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{\Large המחלקה למתמטיקה, בן-גוריון}

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{\Huge הסתברות ותורה ארגודית}\\[0.2\baselineskip]

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\textbf{ב}\emph{יום שלישי, 25 באפריל, 2017}
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\textbf{בשעה} \emph{10:50 -- 12:00}
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\textbf{ב}\emph{Math -101}

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ההרצאה

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{\Large\bfseries Approximate groups and applications to the growth of groups\par}
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תינתן על-ידי
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{\large\scshape Matthew Tointon}
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\textbf{תקציר:}
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Given a set A in a group, write \$A\^{}n\$ for the set of all products \$x\_1…x\_n\$ with each \$x\_i\$ belonging to A. Roughly speaking, a set A for which \$A\^{}2\$ is ''not much larger than`` A is called an ''approximate group``. The ''growth`` of A, on the other hand, refers to the behaviour of \$ & A\^{}n & \$ as n tends to infinity. Both of these have been fruitful areas of study, with applications in various branches of mathematics.\\
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Remarkably, understanding the behaviour of approximate groups allows us to convert information \$A\^{}k\$ for some single fixed k into information about the sequence \$A\^{}n\$ as n tends to infinity, and in particular about the growth of A.

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In this talk I will present various results describing the algebraic structure of approximate groups, and then explain how to use these to prove new results about growth. In particular, I will describe work with Romain Tessera in which we show that if \$ & A\^{}k & \$ is bounded by \$Mk\^{}D\$ for some given M and D then, provided k is large enough, & A\^{}n & is bounded by \$M`n\^{}D\$ for \emph{every} n \textgreater{} k, with M` depending only on M and D. This verifies a conjecture of Itai Benjamini.\\
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