a treatment here. And I agree: if you're trying to convince someone that there is really something to this "polynomial method" business, I think it would be hard to do better than this beautiful result.

Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Added by PLC: Yes, this a very nice proof to show to undergraduates, especially those who know about the Chevalley-Warning Theorem. (A little searching on this site and elsewhere will reveal that Chevalley-Warning is one of my very favorite results in undergraduate number theory.) I was extremely taken by Dvir's proof when it came out and wrote up a treatment here. And I agree: if you're trying to convince someone that there is really something to this "polynomial method" business, I think it would be hard to do better than this beautiful result.

Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Added by PLC: Yes, this a very nice proof to show to undergraduates, especially those who know about the Chevalley-Warning Theorem. (A little searching on this site and elsewhere will reveal that Chevalley-Warning is one of my very favorite results in undergraduate number theory.) I was extremely taken by Dvir's proof when it came out and wrote up a treatment here. And I agree: if you're trying to convince someone that there is really something to this "polynomial method" business, I think it would be hard to do better than this beautiful result.

Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Added by PLC: Yes, this a very nice proof to show to undergraduates, especially those who know about the Chevalley-Warning Theorem. (A little searching on this site and elsewhere will reveal that Chevalley-Warning is one of my very favorite results in undergraduate number theory.) I was extremely taken by Dvir's proof when it came out and wrote up a treatment here. And I agree: if you're trying to convince someone that there is really something to this "polynomial method" business, I think it would be hard to do better than this beautiful result.