THE CATHOLIC UNIVERSITY OF AMERICA
Washington, DC 20064
SEMINAR IN FUNCTIONAL ANALYSIS
AND RELATED AREAS
Wednesday, April 29, 2026
5:15 p.m. - 6:15 p.m. 

 1. SPEAKER: Gabriel Radich
                    The Catholic University of America

TITLE: Attention to Detail: A Modern Mathematical Approach to Image-based Tasks in AI

ABSTRACT: When people, especially in mainstream media, talk about AI and its potential to ‘reason’ independently, have its own agenda and turn against humanity, oftentimes what they fail to recognize is that AI is simply a complicated probability model comprised of millions of parameters, matrix operations, and convenient mathematical formulas such as softmax. In this talk, I intend to unveil the mathematics that governs the components of the Transformer model, which undergirds ChatGPT. My primary interest is not the Transformer’s impact on ‘understanding’ human language, but rather, it’s impact on teaching machines to ‘see’ like humans, or even better than humans. This has spurred an effort to incorporate AI into remote sensing tasks, such as infrared and visible image fusion. In this talk, I will demonstrate how one would go about defining a neural network (with Transformer Encoder and Decoder) mathematically to be used in these image fusion tasks.

 2. SPEAKER: Thomas Aldredge
                    The Catholic University of America

TITLE: Limiting Tangents to Horocycles: The Connection between the Horocycle, the Golden Ratio, and the Ideal Triangle

ABSTRACT: This talk considers the problem of drawing tangents to the horocycle (also called “oricycle”) in the hyperbolic plane H^2, and then examines the relationship of the tangency problem to the ideal triangle. The horocycle is defined as a locus whose chords have perpendicular bisectors which are all parallel. In order to solve the tangent problem, we first consider the elementary principles of hyperbolic geometry. Next, we introduce more modern identities of hyperbolic trigonometry, and explain the relationship of “hyperbolic trigonometry” and “hyperbolic geometry,” which is not often discussed in the literature, and helps illuminate the use of the term “hyperbolic.” With these preliminaries in place, we outline the proof of tangent construction, and obtain a formula for the chord length of a tangent in terms of distance to the curve. Meditating on limiting behavior of chords leads to an analogy with the circle, where the limiting chord has length sqrt.2. We then consider the hyperbolic case, and show how the limiting chord on the horocycle is 2ln(φ), where φ is the Golden Ratio. A classic result in the field, however, shows that the ideal triangle has a unique in-circle, whose points of tangency define an equilateral triangle with side length 2ln(φ). We illustrate the connection between these two constructions, and conclude with some interesting corollaries.  

 PLACE:  Aquinas Hall, Room 108. The talk will also be held on Zoom from 5:15 p.m. to 6:15 p.m. (ET). See the information about the corresponding link below.

ORGANIZERS: V. Bogdan (The Catholic University of America), P. Kainen (Georgetown University), R. Kalpathy (The Catholic University of America), and A. Levin (The Catholic University of America).

Tel: 202-319-5221  E-mail:levin@cua.edu

Web page:  https://mathematics.catholic.edu/faculty-and-research/mathematics-seminar/index.html

The talk will be on Zoom as well (from 5:15 p.m. to 6:15 p.m. ET).   The following is the information about the corresponding link.