Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, August 26, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, August 26, 2014

2:00 pm in 241 Altgeld Hall,Tuesday, August 26, 2014

#### Three Graph Coloring Problems

###### Sarah Loeb (Department of Mathematics, University of Illinois)

Abstract: I will discuss three graph coloring problems. I will first discuss bounds on the maximum 3-dynamic chromatic number of planar graphs; this is joint work with Thomas Mahoney, Benjamin Reiniger, and Jennifer Wise. Next, I'll talk about some ongoing research on backbone coloring of planar graphs. Finally, I'll talk about finding an I,F-partition in sparse graphs.

3:00 pm in 243 Altgeld Hall,Tuesday, August 26, 2014

#### Morse theory and moduli of Higgs bundles

###### Tom Nevins (UIUC Math)

Abstract: The moduli space (or stack) of Higgs bundles on a smooth projective curve plays a central role in aspects of geometric representation theory, the study of fundamental groups of Riemann surfaces, and integrable systems. The dichotomy between semistable and non-semistable Higgs bundles crucially underlies the study of these spaces. I will discuss a refinement of this dichotomy, the Harder-Narasimhan stratification, and explain how a Morse-theoretic study of it, similar to the classical Atiyah-Bott analysis but “one categorical level higher,” leads to new topological information about these spaces. If there is time, I will also explain some new conclusions for the geometric Langlands program. The talk will not assume familiarity with any of the objects or tools mentioned above except curves/Riemann surfaces. This is based on joint work with Kevin McGerty.

3:00 pm in 241 Altgeld Hall,Tuesday, August 26, 2014

#### Disjoint Cycles and Equitable Coloring

###### Elyse Yeager   [email] (UIUC Math)

Abstract: In 1963, Corrádi and Hajnal famously proved the following: If a graph has minimum degree at least $2k$ and at least $3k$ vertices, then it contains a set of $k$ vertex-disjoint cycles. This degree bound is sharp, but has been improved by considering conditions other than minimum degree. We will discuss several such improvements, as well as Corrádi-Hajnal-type results guaranteeing the existence of graphs other than cycles. An equitable coloring of a graph is a proper vertex coloring where no two color classes differ in size by more than one. The most obvious relation between equitable coloring and the problem of finding disjoint cycles is this: A graph $G$ on $3k$ vertices contains a set of $k$ disjoint cycles if and only if the complement of G is equitably $k$-colorable. Chen, Lih, and Wu conjectured in 1994 that a connected graph $G$ is $\Delta(G)$-equitably colorable if it is different from $K_m$, $C_{2m+1}$, and $K_{2m+1,2m+1}$ for every $m \geq 1$. We discuss an Ore-type analog to this conjecture: that every $k$-colorable graph $G$ with maximum degree sum of adjacent vertices at most $2k + 1$ is equitably $k$-colorable unless it contains $K_{1,2k}+K_{k-1}$; $K_{c,2k-c}+K_k$ for odd $c$; or a third graph in the case $k=3$. This is joint work with Alexandr Kostochka, H.A. Kierstead and Theodore Molla