Gorenstein algebras can be found in many different areas in mathe- matics, although it is rather subtle to determine the ones satisfying specific prop- erties. In this talk, we discuss Hilbert functions of Artinian Gorenstein algebras. Artinian algebras with the Lefschetz properties have more restrictive Hilbert func- tions. In 1995, T. Harima characterized Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property and proved that they are, in fact, Stanley-Iarrobino (SI)-sequences. Jordan types are generalizations of the Lefschetz properties which provide more information on the Hilbert function of Artinian al- gebras. We will introduce Lefschetz properties and Jordan types for Gorenstein algebras and mention some of the known results and open problems. We general- ize T. Harima's result and prove that SI-sequences classify the Hilbert functions of Artinian Gorenstein algebras with the strong Lefschetz property. We also explain how one can determine the Jordan type for an Artinian Gorenstein algebra and its connection to the higher Hessians associated with the algebra.

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