1, first of all, you must do a good job in the math class of this science, and it is best to get 90/ 100. Choose your elective courses as far as possible in the mathematics department, so there may be 10 courses, and it doesn't matter if some courses are slightly lower. According to my tutor, it won't look fake. A little hierarchical, right? For example, I have a course with only 3.3, which makes my GPA of mathematics become 3.95.
2, it is best to take several exams, don't look at a lot of exams. In fact, there are only two: mathematical analysis and advanced algebra (advanced mathematics, linear algebra). Prepare for 2 months. More than 95% is not difficult.
3. The key is interest. Interest doesn't mean you tell the professor:? I am interested in * * *? It means that you really understand * * *, read many related articles, know where the problem lies, know the current progress and who has reached what level, and have your own unreasonable, reasonable and better preliminary ideas, which the professor will see when you are magnetized.
Since you want to transfer, you should teach yourself basic courses, and it is best to use some good teaching materials such as GTM LMSST LNM. In this way, when you befriend the professor, he can immediately see whose book you have read from your text, because I applied for algebra (including many directions, ring theory, group theory, group representation, Lie group, representation theory, algebraic number theory, algebraic geometry), so talk about the books that algebra should read.
5, the most basic advanced algebra, abstract algebra will not be said. But you must be familiar with it, and you'd better write a handout about these things yourself. Write it, make it into a PDF, and send it to the professor you want. It could be a surprise. You should read some commutative algebra, module theory and ring theory. No matter which branch of algebra, it will be used a lot. Then Li Qun said that no matter which branch of mathematics must be learned. Then read category theory and homology algebra. After reading these, you can start. In this way, you will have an overall sense of the direction of your application, so that when you write a group of words, you will have a definite aim. How do you show that you have learned these things? The first is to write handouts. I estimate that if you write all this down, your skills will be greatly improved, because you have proved all the details (books such as LNM are generally incomplete and many are easy to see). Secondly, at least 700-800 pages will make you very distinctive, and the professor will think that you are hardworking, interested and inspiring.
Let's talk about scientific research background first. It is impossible for an undergraduate to hand out a math paper. Even if it is published, it is not necessarily a good article, because an undergraduate can learn too little. Even if you get 30 GTMs, it can only be considered as an introduction, because most GTMs are just basic textbooks. Scientific research background is not necessarily the embodiment of the article. You can also attend seminars and take classes with graduate students. For example, if you take a lot of postgraduate courses and pass the exam, even if the score is not high, even if the school can't issue a formal score, you can take one. A score is better than nothing. Because it is your tutor's report card, the tutor has credibility in it. You attended a discussion class and gave lectures by yourself, which can be reflected in ps and letters of recommendation. In addition, there are many workshops all over the country every year, which you can attend. There are also summer schools, such as Morningside of the Academy of Sciences, Mathematics Center of Zhejiang University and Peking University. For example, you apply for algebra. East China Normal University has many discussion classes every year, so you can sign up to listen to the report. There are also many meetings, which can increase the background. The most important thing is to communicate with Daniel. Sometimes Daniel's words are more useful than reading. Maybe Daniel thinks you have to do it.
Of course, you can also send some papers on applied mathematics and computational mathematics, which is better for the papers. For example, an efficient algorithm in a specific problem (physics, chemistry, computer, biology) can be used in many articles, such as a good algorithm for calculating some physical phenomena. It is also useful to send it to non-mathematical magazines, at least to show your critical computing ability and good analytical foundation. It is enough to send three or four articles.
While laying a good foundation, read more articles in the direction you are interested in, such as tools of algebra and category, homology algebra and algebraic geometry, which are all very hot. You can see clusters, layers and cohomology everywhere. You can read these articles. This will broaden your horizons and improve the quality of magnetism.
Through the above introduction of applying for basic majors in American universities, I believe that your application will be enlightening.