The Dimension Theorem for vector spaces is a fundamental result in linear algebra that relates the dimensions of certain components of vector spaces and their subspaces.

The Latimer–MacDuffee theorem is a result in the field of algebra, specifically concerning finite abelian groups and their decompositions. It states that any finite abelian group can be expressed as a direct sum of cyclic groups, and the number of different ways to express a finite abelian group as such a direct sum is given by a specific combinatorial expression related to its invariant factors.

Segal's conjecture is a significant statement in the field of algebraic topology, particularly in the study of stable homotopy theory. Formulated by Graeme Segal in the 1960s, the conjecture concerns the relationship between the stable homotopy groups of spheres and the representation theory of finite groups.

Strassmann's theorem is a result in complex analysis that provides conditions under which a sequence of complex functions converges uniformly on compact sets. Specifically, it addresses the uniform convergence of power series in the context of multivariable functions, but it also applies to single-variable functions.

The Riemann-von Mangoldt formula is an important result in analytic number theory that provides an asymptotic expression for the number of prime numbers less than or equal to a certain value \( x \). More formally, it relates the distribution of prime numbers to the Riemann zeta function, a central object of study in number theory.

The Kodaira embedding theorem is a fundamental result in complex differential geometry that provides a criterion for when a compact complex manifold can be embedded into projective space as a complex projective variety. The theorem tackles the interplay between the geometry of a compact complex manifold and the algebraic properties of holomorphic line bundles over it.

The Krein–Milman theorem is a fundamental result in convex analysis and functional analysis, particularly dealing with convex sets in the context of topological vector spaces. The theorem essentially provides a characterization of convex compact sets.

Savitch's theorem is a result in computational complexity theory that relates the complexity classes \( \text{NL} \) (nondeterministic logarithmic space) and \( \text{L} \) (deterministic logarithmic space).

The Sipser–Lautemann theorem is a result in the field of computational complexity theory that addresses the relationship between complexity classes, particularly focusing on the class of languages recognized by nondeterministic polynomial time machines (NP) and certain probabilistic polynomial time machines (BPP).

G. Evelyn Hutchinson (1903–1991) was a prominent British ecologist and limnologist, widely regarded as one of the founders of modern ecology. He is best known for his significant contributions to the understanding of ecosystems, population dynamics, and biogeochemistry. Hutchinson's work helped lay the foundations for the study of freshwater ecosystems and the interactions between organisms and their environments.

The Valiant–Vazirani theorem is a result in theoretical computer science concerning the complexity of certain problems in the context of randomized algorithms. Specifically, it relates to the complexity class NP (nondeterministic polynomial time) and the concept of zero-knowledge proofs.

Beck's theorem, in the context of geometry, generally refers to a result in the field of combinatorial geometry related to point sets and convex shapes. More specifically, it states that for any finite set of points in the plane, there exists a subset of those points that can be covered by a convex polygon of a certain size, where the size is influenced by the dimension of the space.

Radon's theorem is a result in convex geometry that deals with the intersection of convex sets. Specifically, it states that: **Radon's Theorem:** If a set of \( d + 2 \) points in \( \mathbb{R}^d \) is given, then it is possible to partition these points into two non-empty subsets such that the convex hulls (the smallest convex sets containing the points) of these two subsets intersect.

The Graph Structure Theorem is a significant result in graph theory that characterizes certain classes of graphs. Specifically, it provides a structural decomposition of a broad class of graphs known as "H-minor-free graphs." This theorem states that if a graph does not contain a fixed graph H as a minor, then it can be decomposed into a bounded number of simpler components that exhibit certain structural properties.

Grinberg's theorem is a result in the field of topology and specifically pertains to the properties of continuous mappings between topological spaces. It is often mentioned in the context of compact spaces and homeomorphisms. The theorem states that if \( X \) is a compact Hausdorff space and \( Y \) is a connected space, then every continuous surjective mapping from \( X \) onto \( Y \) is a quotient map.

Schnyder's theorem, or Schnyder's realizability theorem, is a result in graph theory that relates to the representation of planar graphs. It states that: **Every simple planar graph can be embedded in the plane such that its vertices can be labeled with numbers from {0, 1, 2, 3} so that the edges of the graph respect certain ordering conditions.

The Edge-of-the-Wedge theorem is a concept from complex analysis, specifically regarding holomorphic functions. It deals with the behavior of these functions on regions in the complex plane that have "wedge-shaped" domains.

The Mermin-Wagner theorem is a result in statistical mechanics and condensed matter physics that addresses the behavior of certain types of physical systems at low temperatures, specifically those defined by continuous symmetry. The theorem, which was formulated by N. D. Mermin and H. Wagner in the 1960s, states that in two-dimensional systems with continuous symmetry, spontaneous symmetry breaking and long-range order cannot occur at finite temperatures.

As of my last knowledge update in October 2023, Clare Yu is a physicist known for her work in the field of experimental condensed matter physics. Her research often involves the study of materials at the nanoscale and their properties. She has contributed to advancing the understanding of different materials, particularly in areas related to quantum phase transitions and topological materials.

The Cayley–Bacharach theorem is a result in algebraic geometry that deals with the intersection of divisors on a projective space. It is particularly relevant in the study of linear systems of divisors and their properties. In its classical form, the theorem states the following: Let \( C \) be a non-singular irreducible curve of degree \( d \) in the projective plane \( \mathbb{P}^2 \).