Algebra Laboratory, Department of Mathematics, FMIPA UGM
1th speaker: Nguyen Bich Van, Ph.D from CMC University, Hanoi, Vietnam on ” Decomposition Theorems of Feature Spaces, Filters, and Irreducible Representations of UT₃(F₃) in Group Equivariant Convolutional Networks ”
2nd speaker: Sekar Nugraheni, S.Si., M.Sc. from UGM on ” Robinson-Colombeau Ring of Generalized Numbers “
Date: Wednesday, September 10, 2025
Time: 09.30 – 12.30 WIB
Location: Multimedia Room, Dept. of Chemistry, FMIPA UGM/ Zoom Meeting
Dr. Nguyen Bich Van opened her presentation by highlighting the motivation behind classical convolutional neural networks (CNNs) and the move towards their modern extension, group-equivariant convolutional neural networks (G-CNNs). She posed a central question: Why is the group UT₃(F₃) particularly useful for research in G-CNNs?
To address this, she reviewed several key algebraic and representation-theoretic tools, including Schur’s Lemma, Maschke’s Theorem, feature maps and feature spaces, equivariant convolution, convolution filters, and steerable convolutions. She emphasized the significance of equivariance in the structure of convolutional layers and its role in preserving symmetries.
Dr. Van then presented her main results, which included:
– Theorems characterizing the decomposition of equivariant feature spaces and filters.
– A structural preservation theorem for invariant subspace chains in G-CNNs.
– Explicit matrix forms of irreducible representations of UT₃(F₃).
Finally, she demonstrated how these theoretical results can be applied to design new G-CNN architectures that respect deep algebraic structures, opening potential applications in symbolic visual learning.
The second talk was delivered by Sekar Nugraheni, who focused on the Robinson-Colombeau Ring of Generalized Numbers. She began with a review of the non-Archimedean property and non-Archimedean fields, setting the stage for the algebraic framework of generalized numbers.
She then introduced the construction of Rρ, defined as the set of ρ-moderate nets (x_ε) in Rᴵ for a given directed poset I. She showed how Rρ can be equipped with a ring structure through suitably defined addition and multiplication on nets.
Next, she introduced an equivalence relation on Rρ, defining the Robinson-Colombeau Ring of Generalized Numbers as the set of equivalence classes under this relation. This construction provides a rich non-Archimedean field with several interesting properties.
Towards the end of her talk, Sekar discussed the sharp topology on this ring, including the notions of sharply closed, sharply open, and sharply bounded sets, all generated through net-based constructions.
