<a href="https://www.youtube.com/watch?v=V4cCI4mJV74">Video</a> of the lecture.<br>
<a href="https://www.youtube.com/watch?v=48sCx-wBs34">Waiting room video</a>: Penrose tilings.<br><br>

Began by showing  <a href="https://rbsc.princeton.edu/sites/default/files/Non-Cooperative_Games_Nash.pdf">John Nash's PhD thesis</a>  in which he uses Brouwer's theorem to prove the existence of (what are now called) <i>mixed Nash equilibria</i>. <br><br>




Defined what is means for two spaces to be <i><b>homotopy equivalent</b></i>. Showed that any two homeomorphic spaces are homotopy equivalent. Also showed that the space <b>C</b>\{0} of non-zero complex numbers is homotopy equivalent to a circle.

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Stated a major theorem. <br><br>
<center><b>Theorem.</b> Homotopy equivalent spaces have the same Euler characteristic.</center>  
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Illustrated this theorem with a few examples.

<br><br>Stated Brouwer's fixed-point theorem: any continuous function f:&Delta;<sup>n</sup> ---> &Delta;<sup>n</sup> has a fixed point.
