# ITP

## Dr Shin Hattori in Durham University, UK (Part I)

[Introduction]

I have been in Durham for about a month. Durham is an old town dating back to the 10th century and located in the northeast area of England. Durham university is also long-established and it is the third oldest university in UK. There are a couple of staffs who are working on number theory and arithmetic geometry, and I discuss mainly with professor Victor Abrashkin. It is very exciting to study mathematics in Durham university and I would like to thank ITP program for its giving me such a nice opportunity.

 a distant view of the town from Durham Station the city center

[My research]

I am studying ramification theory of torsion Galois representations of local fields. It often happens that if we know the local informations of a variety, then we can also obtain a global information of the variety. So a local theory has not only local but also global significance. Ramification theory is a local theory in the context of arithmetic geometry and deals with a very subtle information of an algebraic variety over a local field, such as their torsion etale cohomology groups (this is a typical example of a torsion Galois representation over a local field).

Not so much had been known about that topic until recently, because it had been difficult to study torsion Galois representations geometrically, especially over an absolutely ramified local field. But, for studying arithmetic of varieties over a number field other than the rational field, we often have to work over a local field with higher absolute ramification. My research goal is to find a way of a better geometric understanding of these representations applicable to any local fields. Professor Abrashkin is one of the masters of ramification theory. We have a meeting per week and discuss and exchange information vigorously about our research.

 Department of Mathematical Sciences, Durham University the visitor's room (my office)

[The city of Durham]

Now I am going to explain how the life is here. Durham is a small town with impressive brick houses and stone pavement. A river runs through the town in a horseshoe shape, and on an upland peninsula surrounded by the winding river, an old castle and cathedral stand, which were built in the 11th century and are very famous in UK. They are even listed as World Heritage Sites.

The cathedral was build to protect the body of St. Cuthbert, one of the most important saints in the northeast England. An attendant of the cathedral told me that the reason why he is worshipped so much is that his body is said to remain intact for ten years after his death and people thought this is mystical. The saint was regarded as so important that bishops of this church had strong powers and been called as "Prince-Bishop". The castle had been a residence of these bishops, but now it is used as a college (student dormitory) of Durham university.

The castle is walled off along the river and we enter inside the wall through small tunnels with brick arches, like movies set in the medieval era. In fact, these relics are also famous as shooting places of the movie series "Harry Potter". When I entered the cloisters of the cathedral, I remembered several scenes of these movies probably shot just at that spot (as I had watched the movie during the long flight to UK).

The cathedral is used for daily duties for Christians even now. In addition, special choir events are held regularly and we can enjoy heavenly choral singings and organ playings in this sublime stone-built heritage with beautiful stained-glass windows.

 the castle, cathedral and Framwellgate bridge dating back to about one thousand years ago the cathedral from a court the castle from its courtyard the cloisters of the cathedral

[Living in this town]

I am staying at a family flat (with two rooms, plus a living-dining-kitchen area) in a college of the university. It is located in a hilly forest area which is a ten-minute walk from the university. The accommodations of the flat are highly satisfactory. I can use a bathroom, oven, grill, microwave, refrigerator, eating utensils for my own use. But since the supermarkets are far from the college, I decided to eat every meals at the restaurant of the college.

Before the departure, many people told me that meals in UK are not so good. But I think it is not so bad. For breakfast, we can choose either a continental or English breakfast. A full English breakfast usually consists of toasts, a fried egg, grilled sausages and bacons, fried mushrooms, grilled tomatoes and baked beans (with tomato sauce). I have read that to eat delicious meals in UK, you have to eat English breakfasts three times a day. This is not an empty boast and it actually tastes good. I liked it very much and I ate it every morning, but as a result I got tired of it after a week, so now I mainly eat cereals and yogurts every morning. We can eat pretty nice quiches, pies, roast beefs, pastas, hamburgers, fried fishes or scampis etc. for lunch and dinner. They are often accompanied with large portion of "chips" (French fries) and I manage to eat it with a great help of white pepper. Chips are famous as one of the ingredients of the traditional "fish-and-chips" and it seems to me that British people like chips very much.

 the college I am staying the living room of the flat English breakfast and cereals at the restaurant of the college the River Wear from the Framwellgate bridge
The climate is comfortable, at least for me. It is cool and people often wear long sleeves even in summer. Temperatures doesn't exceed 20 degrees Celsius usually in July. It rained almost everyday this July, but not so heavily and usually it was a mix of sun and clouds. The blueness of the sky is not so lucid here unlike the Mediterranean region and looks very much similar to that of Japan. The town is surrounded by forests and it is suitable to walk around. Now the students of Durham university are on vacation and there is almost nobody around the college, so it is very quiet at present. Like many countries in Europe, we have very long daytimes in summer. The sky brightens up at five or six am and nighttimes don't come until around ten pm. I heard that in this area it doesn't snow so much in winter and it is like autumn is continued until spring comes.

Apparently Durham is a safe place. I think it is because for one thing the town is in the countryside, and for another thing the restrictions on guns are very strict in UK as is in Japan.