Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 16, 2016.

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Tuesday, February 16, 2016

12:00 pm in Altgeld Hall 243,Tuesday, February 16, 2016

Higgs bundles and flexibility of surface group representations

Steven Bradlow (UIUC Math)

Abstract: Among their many virtues, Higgs bundles reveal features of surface group representations that can be difficult to detect by other means. This will be illustrated in situations where the image of a representation into a group G lies in a subgroup of G, with special attention given to the case where G is a non-compact real form such as U(p,q) or Sp(2n,R).

3:00 pm in 241 Altgeld Hall,Tuesday, February 16, 2016

Strengthening theorems of Dirac and Erdös on disjoint cycles

Andrew McConvey (UIUC Math)

Abstract: Let $k$ be a positive integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. A seminal result of Corrádi and Hajnal states that a graph $G$ with at least $3k$ vertices and minimum degree at least $2k$ contains $k$ disjoint cycles. In 1963, Dirac and Erdös considered the case that $\delta(G) < 2k$. In particular, they proved if $k \geq 3$ and $|H_{k}(G)| - |L_{k}(G)| \geq k^{2} + 2k - 4$, then $G$ contains $k$ disjoint cycles. In this talk, we prove the following stronger result. If $k \geq 2$ and $|H_{k}(G)| - |L_{k}(G)| \geq 3k$, then $G$ contains $k$ disjoint cycles. In the special case that $V(G) = H_{k}(G)$, this reduces to the theorem of Corrádi and Hajnal for $k \geq 2$. This is joint work with Hal Kierstead and Alexandr Kostochka.

4:00 pm in 243 Altgeld Hall,Tuesday, February 16, 2016

$O(2,\mathbb{C})$-bundles, spin structures and theta characteristics on algebraic curves

Brian Collier (UIUC Math)

Abstract: We will discuss some of the contents of two separate papers (one by Atiyah and one by Mumford) which reprove some classical results of Riemann concerning Theta characteristics of an algebraic curve. This will be related to both holomorphic rank 2 orthogonal bundles and spin structures.