Question: At the end of which year did the tribes in the area where the Ashkenazi Jews can trace their ancestry to Jews who settled there invade the roman empire?

Knowledge 1: Ashkenazi Jews
In an ethnic sense, an Ashkenazi Jew is one whose ancestry can be traced to the Jews who settled in Central Europe. For roughly a thousand years, the Ashkenazim were a reproductively isolated population in Europe, despite living in many countries, with little inflow or outflow from migration, conversion, or intermarriage with other groups, including other Jews. Human geneticists have argued that genetic variations have been identified that show high frequencies among Ashkenazi Jews, but not in the general European population, be they for patrilineal markers (Y-chromosome haplotypes) and for matrilineal markers (mitotypes). However, a 2013 study of Ashkenazi mitochondrial DNA, from the University of Huddersfield in England, suggests that at least 80 percent of the Ashkenazi maternal lineages derive from the assimilation of mtDNAs indigenous to Europe, probably as a consequence of conversion. Since the middle of the 20th century, many Ashkenazi Jews have intermarried, both with members of other Jewish communities and with people of other nations and faiths. [SEP] Knowledge 2: Galicia (Spain)
In the early 5th century, the deep crisis suffered by the Roman Empire allowed different tribes of Central Europe (Suebi, Vandals and Alani) to cross the Rhine and penetrate into the rule on 31 December 406. Its progress towards the Iberian Peninsula forced the Roman authorities to establish a treaty (foedus) by which the Suebi would settle peacefully and govern Galicia as imperial allies. So, from 409 Galicia was taken by the Suebi, forming the first medieval kingdom to be created in Europe, in 411, even before the fall of the Roman Empire, being also the first Germanic kingdom to mint coinage in Roman lands. During this period a Briton colony and bishopric (see Mailoc) was established in Northern Galicia (Britonia), probably as foederati and allies of the Suebi. In 585, the Visigothic King Leovigild invaded the Suebic kingdom of Galicia and defeated it, bringing it under Visigoth control.
Knowledge 1: Ashkenazi Jews
A 2010 study on Jewish ancestry by Atzmon-Ostrer et al. stated "Two major groups were identified by principal component, phylogenetic, and identity by descent (IBD) analysis: Middle Eastern Jews and European/Syrian Jews. The IBD segment sharing and the proximity of European Jews to each other and to southern European populations suggested similar origins for European Jewry and refuted large-scale genetic contributions of Central and Eastern European and Slavic populations to the formation of Ashkenazi Jewry", as both groups – the Middle Eastern Jews and European/Syrian Jews – shared common ancestors in the Middle East about 2500 years ago. The study examines genetic markers spread across the entire genome and shows that the Jewish groups (Ashkenazi and non Ashkenazi) share large swaths of DNA, indicating close relationships and that each of the Jewish groups in the study (Iranian, Iraqi, Syrian, Italian, Turkish, Greek and Ashkenazi) has its own genetic signature but is more closely related to the other Jewish groups than to their fellow non-Jewish countrymen. Atzmon's team found that the SNP markers in genetic segments of 3 million DNA letters or longer were 10 times more likely to be identical among Jews than non-Jews. Results of the analysis also tally with biblical accounts of the fate of the Jews. The study also found that with respect to non-Jewish European groups, the population most closely related to Ashkenazi Jews are modern-day Italians. The study speculated that the genetic-similarity between Ashkenazi Jews and Italians may be due to inter-marriage and conversions in the time of the Roman Empire. It was also found that any two Ashkenazi Jewish participants in the study shared about as much DNA as fourth or fifth cousins. [SEP] Knowledge 2: Ashkenazi Jews
Sporadic epigraphic evidence in grave site excavations, particularly in Brigetio (Szőny), Aquincum (Óbuda), Intercisa (Dunaújváros), Triccinae (Sárvár), Savaria (Szombathely), Sopianae (Pécs), and Osijek in Croatia, attest to the presence of Jews after the 2nd and 3rd centuries where Roman garrisons were established, There was a sufficient number of Jews in Pannonia to form communities and build a synagogue. Jewish troops were among the Syrian soldiers transferred there, and replenished from the Middle East, after 175 C.E. Jews and especially Syrians came from Antioch, Tarsus and Cappadocia. Others came from Italy and the Hellenized parts of the Roman empire. The excavations suggest they first lived in isolated enclaves attached to Roman legion camps, and intermarried among other similar oriental families within the military orders of the region.Raphael Patai states that later Roman writers remarked that they differed little in either customs, manner of writing, or names from the people among whom they dwelt; and it was especially difficult to differentiate Jews from the Syrians. After Pannonia was ceded to the Huns in 433, the garrison populations were withdrawn to Italy, and only a few, enigmatic traces remain of a possible Jewish presence in the area some centuries later.
Knowledge 1: Ashkenazi Jews
A study of haplotypes of the Y-chromosome, published in 2000, addressed the paternal origins of Ashkenazi Jews. Hammer et al. found that the Y-chromosome of Ashkenazi and Sephardic Jews contained mutations that are also common among Middle Eastern peoples, but uncommon in the general European population. This suggested that the male ancestors of the Ashkenazi Jews could be traced mostly to the Middle East. The proportion of male genetic admixture in Ashkenazi Jews amounts to less than 0.5% per generation over an estimated 80 generations, with "relatively minor contribution of European Y chromosomes to the Ashkenazim," and a total admixture estimate "very similar to Motulsky's average estimate of 12.5%." This supported the finding that "Diaspora Jews from Europe, Northwest Africa, and the Near East resemble each other more closely than they resemble their non-Jewish neighbors." "Past research found that 50–80 percent of DNA from the Ashkenazi Y chromosome, which is used to trace the male lineage, originated in the Near East," Richards said. [SEP] Knowledge 2: Galicia (Spain)
In the early 5th century, the deep crisis suffered by the Roman Empire allowed different tribes of Central Europe (Suebi, Vandals and Alani) to cross the Rhine and penetrate into the rule on 31 December 406. Its progress towards the Iberian Peninsula forced the Roman authorities to establish a treaty (foedus) by which the Suebi would settle peacefully and govern Galicia as imperial allies. So, from 409 Galicia was taken by the Suebi, forming the first medieval kingdom to be created in Europe, in 411, even before the fall of the Roman Empire, being also the first Germanic kingdom to mint coinage in Roman lands. During this period a Briton colony and bishopric (see Mailoc) was established in Northern Galicia (Britonia), probably as foederati and allies of the Suebi. In 585, the Visigothic King Leovigild invaded the Suebic kingdom of Galicia and defeated it, bringing it under Visigoth control.

Answer: 406