Abstract: Let f be a continuous map from a "nice" compact topological space X to itself. Then f induces an endomorphism H^i(f) of the cohomology groups H^i(X,Q) of X for each i, and the classical Lefschetz trace formula asserts that the virtual trace \sum_i (-1)^i Tr(H^i(f)) can be described in terms of the fixed points of f.

This result has various applications. For example, it gives a one line proof of the famous Brouwer's fixed point theorem.

In the 60's Grothendieck et al. showed that analogs of the Lefschetz trace formula also hold in algebraic geometry. As a consequence, he proved famous Weil conjectures on the number of points of algebraic varieties over finite fields, which is currently one of the most important results in the area.

In my lecture I will describe these results, give some of their applications and discuss recent developments.