Algebraic structures for topological summaries of data
Báo cáo viên: Giáo sư Ezra Miller (Duke University)

Thời gian: 9h30, Thứ 6, ngày 7 tháng 6 năm 2019
Địa điểm: Hội trường Tầng 2, Nhà A6, Viện Toán học, 18 Hoàng Quốc Việt, Cầu Giấy, Hà Nội
Tóm tắt: This talk introduces an algebraic framework to encode, compute, and analyze topological summaries of data. The main motivating problem, from evolutionary biology, involves statistics on a dataset comprising images of fruit fly wing veins, which amount to embedded planar graphs with varying combinatorics. Additional motivation comes from statistics more generally, the goal being to summarize unknown probability distributions. The algebraic structures for topological summaries take their cue from graded polynomial rings and their modules, but the theory is complicated by the passage from integer exponent vectors to real exponent vectors. The key to making the structures practical for data science applications is a finiteness condition that encodes topological tameness -- which occurs in all modules arising from data -- robustly, in equivalent combinatorial and homological algebraic ways. Out of the tameness condition surprisingly falls much of ordinary commutative algebra, crucially including concepts of minimal generator and primary decomposition.

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