Coalescent theory
Coalescent theory is a model of how alleles sampled from a population may have originated from a common ancestor. In the simplest case, coalescent theory assumes no recombination, no natural selection, and no gene flow or population structure, meaning that each variant is equally likely to have been passed from one generation to the next. The model looks backward in time, merging alleles into a single ancestral copy according to a random process in coalescence events. Under this model, the expected time between successive coalescence events increases almost exponentially back in time (with wide variance). Variance in the  model comes from both the random passing of alleles from one generation to the next, and the random occurrence of mutations in these alleles. The mathematical theory of the coalescent was developed independently by several groups in the early 1980s as a natural extension of classical population genetics theory and models, but can be primarily attributed to John Kingman. Advances in coalescent theory include recombination, selection, overlapping generations and virtually any arbitrarily complex evolutionary or demographic model in population genetic analysis. The model can be used to produce many theoretical genealogies, and then compare observed data to these simulations to  test assumptions about the demographic history of a population. Coalescent theory can be used to make inferences about population genetic parameters, such as migration, population size and recombination. Theory. Time to coalescence. Consider a single gene locus sampled from two haploid individuals in a population. The ancestry of this sample is traced backwards in time to the point where these two lineages "coalesce" in their most recent common ancestor (MRCA). Coalescent theory seeks to estimate the expectation of this time period and its variance. The probability that two lineages coalesce in the immediately preceding generation is the probability that they  share a parental DNA sequence. In a population with a constant effective population size with 2"N" copies of each locus, there are 2"N" "potential parents" in the previous generation. Under a random mating model, the probability that two alleles originate from the same parental copy is thus 1/(2"N") and, correspondingly, the probability that they do "not" coalesce is 1 − 1/(2"N"). At each successive preceding generation, the probability of coalescence is geometrically distributed—that is, it is the probability of "non"coalescence at the "t" − 1 preceding generations multiplied by the probability of coalescence at the generation of interest: For  sufficiently large values of "N", this distribution is well approximated by the continuously defined exponential distribution This is mathematically convenient, as the standard exponential distribution has both the expected value and the standard deviation equal to 2"N". Therefore, although the "expected" time to coalescence is 2"N", actual coalescence times have a wide range of variation. Note that coalescent time is the number of preceding generations where the coalescence took place and not calendar time, though an estimation of the latter can be made multiplying 2"N" with the average time between generations. The above calculations apply equally to a diploid population  of effective size "N" (in other words, for a non-recombining segment of DNA, each chromosome can be treated as equivalent to an independent haploid individual; in the absence of inbreeding, sister chromosomes in a single individual are no more closely related than two chromosomes randomly sampled from the population). Some effectively haploid DNA elements, such as mitochondrial DNA, however, are only passed on by one sex, and therefore have one quarter the effective size of the equivalent diploid population ("N"/2) Neutral variation. Coalescent theory can also be used to model the amount of variation in DNA sequences expected from  genetic drift and mutation. This value is termed the mean heterozygosity, represented as formula_3. Mean heterozygosity is calculated as the probability of a mutation occurring at a given generation divided by the probability of any "event" at that generation (either a mutation or a coalescence). The probability that the event is a mutation is the probability of a mutation in either of the two lineages: formula_4. Thus the mean heterozygosity is equal to For formula_6, the vast majority of allele pairs have at least one difference in nucleotide sequence. Extensions. There are numerous extensions to the coalescent model, such as  the Λ-coalescent which allows for the possibility of multifurcations. Graphical representation. Coalescents can be visualised using dendrograms which show the relationship of branches of the population to each other. The point where two branches meet indicates a coalescent event. Applications. Disease gene mapping. The utility of coalescent theory in the mapping of disease is slowly gaining more appreciation; although the application of the theory is still in its infancy, there are a number of researchers who are actively developing algorithms for the analysis of human genetic data that utilise coalescent theory. A considerable number of human diseases can be  attributed to genetics, from simple Mendelian diseases like sickle-cell anemia and cystic fibrosis, to more complicated maladies like cancers and mental illnesses. The latter are polygenic diseases, controlled by multiple genes that may occur on different chromosomes, but diseases that are precipitated by a single abnormality are relatively simple to pinpoint and trace – although not so simple that this has been achieved for all diseases. It is immensely useful in understanding these diseases and their processes to know where they are located on chromosomes, and how they have been inherited through generations of a family, as can be accomplished  through coalescent analysis. Genetic diseases are passed from one generation to another just like other genes. While any gene may be shuffled from one chromosome to another during homologous recombination, it is unlikely that one gene alone will be shifted. Thus, other genes that are close enough to the disease gene to be linked to it can be used to trace it. Polygenic diseases have a genetic basis even though they don't follow Mendelian inheritance models, and these may have relatively high occurrence in populations, and have severe health effects. Such diseases may have incomplete penetrance, and tend to  be polygenic"," complicating their study. These traits may arise due to many small mutations, which together have a severe and deleterious effect on the health of the individual. Linkage mapping methods, including Coalescent theory can be put to work on these diseases, since they use family pedigrees to figure out which markers accompany a disease, and how it is inherited. At the very least, this method helps narrow down the portion, or portions, of the genome on which the deleterious mutations may occur. Complications in these approaches include epistatic effects, the polygenic nature of the mutations, and environmental factors. That  said, genes whose effects are additive carry a fixed risk of developing the disease, and when they exist in a disease genotype, they can be used to predict risk and map the gene. Both regular the coalescent and the shattered coalescent (which allows that multiple mutations may have occurred in the founding event, and that the disease may occasionally be triggered by environmental factors) have been put to work in understanding disease genes. Studies have been carried out correlating disease occurrence in fraternal and identical twins, and the results of these studies can be used to inform coalescent modeling. Since  identical twins share all of their genome, but fraternal twins only share half their genome, the difference in correlation between the identical and fraternal twins can be used to work out if a disease is heritable, and if so how strongly. The genomic distribution of heterozygosity. The human single-nucleotide polymorphism (SNP) map has revealed large regional variations in heterozygosity, more so than can be explained on the basis of (Poisson-distributed) random chance. In part, these variations could be explained on the basis of assessment methods, the availability of genomic sequences, and possibly the standard coalescent population genetic model . Population genetic influences could have a major influence on this variation: some loci presumably would have comparatively recent common ancestors, others might have much older genealogies, and so the regional accumulation of SNPs over time could be quite different. The local density of SNPs along chromosomes appears to cluster in accordance with a variance to mean power law and to obey the Tweedie compound Poisson distribution. In this model the regional variations in the SNP map would be explained by the accumulation of multiple small genomic segments through recombination, where the mean number of SNPs per segment would be gamma  distributed in proportion to a gamma distributed time to the most recent common ancestor for each segment. History. Coalescent theory is a natural extension of the more classical population genetics concept of neutral evolution and is an approximation to the Fisher–Wright (or Wright–Fisher) model for large populations. It was discovered independently by several researchers in the 1980s. Software. A large body of software exists for both simulating data sets under the coalescent process as well as inferring parameters such as population size and migration rates from genetic data.