HalvarF already said it in his answer: Letters with umlauts (ä, ö, ü) are distinct letters in German language. Historically they derived from the letters without dots, and there are still connections (plural of »Mutter« is »Mütter«, diminutive of »Paar« is »Pärchen«, etc.) but they still are distinct letters, with their own pronunciations.
The German letter Ö has two different pronunciations, and they are different from the pronunciations of the letter O which also has two different pronunciations. Which of the two possibilities you should use depends in both cases on the length of the vowel. (How long does it take when you speak the sound?)
IPA = International Phonetic Alphabet
in these words:
Bö, Höhle, böse, Gröbner, ...
Name of this sound (and link to Wikipedia article about this sound): Close-mid front rounded vowel
This sound exists also in some variations of English (like in New Zealand and South Africa). There you can hear this sound in these words:
in these words:
Hölle, plötzlich, Göttin, Höcker, ...
Name of this sound (and link to Wikipedia article about this sound): Open-mid front rounded vowel
This sound is very similar to [ø]. Many German native speakers can't notice a difference between the two sounds (except the length). When you hear this sound in English, it is in the same regions and in the same words (bird, word, ...)
Just für completeness:
Boot, Tor, rot, ...
Name and Wikipedia-article: Close-mid back rounded vowel
Received pronunciation, Cockney, but also Australia, New Zealand and South Afrika: "yawn"
India, Pakistan, Singapore: "go"
offen, Tonne, toll, Klotz, ...
Name and Wikipedia: Open-mid back rounded vowel
Received Pronunciation, Australia, New Zealand: "not"
General American, Scotish: "thought"
When is a vowel to be pronounced long and when short?
There is a rule of thumb, but it has many exceptions:
- If the vowel is the last sound of a syllable
- If there is only one consonant after the vowel within the same syllable
- If there is an explicit length marker (double vowel, e after i, h after any vowel)
- If there are two or more consonants after the vowel within the same syllable
- double consonants count as two consonants
These rules apply only to words that have been part of the German vocabulary for centuries. They do not apply to foreign words and many loan words.
In the name Gröb-ner there is only one consonant after the vowel in the same syllable, so the vowel ö has to be pronounced long.
Btw: The second syllable in Gröb-ner is a reduction syllable, and this means it is even weaker stressed than a normal unstressed syllable (so, the stressed syllable in Gröbner is »gröb«), and the spoken vowel of a reduction syllable is in most cases (like in Gröbner) a schwa sound, like in the last syllable of teacher and mister in received pronunciation. The pronunciation of the consonants in Gröbner ist just strait forward, like in English. Note, that the r at the end of the word is silent, so the last sound of Gröbner is an unstressed schwa sound.
So, this is the correct pronunciation of Gröbner in IPA symbols:
add-on (reaction to comments)
Wolfgang Gröbner was an Austrian mathematician. He was born in 1899 in a small village in South Tyrol, which at that time was part of the Austro-Hungarian Empire and today is part of Italy. He attended school in Feldkirch/Vorarlberg, in the very west of Austria, and attended university in Graz in the south-east of Austria and later in Vienna, the capital city of Austria. Then he spent a few years in Göttingen in the center of Germany, but soon he went back to Austria, then he was in Rome, went back to Vienna, then to Braunschweig, then to Vienna again, and then to Innsbruck in the west of Austria. So, he was a native Austrian, he was fully socialized in Austria, and he spent most of his life in Austria. And this is why the Austrian pronunciation of his name is correct, which I already posted above: [ˈɡʁøːbnɐ] with a long and closed Ö and a voiced B (German name: »weiches B« = soft B).
The name Gröbner is a typical Austrian name. The region in Germany with the most people named Gröbner is Traunstein in the south-east of Bavaria, next to the Austrian border. So, you will not find many people with this name in the north of Germany. But those who live there probably might pronounce their name differently:
Here we have a short and open Ö and the consonant after it is a voiceless P (German name: »hartes P« = hard P).
I have no explanation why people in northern regions tend to pronounce some vowels shorter than in the south, but at least the devoicing of some consonantes at the end of syllables (b→p, d→t, g→k) that happens in middle and northern regions of Germany (but not in southern regions like Austria) has a name: Auslautverhärtung (in English: Final-obstruent devoicing). This usually happens only at the end of a word, but also before some reduction syllables, and -ner seems to be one of them. But this topic is already far beyond the scope of the question, so, if you want to learn more about it, please search here in German.stackexchange or where ever you want for the given terms.