January 17

Planning meeting

January 24

Speaker: Zach Teitler

Title: Frobenius’s linear determinant preserver theorem

Abstract: In 1897 Frobenius gave a beautiful characterization of those linear transformations of matrices preserving the determinant: If T is linear and det(TX)=det(X) then TX=PXQ for some P,Q with det(PQ)=1, or similarly with the transpose of X. The proof is elementary. This theorem was part of the origin of representation theory, and led also to the field of linear preserver problems.

January 31

No meeting

February 7

Speaker: Jens Harlander 

Title: Left orderable groups are locally indicable

February 14

Speaker: Zach Teitler

Title: Power sum decomposition of determinant

Abstract: I'll report on a new expression for the determinant, as a sum of powers, using exponentially fewer terms than previously known power sum decompositions. This is a significantly improved upper bound for the Waring rank of the determinant. The decomposition is highly symmetric; I'll describe some of the symmetries, but the full symmetry group hasn't been worked out yet. This is work in progress jointly with Garritt Johns.

February 21

Speaker: Zach Teitler

Title: Alese's two cut theorem

Abstract: If a 2D curve turns at least once, then it can be cut twice and the resulting three pieces rearranged to form a closed loop. The proof only uses tools from elementary topology. This is recent work of Leonardo Alese. There are a number of generalizations and open questions, which could be student projects.

February 28

Speaker: Jens Harlander

Title: The Freiheitssatz for 1-relator groups

Abstract: The Freiheitssatz (freedom theorem) states that if G=< x_1,...,x_n | r >, where r is cyclically reduced and involves all generators, then any subgroup generated by a proper subset of the generators is free. I will present a short proof of the Freiheitssatz using the known (but highly non-trivial) fact that 1-relator groups are orderable.

March 6

Speaker: Kennedy Courtney

Title: Thesis Defense: The Directed Forest Complex of Cayley Graphs

Abstract: Let Gamma be a directed graph. The directed forest complex DF(Gamma) is a simplicial complex whose vertices are the edges of Gamma and whose simplices are sets of edges that form a directed forest in Gamma. We study the directed forest complex of Cayley graphs Gamma of finite groups. The homology of DF(Gamma) contains information about the graph, Gamma and about the group, G. The ultimate goal is to classify DF(Gamma) up to homotopy, compute its homology, and interpret the findings in terms of properties of Gamma. In this talk, we present progress made toward this goal.

March 13

Speaker: Allison Arnold-Roksandich

Title: TBA

March 20

Speaker: Jarosław Buczyński

Title: TBA

March 27

No meeting (Spring Break)

April 3

April 10

April 17

April 24

May 1

No meeting (last day of classes)