Title:  Non-Vanishing Homology of the Matching Complex

Abstract: A matching on a graph G is any subgraph where the maximum vertex degree is 1.  Since edge-deletion preserves the property of being a matching, the set of all matchings on G forms a simplicial complex M(G).  We will survey results on the lowest non-vanishing homology group for M(K_n) and discuss the extension of these results to more general graphs, specifically the r-stable ones.  Prior familiarity with simplicial complexes and homology is assumed.

Title:  TBA

Abstract:  TBA

Title:  The number of representations of an integer as a sum of three squares

Abstract:  I will talk about a paper written by Andre Weil that describes the number of representations of an integer m as a sum of three squares. The talk will focus on the case when m is congruent to 3 modulo 8. Along the way we will discuss briefly similar results for the number of representations of an integer as a sum of two or four squares.

In this talk Shu Gu will present the proof of the boundary Lipschitz estimates with the Dirichlet boundary condition for second-order elliptic systems with periodic coefficients, arising in the theory of homogenization. The proof is due to M. Avellaneda and F. Lin. Future research in the area will also be discussed.