①【Inspirations for Moduli Spaces from Counting Problems　/　Motohico Mulase】

In this talk I will weave a story of a simple counting
problem about cell-decompositions of a closed topological surface.
Despite the elementary formulation of the problem, the results as
concrete formulas exhibit unexpected connections to, and new
interpretations of, the topological properties of the moduli
stacks of stable algebraic curves. The key idea of proving all
these formulas lies in "topological recursion," the Laplace
transform of an elementary combinatorial relation. The story
spotlights the hidden "spectral curve" and its quantization known
as an "oper." Finally we switch to a new, still developing story of
the geometry translating Apéry's irrationality proof of $\zeta(3)$.
In this new situation, we have an oper and well-understood moduli
problem. Yet we do not know what the spectral curve is, which is
expected to be the (semi) classical limit of the oper.

The first part of the talk is based on my joint papers with
Olivia Dumitrescu, Bertrand Eynard, Paul Norbury, Sergey Shadrin,
Piotr Su\lkowski, and others.

②【3 次元球面の種数が 3 と 4 の Heegaard 分解に対する Powell 予想　/　古宇田 悠哉】

Powell 予想とは，3 次元球面の Heegaard 分解の写像類群の具体的な
有限生成系を提唱するものであり，3 次元多様体，結び目，写像類群の研究に関連す
る未解決問題である．本講演では，この予想の背景を概説した後，Heegaard 分解の種
数が3 と 4 の場合にこの予想が正しいことを証明する．種数が 3 の場合の結果は
Freedman--Scharlemann によるものの別証明であり，種数が 4 の場合の結果は新事実
となっている．本講演の内容は，Sangbum Cho 氏 (Hanyang University・KIAS),
Jung Hoon Lee (Jeonbuk National University) との共同研究に基づく．

(大談話会)