- We are by now accustomed to seeing marvelous facts in combinatorics (and related subjects) for which the proofs are simple, yet unenlightening. For example, the identities associated with Pascal’s triangle are well-known, but the usual proofs don’t shed much light on how it’s all connected. Thankfully, there are often elegant proofs for these facts which showcase the geometry of the underlying mathematics and help us better understand what’s really going on. In this talk, we’ll discuss some pretty examples of these proofs and apply the techniques to a recent problem in combinatorics. No background in combinatorics is needed.

- The symmetric group can be viewed as a factor group of the braid group by forgetting which strand is on top in each crossing. Algebraically, this can be achieved by giving the generators corresponding to crossing adjacent strands an order of 2. What happens when we give such crossings order n instead? In particular, when will such a group be finite? Considering this question will involve platonic solids, Coxeter groups, braid groups, Artin groups, complex reflection groups and hyperplane arrangements. However, no previous knowledge of these concepts will be assumed.

- A flow on an oriented graph is a labeling of the edges such that the sum of the labelings directed into each each edge
*e*is equal to the sum of the labelings directed out of*e*. We will focus on counting the number of flows using integers between -*k*and*k*. Specifically, a polytope with integral vertices can be defined to represent the constraints on allowable flows. Then, there is a well-defined theory in which we can associate a polynomial to this polytope that counts the number of interior lattice points, which corresponds exactly to the number of labelings. Lastly, this polynomial satisfies a combinatorial reciprocity law that incorporates the totally cyclic orientations of a graph*G*. No background in combinatorics or graph theory is necessary.

- A Young tableau is a collection of left-justified rows of boxes with the row length weakly decreasing such that each box is labelled with a positive integer. These combinatorial objects give rise to a large collection of representations of the symmetric group. In particular, using Young tableaux, there is a natural bijection between the set of partitions of n and the set of isomorphism classes of irreducible representations of the symmetric group on n letters. In this talk we will explore this bijection and provide combinatorial answers to algebraic questions about these representations (e.g. the branching rule).

- In this talk we will look at embeddings of graphs in manifolds, and show that the only really interesting case is that of graphs embedding in surfaces. This motivates the definition of the genus of a graph, which is the smallest genus surface into which the given graph can be embedded. We will calculate some naive bounds on the genus, and discuss a procedure by which the genus of a graph can be found. No knowledge of graph theory is required.

- Let M be a 2 or 3-manifold and G be a Lie group; in this talk we will study the set R(M,G) consisting all representations from the fundamental group of M into G. R(M,G) is about as non-discrete an object as you can find and so you might feel that it has no place in this seminar. However, we will see that by studying combinatorics of triangulations of M we can find nice coordinates on (large chunks of) R(M,G) that have a geometric meaning. Furthermore, these coordinates shed light on an interesting relationship between various relationships between 2 and 3 manifolds. This talk should be mostly expository and focus on the cases of G=PGL_2(C) and PGL_3(C).

*Abstract omitted.*