Research Experiences for Undergraduates

Actuarial science and financial mathematics have evolved significantly over the last couple of decades. The level of sophistication of their mathematical and financial content, along with their potential range of applications, have increased enormously. Recent economic and financial events and uncertainties reflect the fact that we live in a stochastic and ever-riskier world -- with risks both natural and man-made -- and that mathematical, financial, and analytical skills are critical for identifying, quantifying, understanding, and managing the impact of those risks.

This summer research experience will provide a select group of undergraduate students with an opportunity to explore and research the mathematical modeling of stochastic processes in an actuarial and/or financial context. We will start by obtaining a theoretical background in several key areas: stochastic processes (including Brownian motion, stochastic differential equations, and Poisson jump processes); risk theory (beginning with the classic Cramér-Lundberg model); and Monte Carlo simulation. Both mathematical theory and hands-on use of state-of-the-art actuarial simulation models will be provided.

Building upon our exposure to these foundational concepts, several possible research directions may be entertained and pursued by the participants. Depending upon participant interests, both mathematical and data-analytical types of research may be possible. Some of the potential research areas include the following: modeling of financial scenarios (e.g., interest rates, stock market movements); dynamic financial analysis (modeling an insurance or other financial organization impacted by stochastic underwriting and investment processes); enterprise risk management (developing a model to reflect risk correlations and interactions across an enterprise); mathematical risk measures; and stochastic control theory and optimization of portfolios and strategic decision-making.

REU in Experimental Number Theory

Experimental mathematics is a term coined a few decades ago to describe a way of doing mathematics in which computation, or "experimentation", enters in a significant manner. Computation can be used to gain insight and intuition, to provide evidence for (or against) known conjectures, and to come up with new conjectures that are suggested by numerical evidence. Experimental mathematics has risen to prominence in recent years, as evidenced by a journal by that name, several books on the subject, and even a well-funded institute, the Centre for Experimental and Constructive Mathematics.

In this REU program we apply the philosophy of experimental mathematics to problems in number theory. The first two weeks of the program will focus on acquiring theoretical background knowledge and computing skills. During the remainder of the program, participants will work on specific projects, usually in small groups. The projects will all involve computer programming to varying degrees. Some may require heavy duty computations using low level languages such as C/C++, while others may only need some light computations using a symbolic software program such as Mathematica, Maple, or Pari/GP.

Aside from the work on specific research projects, a key goal of the program is to introduce participants to the "research infrastructure" in mathematics. We will cover topics such as using library resources, using the mathematical literature, and getting research published, and the participants will learn how to use LaTeX to produce professionally looking mathematical documents. Additional program details are posted at www.math.uiuc.edu/~hildebr/reu/

Participants should have some basic knowledge of number theory at the level of an undergraduate number theory course or be prepared to acquire that knowledge on their own before the start of the program. Strong computer skills (such as proficiency in C/C++) are desirable, but not required. Unix/Linux geeks and perl wizards are especially welcome. As with all programs of this nature, a high degree of talent and aptitude for solving mathematical problems is required, and the selection to the program is based primarily on evidence of such talent.

The University of Illinois Department of Mathematics has a long tradition of strength in number theory. The Illinois number theory graduate program is probably the largest of its kind in the United States. More than a dozen Illinois faculty members work in number theory, with interests covering the full spectrum of number theory, and approximately two dozen graduate students are currently working on a thesis in number theory. More information about the number theory program can be found at the Illinois Number Theory Page.

How to Apply

To apply to the program, please send:

• A completed application form (you may print out the application form in pdf format or complete the online form.
• Statement of interest (1-2 typed pages).
• A list of math courses completed or currently being taken, with grades for those completed.
• Two letters of recommendation from professors who will comment on your mathematical potential, motivation and work habits.
• Send all application materials electronically to reu@math.uiuc.edu or by U.S. mail to:

  Joseph Miles
  Director of Undergraduate Studies
  Department of Mathematics
  University of Illinois at Urbana-Champaign
  313 Altgeld Hall
  1409 W. Green Street
  Urbana, IL 61801

Questions

See the REU Frequently Asked Questions page. If you have further questions, contact Professor Joseph Miles at reu@math.uiuc.edu.

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