Коллоквиум. How to realize Hecke algebras in many different ways?

Hecke algebra is a rich object that has amazing connections to various subjects such as representation theory, number theory, geometry and categorification. It has notable applications on local Langlands correspondence and Kazhdan-Lusztig polynomials. In this talk, we shall take a tour on the development on Hecke algebras, starting from classical perspectives to modern approaches and applications, including several ways to realize Hecke algebras.