- Date: 15 Oct. (Thu.)
- Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
- Time: 16:50-19:00
- Speaker 1: Riki Kitano（Ritsumeikan University）
- Time:16:50-17:50
- Title: パスグラフ上の反射ウォークとゼータ関数
- Abstract:
有限パスグラフ上の反射壁ランダムウォークと、それに対応する Szegedy ウォークに付随するゼータ関数を考える。推移確率が対称でない場合も含めた一般化反射原理を考え、ランダムウォークのゼータ関数を、境界部分と、二つの境界反射が生成する無限二面体群の各元に対応する局所対応部分との積として表す。さらに、古典系と量子系のゼータ関数を結ぶ Konno-Sato 対応を局所対応部分ごとに定め、この分解を量子ウォークへ移す。本講演では、これらの構成やトレース公式を紹介し、得られる分解と極限における構造との関係にも触れる。
- Speaker 2: Akinari Hoshi（Nigata University）
- Time:18:00-19:00
- Title: Rationality problem for fields of invariants, norm one tori and Hasse norm principle
- Abstract:
The aim of this talk is to give a survey of recent developments in the rationality problem
for fields of invariants, e.g. Noether’s problem, rationality problem for algebraic tori,
norm one tori and quasi-monomial actions.
I will give some explicit examples of (non-)stably (resp. retract) rational fields
which are weaker concepts than the rationality.
Negative results may be obtained by using (stable) birational invariants, e.g. flabby class,
unramified Brauer (cohomology) group which is an avatar of the Artin-Mumford invariant.
By Colliot-Thelene and Voisin’s result (2012, Duke Math. J.), an explicit example of
non-vanishing of the unramified cohomology group of degree three provides an explicit
counter-example of the integral Hodge conjecture.
A counter-example of Noether’s problem for N\rtimes A_6 over C is also given
where A_6 is the alternating group of degree 6 and N=(Z/10Z)^9.
As an application, we present recent results about Hasse norm principle for K/k
where K/k is a finite separable field extension of a global field k.
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