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Ben Lund, Furstenberg sets over finite fields

Wednesday, January 15, 2020 @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)


Ben Lund
IBS Discrete Mathematics Group

An important family of incidence problems are discrete analogs of deep questions in geometric measure theory. Perhaps the most famous example of this is the finite field Kakeya conjecture, proved by Dvir in 2008. Dvir’s proof introduced the polynomial method to incidence geometry, which led to the solution to many long-standing problems in the area.
I will talk about a generalization of the Kakeya conjecture posed by Ellenberg, Oberlin, and Tao. A $(k,m)$-Furstenberg set S in $\mathbb F_q^n$ has the property that, parallel to every affine $k$-plane V, there is a k-plane W such that $|W \cap S| > m$. Using sophisticated ideas from algebraic geometry, Ellenberg and Erman showed that if S is a $(k,m)$-Furstenberg set, then $|S| > c m^{n/k}$, for a constant c depending on n and k. In recent joint work with Manik Dhar and Zeev Dvir, we give simpler proofs of stronger bounds. For example, if $m>2^{n+7}q$, then $|S|=(1-o(1))mq^{n-k}$, which is tight up to the $o(1)$ term.


Wednesday, January 15, 2020
4:30 PM - 5:30 PM KST
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Room B232
IBS (기초과학연구원)


Sang-il Oum (엄상일)
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IBS 이산수학그룹 Discrete Mathematics Group
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209
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