Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, August 24, 2021.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
      July 2021             August 2021           September 2021
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3    1  2  3  4  5  6  7             1  2  3  4
4  5  6  7  8  9 10    8  9 10 11 12 13 14    5  6  7  8  9 10 11
11 12 13 14 15 16 17   15 16 17 18 19 20 21   12 13 14 15 16 17 18
18 19 20 21 22 23 24   22 23 24 25 26 27 28   19 20 21 22 23 24 25
25 26 27 28 29 30 31   29 30 31               26 27 28 29 30



Tuesday, August 24, 2021

1:00 pm in 345 Altgeld Hall,Tuesday, August 24, 2021

#### Cycles in Color-Critical Graphs

###### Douglas B. West (Zhejiang Normal University and UIUC)

Abstract: Tuza [1992] proved that a graph $G$ with no cycles of length congruent to $1$ modulo $k$ is $k$-colorable. For $2\le r\le k$, we prove that if $G$ has an edge $e$ such that $G-e$ is $k$-colorable and $G$ is not, then $e$ lies in at least $(k-1)!/(k-r)!$ cycles of length $1{\,\mathrm{mod}\,} r$ in $G$, and $G-e$ contains at least $(1/2)(k-1)!/(k-r)!$ cycles of length $0 {\,\mathrm{mod}\,} r$.

A $(k,d)$-coloring of $G$ is a homomorphism from $G$ to the graph with vertex set $Z_k$ defined by making $i$ and $j$ adjacent if $d\le j-i \le k-d$. Assume $\gcd(k,d)=1$, and let $s=d^{-1}{\,\mathrm{mod}\,} k$. Zhu [2002] proved that $G$ is $(k,d)$-colorable when $G$ has no cycle $C$ with length congruent to $is$ modulo $k$ for any $i\in \{1,...,2d-1\}$. In fact, only $d$ classes need be excluded: we prove that if $G-e$ is $(k,d)$-colorable and $G$ is not, then $e$ lies in at least one cycle with length congruent to $is {\,\mathrm{mod}\,} k$ for some $i$ in $\{1,...,d\}$.

These results are joint work with Benjamin R. Moore.

2:00 pm in 347 Altgeld Hall,Tuesday, August 24, 2021

#### Heat kernel upper bounds for symmetric Markov semigroups

###### Panki Kim (Seoul National University)

Abstract: It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this talk, we discuss the equivalence among these and off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. This talk is based on a joint work with Zhen-Qing Chen, Takashi Kumagai and Jian Wang.