Larusson, Finnur; Matematični kolokvij 2010

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Homotopy theory in complex geometry

Finnur Lárusson

University of Adelaide, Avstralija

2. december 2010

Homotopy theory started in the mid-20th century from the idea to treat two continuous maps that can be deformed to each other as if they were the same map. Of central concern in homotopy theory are lifting and extension problems and obstructions to solving them. Around 1970, D. Quillen abstracted the key ideas of homotopy theory into a category-theoretic framework called a model structure.

Model structures are used in algebraic topology, algebraic geometry, higher category theory, and even in theoretical computer science. Several years ago, model structures were discovered in complex geometry. This was part of a development set in motion by a seminal paper of M. Gromov in 1989. The area in question, now called Oka theory, is concerned with a tight relationship between homotopy theory and complex analysis in a geometric setting.

We will begin with a brief introduction to model structures, with examples of how they arise in practice, and then focus on Oka theory from a homotopy-theoretic perspective.

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