# Mathematics Department

## Student Colloquia

Most Tuesday afternoons during the academic year, the Mathematics students host a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels. Refreshments are served before the talks.

Please email Ricardo Rojas-Echenique and Forrest Glebe, if interested in signing up for a talk.

### 2016-17 Schedule

#### Fall

4:40 in Physics 123 (unless marked otherwise).

Sept 6 Graduate School Q&AAndrew Bray, Adam Groce, Angélica Osorno, Mathematics Department, Reed College Linear Systems on Graphs (Winning the Dollar Game)Forrest Glebe and Palak Jain, Reed CollegeThe chip-firing game, starts with some, possibly negative, number of chips at each vertex of $$G$$. A vertex $$v$$ can fire, sending one chip along each outgoing edge to a neighbouring vertex (in turn losing/subtracting that many chips from itself). After firing a sequence of vertices, the game is "won" if none of the vertices have a negative number of chips on them. Our research this summer dealt with the "winning" configurations for any particular chip-firing game.    We found that there are a few equivalent ways for counting these. One involves expressing all divisors of a particular type  uniquely as linear combination of a finite class of divisors. Another uses a bijection between the winning divisors and invariant polynomials. Bijections Between the Recurrent Sandpiles of an Eulerian Digraph and its ReverseRicardo Rojas-Echenique, Reed CollegeAn Eulerian digraph is a directed graph where the indegree of each vertex is equal to the outdegree.  After designating a single vertex of an Eulerian digraph as a sink, the set of recurrent sandpiles on the graph forms a finite abelian group isomorphic to the cokernel of the Laplacian matrix of the graph. Given a digraph, we can also construct its reverse graph by reversing the direction of each edge. In the case of Eulerian digraphs we have the salient property that the Laplacian of the reverse graph is equal to the transpose of the Laplacian of the original graph.  This allows us to conclude the two groups of recurrent sandpiles are isomorphic.  While this isomorphism shows that the number of recurrent sandpiles on an Eulerian digraph and its reverse are equal, the two sets of sandpiles can look quite different. We discuss a few possible natural bijections between the recurrent sandpiles on an Eulerian digraph and its reverse. Constraints on Friends of 15Niko Terry, Reed CollegeThe abundancy index is a measure of how many factors a number has, proportional to its size. In this talk, we examine the constraints on "friends" of 15, that is, numbers that share its abundancy index of 8/5. We show through elementary arguments that any friend of 15 must have certain properties, in fact it must have 5 as its smallest prime factor, raised to an even power, must have more than three prime factors, and must fulfill certain congruences in the powers of its prime factors depending on which of several congruences those prime factors fulfill. Behaviour of Schur Functions Under Random Ring HomomorphismsJesse Kim, Reed CollegeThe ring of symmetric functions plays an important role in algebraic combinatorics. The Schur functions form a linear basis for this ring. In order to better understand the structure of the Schur function basis, in this talk we consider random homomorphisms from the ring of symmetric functions to a finite field, and examine the behaviour of Schur functions under them. Combinatorics of the $$\times n$$ quantum GrassmannianChris Keane, Reed CollegeThe algebra called the quantum Grassmannian" has become a recent object of focus for mathematicians working in combinatorics as well as in partial differential equations. Postnikov recognized multiple bijections between the torus-invariant prime ideals of the algebra and other combinatorial objects that opened up several new directions of investigation. We discuss one such direction that involves putting several partial orders on the set of generators of the algebra, and recognize a way to obtain information about the torus-invariant primes from the poset structures. We discuss how the obtained data can be used to compute generating sets for the $$2\times n$$ primes, which was the focus of my REU project this summer.

#### Spring

4:40 in Physics 123 (unless marked otherwise).

Feb 14 Different Constructions of the Stone-Čech compactification of the Natural NumbersForrest Glebe, Reed CollegeThe Stone-Čech compactification is a way of embedding a topological space $$X$$ into a compact space $$\beta X$$, so that continuous maps from $$X$$ to other compact spaces can be preserved. While the Stone-Čech compactification can be completely characterized by its mapping properties the elements of the set $$\beta N$$ , under certain constructions are interesting in thier own right. I will go over different equivalent ways of constructing the Stone-Čech compactification of the natural numbers, $$\beta N$$ . One involves embedding the natural numbers in product of real intervals. Another way puts a topology on the set of ring homomorphisms from $$C^*(N)$$ , the ring of bounded real sequences, to the real numbers. In particular we give $$R^{C^*(N)}$$ the product topology. Finally, we will observe that all the ring homomorphisms are completely characterized by their behavior on characteristic functions, so the Stone-Čech compactification can be made as a set of subsets of $$N$$ called ultrafilters. The cool but weird topological properties of $$\beta N$$ can be seen most clearly this way. Constructing the Burnside and Grothendieck-Witt CategoriesRicardo Rojas-Echenique, Reed College A Gysin functor is a contravariant functor to the category of commutative rings along with a covariant functor to the category of abelian groups satisfying a few axioms. Following Dugger, we will introduce the theory of Gysin functors in order to construct the Burnside and Grothendieck-Witt categories over a field k and a natural functor between these categories. If time permits, we will look at the specific case of finite fields and discuss methods of extending these results to local fields. Cohomological Lego BricksSam Johnston, Reed CollegeTake your favorite group G and an abelian group A. A classic question is classifying the groups that can be “built” out of G and A. That is, what are all the groups E which have a normal subgroup A, with quotient group G? This question corresponds to calculating classes in degree 2 group cohomology, but as will be shown, this question is also nicely handled by considering objects called 2-groups, which can be thought of as 2-dimensional groups. In fact, equivalence classes of groups E described above correspond to equivalence classes of morphisms from a certain 2-group corresponding to G, and a 2-group corresponding to the automorphisms of A. This talk will give background on the extension problem and 2-groups, provide a sketch of the above result, and demonstrate how this perspective allows for computation.