Hrant Hakobyan Department of Mathematics University of Toronto 40 St.George Street Toronto, ON M5S 2E4 Email: hhakob@math.toronto.edu Phone: 416-978-5214 Office: Bahen Building 6-172

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Preprints

• Conformal dimension: Cantor sets and Moduli, In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension $1$ thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede. It implies in particular that there are many sets $E\subset\mathbb{R}$ of zero length such that $X\times Y$ is minimal for conformal dimension for every compact $Y$.

Papers

• Cantor sets which are minimal for quasisymmetric maps, Journal of Contemporary Mathematical Analysis, 41 (2006), no. 2, 5-13.
  
We show that middle interval Cantor sets of Hausdorff dimension 1 are minimal for quasisymmetric mappings of the real line. This gives first examples of such sets of a line of 0 length. Combinig this with a theorem of Jang Mei Wu one gets that there are "rigid" Cantor sets, i.e. sets whose every quasisymmetric image has length 0 and Hausdorff dimension 1.

• Euclidean Quasiconvexity (with David Herron ), Ann. Acad. Sci. Fenn., 33 (2008), 205-230
  
We exhibit compact tottaly disconnected sets in R^n of Hausdorff dimension n-1 whose complements fail to be quasiconvex, and similar sets with positive n-measure whose complements are quasiconvex. We characterize the finitely connected quasiconvex plane domains.

• A Central set of dimension 2 (with Christopher Bishop ), Proc. Amer. Math. Soc.136 (2008), 2453-2461.
  
The central set C(D) of a domain D is the set of centers of maximal discs included in D. Fremlin showed that a central set has to have zero area and asked wether it can have Hausdorff dimension strictly larger than 1. We answer this affirmatively by constructing a domain with central set of Hausdorff dimension 2. In fact, given any Hausdorff measure function h(t) such that h(t)/t^2 tends to infinity, D can be chosen so that H_h(C(D)) > 0.

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Lectures

• A QC invariant notion of dimension.
  
Slides from a talk in the Ahlfors-Bers colloquium.