Convex-Biembeddable Linear Orders
Convex-Biembeddable Linear Orders
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The shuffle of a non-empty countable set of linear orders (the shufflands) is the linear order obtained by replacing each rational in ℚ with an isomorphic copy of one of the shufflands in a homogenous manner.
Collaborating with a fourth-year undergraduate, I co-authored a paper proving that any two countable shuffle linear orders that embed as intervals within each other are order isomorphic.
Image caption: The six columns in this image represent linear orders. Our result asserts that if the first three columns represent isomorphic linear orders, and the same holds for the other three columns, then all six columns are isomorphic
Image created using Asymptote on LaTeX.
Recognised and explored a connection between finite automata and a class of linear orders arising in logic and representation theory, discovering a new technique for hammock computation for string algebras
Image caption: The red points on the binary tree consitute the language of the automaton on top. Under inorder, the red points form an isomorphic copy of the linear order ω + (ω* × ω*) + ω × (ω* + ω).
Image created using TikZ on LaTeX and Turtle on Python.
Stable Ranks for String Algebras
Stable Ranks for String Algebras
A research project with Dr, Amit Kuber and Vinit Sinha in the representation theory of string algebras. We showed that the stable rank of any string algebra is strictly less than ω².
Image caption: A pictorial representation of a 2-hammock in the cartesian lattice (ω + Ξ(ω, ω*, ζ) + ω*) × (ω + Ξ(ω, ω*, ζ) + ω*). Each color region is a maximal scattered box, with the color encoding its order type.
Image created using Asymptote on LaTeX.
An undergraduate project undertaken under the supervision of Dr. Ashutosh Kumar, in which I generalized the von Neumann hierarchy to a version of set theory with two types of memberships and extended the Conway normal form to base-κ for regular uncountable κ.
Image caption: The surreal number tree
(Image by Lukáš Lánský, distributed under a CC BY-SA 3.0 license.)
Scott Complexities of Linear Orders
Scott Complexities of Linear Orders
A research project with Dr. Amit Kuber to compute the Scott complexities of finitely presented linear orders, a class of finite Hausdorff rank linear orders first introduced by Läuchli and Leonard. We obtained preliminary results for linear orders with low Hausdorff rank.
Image caption: The 3-signed tree at the top of the picture is a finite presentation of the linear order at the bottom.
Image created using Asymptote and TikZ on LaTeX.
A reading project on infinite combinatorics and descriptive set theory undertaken under the supervision of Dr. Ashutosh Kumar during my second year at IITK.
Last updated: 20 November 2024
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