ITP

Dr Daisuke Hirose in Michigan University (Part I)

[Introduction]

I arrived at the Detroit Metropolitan airport in the USA on 1st of September, 2009. More precisely, the airplane I took stopped at the Dallas airport first, After that, I left there for the Detroit airport by a domestic plane. This was my first trip going abroad. So everything was new for me. As the first report, I would like to talk about the area I live.

[Ann Arbor]

I live in Ann Arbor now. Ann Arbor is a country town in Michigan States. It is known as a city where Michigan University is located. The Domino's pizza's headquarter is in Ann Arbor. (Unfortunately, I have not eaten its pizza yet.) The Border's book store, which is the second-biggest book store chain in the US, is located in Ann Arbor, too. In fact, its first store is within 1-minute-walk from my apartment! Furthermore, you can see the picture of it at the wikipedia.
http: //ja.wikipedia.org/wiki/ボーダーズ・グループ

Ann Arbor is also known as a safe city in the US. So I have not had accidents yet.

[Michigan University]

Michigan University is the oldest state's university. It has a wide campus area. You can see big and old-fashioned buildings there. This picture is one of the buildings in campus on my way home from my department. My office mate said that the department of mathematics used to be in this building. He also said that looking at it from outside is better than working in it.

I found a big national flag in a park. This park is a place where some communities do various activities in the university. I attached the picture of it. In the picture, you can see a clock tower on the left side of the flag. It is a landmark in Ann Arbor, called the bell tower.

A library for undergraduate students is on the opposite side of the flag in the park. It is also a beautiful building, so I took a picture of it at night.

[Department of Mathematics]

Many mathematicians are in the department of mathematics, at the university. It has 96 members of tenured and tenure-track faculty! I counted it from
http://www.math.lsa.umich.edu/people/facultyResAreaTenure.php
In particular, many mathematicians study algebraic geometry or commutative algebra. This implies that many young mathematicians in these research areas are also there. Those facts give me the best environment to do my research. And this Department has many seminars every week. For example, Algebraic Geometry seminar, Commutative Algebra seminar, Topics of Algebraic Geometry seminar,and Student Algebraic seminar, etc. So I join them every day, to be familiar to members there and to improve my listening skill.

[My research]

I would like to write about my mathematical research here. I study the tight closure theory, which deals with a commutative ring of positive characteristic, using the Frobenius map of it. This was started by Professor Hochster and Professor Huneke. The former mathematician is the chair man of this department! I am interested in a test ideal of the ring, which measures how the ring behaves well in terms of the theory. It might be a positive characteristic analog of a multiplier ideal. The multiplier ideal is an ideal sheaf of an algebraic variety over the complex number field, which can give information of a singularity of the variety. It is known that Professor Hara and Professor Smith proved the connection independently. And the latter mathematician is also in the department! So the test ideal is interesting and mysterious object, and this department is an idealistic environment to study my research area. In particular, I was focusing an invariant related to a test ideal, called the F-threshold. This invariant was defined by Professor Mustata, Professor Takagi and Professor Watanabe. I studied it on toric rings, give a formula of it, and write my dissertation.

[First contact with Professor Mustata]

Professor Mustata is my advisor in the university. Before I arrived here, he helped me a lot of to prepare for my visit, such as deciding my accommodation, and so on. I made an appointment to meet him to talk about my mathematical interest in the first week of my visit. At the meeting, I discussed what I proved, what I was studying, and what I want to research. He saw it fast and guided me. The point in his advice was to find a particular example. The meeting ended.

I would put a period of my report here. My visit has not ended yet, but here I express my thanks to the ITP committee for giving me a great opportunity to study my research in the university of Michigan, to arrive in Ann Arbor, the US. I might present a continuing report with funny animals.

ITP