20070124Sun Zhiwei

First Talk:

Title: Problems and Results on Restricted Sumsets

Time&Place: Jan. 24 9:30am-11:30am (Beijing Normal University, 北师大教八楼（数学楼）214室)

Time&Place: Jan. 24 9:30am-11:30am (Beijing Normal University, 北师大教八楼（数学楼）214室)

Speaker: Zhi-Wei Sun (Nanjing University) http://pweb.nju.edu.cn/zwsun/

Abstract: Additive number theory is currently an active field related to combinatorics. A sumset has the form A+B={a+b: a in A, b in B}. Freiman's theorem on sumsets plays an important role in Gowers' famous work on Szemeredi's theorem; however, methods in that direction cannot be used to get sharp results on restricted sumsets such as {a+b: a in A, b in B and a not=b}. In this talk we give a mordern survey of problems and results concerning lower bounds for cardinalities of various restricted sumsets with elements in a field or an abelian group. The most powerful tool in this field is Alon's Combinatorial Nullstellensatz which provides an algebraic technique to handle combinatorial probelms. We will present various applications of the Combinatorial Nullstellensatz to the Erdos-Heilbronn conjecture (which states that if p is a prime and A is a subset of Z/pZ then there are at least min{p,2|A|-3} elements of Z/pZ in the form a+b with a,b in A and a not=b), Snevily's conjecture (which asserts that if A and B are subsets of an abelian group of odd order with |A|=|B|=n then there is a numbering a_1,...,a_n of the elements of A and a numbering b_1,...,b_n of the elements of B such that a_1+b_1,...,a_n+b_n are distinct), Lev's conjecture, and their further generalizations. We will also mention the speaker's recent additive result (a 3-dimensional analogue of Snevily's conjecture) related to Latin transversals in an additive Latin cube.

Abstract: Additive number theory is currently an active field related to combinatorics. A sumset has the form A+B={a+b: a in A, b in B}. Freiman's theorem on sumsets plays an important role in Gowers' famous work on Szemeredi's theorem; however, methods in that direction cannot be used to get sharp results on restricted sumsets such as {a+b: a in A, b in B and a not=b}. In this talk we give a mordern survey of problems and results concerning lower bounds for cardinalities of various restricted sumsets with elements in a field or an abelian group. The most powerful tool in this field is Alon's Combinatorial Nullstellensatz which provides an algebraic technique to handle combinatorial probelms. We will present various applications of the Combinatorial Nullstellensatz to the Erdos-Heilbronn conjecture (which states that if p is a prime and A is a subset of Z/pZ then there are at least min{p,2|A|-3} elements of Z/pZ in the form a+b with a,b in A and a not=b), Snevily's conjecture (which asserts that if A and B are subsets of an abelian group of odd order with |A|=|B|=n then there is a numbering a_1,...,a_n of the elements of A and a numbering b_1,...,b_n of the elements of B such that a_1+b_1,...,a_n+b_n are distinct), Lev's conjecture, and their further generalizations. We will also mention the speaker's recent additive result (a 3-dimensional analogue of Snevily's conjecture) related to Latin transversals in an additive Latin cube.

The Second Talk

Title: Combinatorial aspects of Szemeredi's theorem

Time: Jan. 30 (Institute of Mathematics)

Speaker: Zhi-Wei Sun (Nanjing University)

Abstract:

Title: Combinatorial aspects of Szemeredi's theorem

Time: Jan. 30 (Institute of Mathematics)

Speaker: Zhi-Wei Sun (Nanjing University)

Abstract:

In the first half of this talk we will review the beatiful combinatorial theory of Ramsey and introduce several important theorems in this field such as Ramsey's theorem, Schur's theorem and Rado's theorem,

van der Waerden's theorem and Shelah's proof of the Hales-Jewett theorem (which is a further extension of van der Waerden's theorem). In the second half of this talk, we focus on the combinatorial aspects of

the famous Szemeredi theorem (which implies van der Waerden's theorem and plays an important role in the proof of the Green-Tao theorem). We will deduce Roth's theorem (Szemeredi's theorem in the case k=3) from the Triangle Removal Lemma (in graph-theoretic language) which is an application of Szemeredi's Regularity Lemma (a powerful tool in graph theory), and show the Balog-Szemeredi-Gowers theorem by a graph-theoretic method. The combinatorial proof of the general theorem of Szemeredi involves the hypergraph versions of the Triangle Removal Lemma and Szemeredi's Regularity Lemma. Gowers' proof of Szemeredi's theorem with explicit bounds uses a deep theorem of Freiman which states that if A is a finite set of integers with |A+A|<c|A| then A is contained in an n-dimensional progression of size at most C|A| where n and C depend only on c. We will give a sketch of Ruzsa's modern proof of Freiman's theorem

which involves Plunnecke's inequality for Plunnecke graphs, Bohr neighborhoods, Freiman isomorphisms and Minkowske's second theorem in the geometry of numbers.

van der Waerden's theorem and Shelah's proof of the Hales-Jewett theorem (which is a further extension of van der Waerden's theorem). In the second half of this talk, we focus on the combinatorial aspects of

the famous Szemeredi theorem (which implies van der Waerden's theorem and plays an important role in the proof of the Green-Tao theorem). We will deduce Roth's theorem (Szemeredi's theorem in the case k=3) from the Triangle Removal Lemma (in graph-theoretic language) which is an application of Szemeredi's Regularity Lemma (a powerful tool in graph theory), and show the Balog-Szemeredi-Gowers theorem by a graph-theoretic method. The combinatorial proof of the general theorem of Szemeredi involves the hypergraph versions of the Triangle Removal Lemma and Szemeredi's Regularity Lemma. Gowers' proof of Szemeredi's theorem with explicit bounds uses a deep theorem of Freiman which states that if A is a finite set of integers with |A+A|<c|A| then A is contained in an n-dimensional progression of size at most C|A| where n and C depend only on c. We will give a sketch of Ruzsa's modern proof of Freiman's theorem

which involves Plunnecke's inequality for Plunnecke graphs, Bohr neighborhoods, Freiman isomorphisms and Minkowske's second theorem in the geometry of numbers.

Refference:

1. T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006.

2. B. Green, Structure theory of set addition (Lecture Notes), available from Green's homepage

http://www.dpmms.cam.ac.uk/~bjg23/

3. M. B. Nathanson, Additive Number Theory--Inverse Problems and the Geometry of Sumsets, Graduate Texts in Math. 165, Springer, 1996.

Y. Ishigami, A simple regularization of hypergraphs,

http://arxiv.org/abs/math.CO/0612838

5. Some of my talks and papers from my homepage

http://pweb.nju.edu.cn/zwsun/

2. B. Green, Structure theory of set addition (Lecture Notes), available from Green's homepage

http://www.dpmms.cam.ac.uk/~bjg23/

3. M. B. Nathanson, Additive Number Theory--Inverse Problems and the Geometry of Sumsets, Graduate Texts in Math. 165, Springer, 1996.

Y. Ishigami, A simple regularization of hypergraphs,

http://arxiv.org/abs/math.CO/0612838

5. Some of my talks and papers from my homepage

http://pweb.nju.edu.cn/zwsun/

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发件人 Sun Zhiwei2007-1 |