The discovery you mentioned regarding a galaxy that appears to be 32 billion light-years away is intriguing, but it's essential to understand the concept of cosmological redshift to resolve the apparent contradiction with the age of the universe.
Cosmological redshift occurs due to the expansion of the universe. As space expands, it stretches the wavelength of light traveling through it, causing it to shift towards longer wavelengths, which is commonly referred to as "redshift." The amount of redshift is directly related to the distance between the source of light and the observer.
In the case of the galaxy you mentioned, its light has indeed traveled a great distance, but the distance isn't solely determined by the time it took for the light to reach us. It is also influenced by the expansion of space during that time. As the light from the galaxy traveled towards us, the space between us and the galaxy continued to expand, stretching the wavelength of the light and causing its redshift.
Due to the expansion of the universe, the rate of expansion has changed over time. The current accepted age of the universe is estimated to be around 13.7 billion years based on various observations and measurements. However, the distance to the galaxy in question is not determined solely by the current age of the universe, but also by the expansion rate at the time the light was emitted.
The expansion of the universe allows light from extremely distant objects to reach us, even though the distance between us and those objects has increased during the travel time of the light. The light we observe from such distant objects has been traveling for a long time, but the expansion of space accounts for the apparent discrepancy between the light-travel time and the age of the universe.
Therefore, the discovery of galaxies that appear to be significantly farther away than the age of the universe is not inconsistent with our understanding of cosmology and the expansion of space. It highlights the remarkable scale and vastness of the universe.