Mathematics
Presenter(s): Melissa Sugimoto
Advisor(s): Dr. Mihaela Vajiac
The Brouwer fixed-point theorem, proven by L.E.J. Brouwer in 1909, is one of the fundamental theorems of topology with widespread applications across multiple fields of mathematics, as well as multiple natural realizations - even appearing in physics and economics [1]. The theorem states that every continuous function has a fixed point, or in other words, every continuous function has a point where the output is the same as the input. In this poster, we provide a proof of the Brouwer fixed-point theorem and present several applications. For example, the theorem is interlinked with the game of Hex [2], and can also be used to show that if you overlay two maps of one region of different sizes, there must always be a point that represents the same place on both maps [3]. Returning to mathematics, one can also use the Brouwer fixed-point theorem to prove that certain kinds of equations have solutions [4]. References: [1] A. Bright. Applications of Brouwer's Fixed Point Theorem. Masters Presentation, 2016. [2] D. Gale. The Game of Hex and the Brouwer Fixed-Point Theorem. American Mathematical Monthly, vol. 86, no. 10, 1979. [3]Brouwer Fixed Point Theorem, brilliant.org. [4] M. Tabata and N. Eshima. Application of the Brouwer and the Kakutani fixed-point theorems to a discrete equation with a double singular structure. Fixed Point Theory and Applications, article 24, 2018.