Question: Which monarch of the Mexican Empire was the 19th-century castle on the Gulf of Trieste Miramare Castle built for?
Context: Castell Coch (] ; Welsh for Red Castle ) is a 19th-century Gothic Revival castle built above the village of Tongwynlais in South Wales.  The first castle on the site was built by the Normans after 1081, to protect the newly conquered town of Cardiff and control the route along the Taff Gorge.  Abandoned shortly afterwards, the castle's earth motte was reused by Gilbert de Clare as the basis for a new stone fortification, which he built between 1267 and 1277 to control his freshly annexed Welsh lands.  This castle was likely destroyed in the native Welsh rebellion of 1314.  In 1760, the castle ruins were acquired by John Stuart, 3rd Earl of Bute, as part of a marriage settlement that brought the family vast estates in South Wales. Peebles Castle was a 12th-century castle built near Peebles, Scotland.  Peebles was created a royal burgh by King David I of Scotland in the 12th century.  The castle, once a royal castle, was built as a motte-and-bailey castle.  Nothing remains above ground. Miramare Castle (Italian: "Castello di Miramare" ; German: "Schloss Miramar" ; Slovene: "Grad Miramar" ) is a 19th-century castle on the Gulf of Trieste near Trieste, northeastern Italy.  It was built from 1856 to 1860 for Austrian Archduke Ferdinand Maximilian and his wife, Charlotte of Belgium, later Emperor Maximilian I and Empress Carlota of Mexico, based on a design by Carl Junker. Carl Junker (18 June 1827 – 17 May 1882) was an Austrian engineer and architect.  His construction projects include Miramare Castle in Trieste and the First Vienna Mountain Spring Pipeline. Kyrenia Castle (Greek: Κάστρο της Κερύνειας Turkish: "Girne Kalesi" ), at the east end of the old harbour in Kyrenia is a 16th-century castle built by the Venetians over a previous Crusader fortification.  Within its walls lies a twelfth-century chapel showing reused late Roman capitals, and the Shipwreck Museum. Barcola is a maritime neighbourhood of Trieste, Italy.  It is a popular tourist place with beaches and long promenade walkways, near to the Habsburg-established Miramare Castle. Montrose Castle was a 12th-century castle built in Montrose, Angus, Scotland.  Montrose was created a royal burgh by King David I of Scotland in the 12th century.  The castle, once a royal castle, was built as a motte and bailey castle.  King Edward I of England accepted John Balliol's surrender of Scotland at the castle on the 10 July 1296.  The castle was destroyed by William Wallace in 1297.  The castle was noted to be in ruins in 1488.  Nothing now remains above ground. Baynard Castle was a moated castle built in the 12th and 13th centuries in the village of Cottingham, East Riding of Yorkshire, England.  It was referred to by terms including the 'castle at Cottingham' or 'Stuteville's castle', the term "Baynard Castle" is common in 19th-century references and later. Vredenburg Castle (Dutch: "(Kasteel) Vredenburg" or "Vredeborch") was a 16th-century castle built by Habsburg emperor Charles V in the city of Utrecht in the Netherlands.  Some remains of the castle, which stood for only 50 years, are still visible on what is now Vredenburg square in Utrecht. Maximilian (Spanish: "Maximiliano"; born "Ferdinand Maximilian Joseph"; 6 July 1832 – 19 June 1867) was the only monarch of the Second Mexican Empire.  He was a younger brother of the Austrian emperor Francis Joseph I.  After a distinguished career in the Austrian Navy, he accepted an offer by Napoleon III of France to rule Mexico.  France (along with the United Kingdom and Spain, who both withdrew the following year after negotiating agreements with Mexico's democratic government) had invaded Mexico in the winter of 1861, as part of the War of the French Intervention.  Seeking to legitimize French rule in the Americas, Napoleon III invited Maximilian to establish a new Mexican monarchy for him.  With the support of the French army, and a group of conservative Mexican monarchists hostile to the liberal administration of new Mexican President Benito Juárez, Maximilian traveled to Mexico.  Once there, he declared himself Emperor of Mexico on 10 April 1864.
Answer: Emperor Maximilian I