Time: 9:30-12:00, Wednesday, November 27th
Venue: E4-233
Speaker: Sean Brendon Lynch
Titile: Two-Dimensional Zeta Functions
Abstract: In the olden days of number theory, some people thought that divisors were more important than ideals and some people thought the opposite. These days, we may take a step back and admire beautiful relations between the two. They are best viewed in the context of more general algebraic geometry, for example, the relationship between Chow theory and K-theory. We present a lesser known example with infinite product formulae relating zeta functions for closed subschemes to zeta functions for 0-cycles. This leads to my recent work with Daniel Chan about zeta functions of maximal orders on arithmetic surfaces. We'll end by explaining how analogous zeta functions on arithmetic curves yield information about the Brauer group.
Speaker: Yichen Tong
Title: On the fundamental groups and the magnitude-path spectral sequence of a directed graph
Abstract: Directed graphs are crucial combinatorial objects and are of great interest in both pure and applied mathematics. In particular, methods from algebraic topology, especially homotopy theory, have been introduced to study (directed) graphs. The fundamental group and the path homology of a directed graph are introduced by Grigor'yan, Lin, Muranov, and Yau, and is related through the Hurewicz theorem. Later, Di, Ivanov, Mukoseev, and Zhang showed that the fundamental group admits a natural sequence of quotient groups called r-fundamental groups, which captures the quasimetric structure of a directed graph that the fundamental group cannot capture. On the other hand, the magnitude-path spectral sequence connects magnitude homology and path homology of a directed graph due to Asao, and it may be thought as a sequence of homology of a directed graph. In this talk, we discuss relations of the r-fundamental groups and the magnitude-path spectral sequence through the Hurewicz theorem and the Seifert-van Kampen theorem. This is joint work with Daisuke Kishimoto.
