The mapping class group is an algebraic invariant of a topological space that detects the symmetries of that space. In the case of surfaces, this group has a deep connection with the fundamental group and the associated Teichmuller space. In this talk I'll give a introduction to the mapping class group along with some examples and intuition for some interesting results surrounding it. Time permitted, I'll also discuss some associated results for so-called "big" mapping class groups.

In the 1940's, Jakob Nielsen set about analyzing and classifying the elements in the mapping class group for closed orientable surfaces. In the 1970's, while working on his famous Geometrization Conjecture, Thurston managed to successfully complete what we now call the Nielsen-Thurston Classification of mapping classes. In this talk I'll motivate the classification on the torus, highlight some parallels with hyperbolic isometries, and present an overview of Thurston's proof for surfaces with genus g>1.

Although 4-manifolds are outside our imagination, there is a very nice way to encode the topological information in the so-called handlebody diagram. In this talk, we'll first start with basic descriptions and we'll see examples of handlebody diagrams for low-dimensional manifolds. Time-permitted, we will see how to obtain more suitable handlebody diagrams. Namely, we will modify it using two fundamental topological operations: handle cancelation/creation and handle sliding.

Last time we worked on handlebody diagrams of 1- and 2-dimensional manifolds. In the second part, we will describe how to obtain more suitable handlebody diagrams via two fundamental topological operations: handle cancelation/creation and handle sliding. In this talk we will see handlebody diagrams of 3- and 4-manifolds as well.

Braid group theory is an interesting and versatile subject with applications in many different fields of mathematics including algebra, topology, and quantum computation. In this talk, I will give an introduction to the braid groups and share my intuition for why and how these groups are used. In particular, I will discuss the representations of the braid groups and some of the motivating open questions that fuel my research. Many of the famous representations of the braid groups are parametrized by a variable $q$ (these representations secretly come from quantum groups). I will share some of my results about choosing careful specializations of $q$ with the aim of structural results about the image of the representation.

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