Stephen Hawking's PhD thesis, "Properties of Expanding Universes," completed in 1966 at the University of Cambridge, has become a cornerstone in the fields of cosmology and the study of black holes. This groundbreaking document not only established Hawking's reputation as one of the most brilliant minds in theoretical physics but also addressed vital questions about the nature of the universe and its expansion.

Hawking’s thesis delves deep into the complex interplay between singularities, general relativity, and the expansion of the universe. At a time when fundamental concepts like the Big Bang theory were burgeoning, his work provided significant theoretical support and sparked intriguing questions that would guide future explorations in physics.

The Context of Cosmic Investigation

By the mid-20th century, scientists had made significant strides in understanding the vast structure of the cosmos. Albert Einstein's General Theory of Relativity had set the groundwork for comprehending the curvature of space-time, while Edwin Hubble's observational evidence confirmed that the universe was expanding. Yet, the mysteries surrounding singularities—points where the laws of physics as we know them cease to operate—remained enigmatic.

Hawking, under the mentorship of Dennis Sciama, undertook the challenge of deciphering these complexities. His objective was not merely to articulate theoretical models but to understand the implications of these models on the fabric of reality itself.

Groundbreaking Contributions of the Thesis

Hawking's analysis in "Properties of Expanding Universes" highlighted several key insights:

1. **Singularities as Physical Realities** Hawking established that singularities in space-time, where gravitational forces become infinite, are integral to our understanding of the universe's origins. His work contended that under the constraints of general relativity, singularities are unavoidable, leading to the compelling conclusion that the universe must have originated from such a point. This framework solidified the theoretical underpinnings of the Big Bang model and revolutionized the way we perceive cosmic beginnings.

2. **Causality in an Expanding Universe** Examining the concept of causality, Hawking's research questioned whether information and energy can traverse paths that defy the cone of causation. With mathematical precision, he illustrated that in an expanding universe, causality holds strong, ruling out the likelihood of paradoxical scenarios like time loops or faster-than-light communication.

3. **Matter’s Influence on Cosmic Expansion** The varying distributions of matter affect how the universe maintains its expansion. Hawking’s thesis assessed both uniformly spread and clumped matter in the cosmos, significantly advancing our understanding of galaxy formation and large-scale structures emerging from a smooth universe.

4. **Boundary Conditions and Universal Shape** By exploring the possible boundary conditions of the universe, Hawking's thesis laid the groundwork for his later conceptualization of the 'no-boundary' proposal with physicist James Hartle. He addressed different topological possibilities, assessing how they could affect the universe’s expansion.

The Mathematical Complexity

Hawking's thesis is rich in mathematical depth, employing sophisticated tensor calculus and differential equations rooted in general relativity—tools essential for framing modern cosmological theories. The clarity of his mathematical approach enabled profound insights into singularities and cosmological structure.

A Turning Point for Cosmology

The publication of "Properties of Expanding Universes" marked a pivotal moment for Stephen Hawking and cosmology as a whole. His insights propelled discussions on the origins of the universe and framed future research directions, especially concerning black holes and quantum gravity.

In subsequent years, his foundational work on singularities laid the groundwork for the collaboration with Roger Penrose, leading to the groundbreaking Penrose-Hawking Singularity Theorems.

Legacy of Inquisitiveness

In 2017, when Cambridge University made Hawking's thesis publicly accessible, it was downloaded millions of times within days. Reflecting on his motivation, he stated, “By making my PhD thesis Open Access, I hope to inspire people around the world to look up at the stars and not down at their feet; to wonder about our place in the universe and to try and make sense of the cosmos.”

Stephen Hawking's "Properties of Expanding Universes" remains a monumental piece of scientific literature, encapsulating a curiosity-driven pursuit to demystify our universe. More than five decades later, it not only continues to influence theories of cosmology but also inspires countless aspiring scientists and thinkers across the globe. Hawking’s legacy serves as a beacon for inquiry into the profound mysteries of existence—showing that human intellect, driven by curiosity, can chase answers to questions that dwell in the cosmos.