Topology 1
a topological space that is similar in some essential ways to a sphere but may not be geometrically identical to it
Homotopy Sphere
a topological space that is locally Euclidean, meaning it resembles Euclidean space in the neighborhood of each point
Topological Manifold
their values are subject to randomness, often described by probability distributions
random variables
a mathematical model describing a sequence of events where the probability of each event depends only on the current event, not on previous events
Markov chain
a mathematical measure of how curves intersect
Algebraic Intersection Number
Two spaces are ... if they can be continuously deformed into one another while preserving certain topological properties.
homotopy equivalent
Techniques that incorporate randomness and probability into mathematical models for systems with uncertainty.
stochastic methods
a computational technique that uses random sampling to estimate complex quantities or solve problems with probabilistic elements
Monte Carlo simulation
a term used metaphorically to describe the process of separating or pushing apart curves to avoid intersections
Social Distancing
objects in four-dimensional topology that are used to understand the structure of four-dimensional spaces
Casson Handles
The process of drawing conclusions or making predictions from data, often involving probabilistic models and uncertainty quantification.
statistical inference
the determination of the value of financial elements, such as stock options or derivatives
option pricing
topologically equivalent
Homeomorphic
two-dimensional spaces bounded by curves that are used in mathematics to facilitate the separation of intersecting curves
Whitney Discs
a type of stochastic process used to model events that occur randomly over time
Poisson Processes
the long-term probability distribution of states in a Markov chain
stationary distribution
a mathematical process that involves continuously deforming one object into another while maintaining certain properties, often used in topology to study deformations of spaces
Isotopy
a topological space that is locally Euclidean, meaning it resembles Euclidean space in the neighborhood of each point
topological manifold
a state in which a system remains constant over time, as in a stationary distribution
equilibrium state
the random movement of particles suspended in a fluid, used as a model in physics
Brownian motion