Nov

23

Tue

Janitha Awedige will discuss the paper

“Inference Rules for Probability Logic” by Marija Boricic.

Feb

24

Thu

The relative class number one problem for function fields

Abstract: Gauss conjectured that there are nine imaginary quadratic fields of class number 1; this was resolved in the 20th century by work of Baker, Heegner, and Stark. In between, Artin had introduced the analogy between number fields and function fields, the latter being finite extensions of the field of rational functions over a finite field. In this realm, the class number 1 problem admits multiple analogues; we recall some of these, one of which was “resolved” in 1975 and then falsified (and corrected) in 2014, and another one of which is a brand-new theorem in which computer calculations (in SageMath and Magma) play a pivotal role.

Mar

10

Thu

Abstract

Topological data analysis (TDA) is a new approach to analyzing complexdata which often helps reveal otherwise hidden patterns by highlightingvarious geometrical and topological features of the data. Persistenthomology is a key in the TDA toolbox. It measures topological featuresof data that persist across multiple scales and thus are robust withrespect to noise. Persistent homology has had many successfulapplications, but there is room for improvement. For large datasets,computation of persistent homology often takes a significant amount oftime. Several approaches have been proposed to try to remedy this issue,such as witness complexes, but those approaches present their owndifficulties.

In this work, we show that one can leverage a well-known data structurein computer science called a cover tree. It allows us to create a newconstruction that avoids difficulties of witness complex and canpotentially provide a significant computational speed up. Moreover, weprove that the persistence diagrams obtained using our novelconstruction are actually close in a certain rigorously defined way topersistent diagrams which we can get using the standard approach basedon Čech complexes. This quantifiable coarse computation of persistentdiagrams has the potential to be used in many applications wherefeatures with a low persistence are known to be less important.

Apr

5

Tue

Speaker: Chuang Xu (Technical University Munich)

Apr

7

Thu

Title:

Analyzing walks with combinatorics and automata theory

Abstract:

The enumeration theorem by Chomsky and Schützenberger revealed

a significant intersection between the theory of automata and

enumerative combinatorics. Since then, much progress has been made in

both fields. However, their intersection remains unchanged in the sense

that no further enumeration theorem emerged beyond that of

Chomsky-Schützenberger. We survey the literature in both fields and

picture what it would look like to expand the intersection between them.

## University of Hawaiʻi at Mānoa