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# 2018 Senior Thesis Topics

## What is a senior thesis?

❶Tortorice has worked with this data in the past and would be happy to advise someone who wants to work with it. Analytic number theory involves applying calculus and complex analysis to the study of the integers.

## Ideas For Starters

Islamic, Chinese, or the development of mathematics in the U. Medical researchers and policy makers often face difficult decisions which require them to choose the best among two or more alternatives using whatever data are available. An axiomatic formulation of a decision problem uses loss functions, various decision criteria such as maximum likelihood and minimax, and Bayesian analysis to lead investigators to good decisions.

Foundations, Concepts and Methods, Springer-Verlag, The power of modern computers has made possible the analysis of complex data set using Bayesian models and hierarchical models. These models assume that the parameters of a model are themselves random variables and therefore that they have a probability distribution.

Bayesian models may begin with prior assumptions about these distributions, and may incorporate data from previous studies, as a starting point for inference based on current data. This project would investigate the conceptual and theoretical underpinnings of this approach, and compare it to the traditional tools of mathematical statistics as studied in Ma It could culminate in an application that uses real data to illustrate the power of the Bayesian approach.

Oxford University Press, New York. Bayesian Statistics for Evaluation Research: Measurements which arise from one or more categorical variables that define groups are often analyzed using ANOVA Analysis of Variance. Linear models specify parameters that account for the differences among the groups.

Sometimes these differences exhibit more variability than can be explained by these "fixed effects", and then the parameters are permitted to come from a random distribution, giving "random effects.

This modeling approach has proved useful and powerful for analyzing multiple data sets that arise from different research teams in different places. For example the "meta-analysis" of data from medical research studies or from social science studies often employs random effects models. This project would investigate random effects models and their applications. MA , with a plus.

Because a computer is deterministic, it cannot generate truly random numbers. A thesis project could explore methods of generating pseudo-random numbers from a variety of discrete and continuous probability distributions. The art of tilings has been studied a great deal, but the science of the designs is a relatively new field of mathematics.

Some possible topics in this area are: The problems in this area are easy to state and understand, although not always easy to solve. The pictures are great and the history of tilings and patterns goes back to antiquity. An example of a specific problem that a thesis might investigate is: Devise a scheme for the description and classification of all tilings by angle-regular hexagons.

Roughly speaking, a contraction of the plane is a transformation f: With a little effort CF can even be made to look like a tree or a flower!! A thesis in this area would involve learning about these contraction mapping theorems in the plane and in other metric spaces, learning how the choice of contractions effects the shape of CF and possibly writing computer programs to generate CF from F. Consider a population of individuals which produce offspring of the same kind. Associating a probability distribution with the number of offspring an individual will produce in each generation gives rise to a stochastic i.

The earliest applications concerned the disappearance of "family names," as passed on from fathers to sons. Modern applications involve inheritance of genetic traits, propagation of jobs in a computer network, and particle decay in nuclear chain reactions. A key tool in the study of branching processes is the theory of generating functions, which is an interesting area of study in its own right.

Branching processes with biological applications. The Poisson Process is a fundamental building block for continuous time probability models. The process counts the number of "events" that occur during the time interval [0, T ], where the times between successive events are independent and have a common exponential distribution.

Incoming calls to a telephone switchboard, decays of radioactive particles, or student arrivals to the Proctor lunch line are all events that might be modeled in this way. Poisson processes in space rather than time have been used to model distributions of stars and galaxies, or positions of mutations along a chromosome.

Starting with characterizations of the Poisson process, a thesis might develop some of its important properties and applications. Wiley, , Chapter 1. Two famous problems in elementary probability are the "Birthday Problem" and the "Coupon Collector's Problem.

For the second, imagine that each box of your favorite breakfast cereal contains a coupon bearing one of the letters "P", "R", "I", "Z" and "E". Now suppose that the "equally likely" assumptions are dropped.

But how does one prove such claims? A thesis might investigate the theory of majorization, which provides important tools for establishing these and other inequalities. This is a modern topic combining ideas from probability and graph theory. A "cover time" is the expected time to visit all vertices when a random walk is performed on a connected graph. Here is a simple example reported by Jay Emerson from his recent Ph. Consider a rook moving on a 2x2 chessboard.

From any square on the board, the rook has two available moves. If the successive choices are made by tossing a coin, what is the expected number of moves until the rook has visited each square on the board?

Reliability theory is concerned with computing the probability that a system, typically consisting of multiple components, will function properly e. The components are subject to deterioration and failure effects, which are modeled as random processes, and the status of the system is determined in some way by the status of the components.

For example, a series system functions if and only if each component functions, whereas a parallel system functions if and only if at least one component functions. In more complicated systems, it is not easy to express system reliability exactly as a function of component reliabilities, and one seeks instead various bounds on performance. Specifically, in order to be Riemann integrable, a function must be continuous almost everywhere.

However, many interesting functions that show up as limits of integrable functions or even as derivatives do not enjoy this property. Certainly one would want at least every derivative to be integrable. To this end, Henri Lebesgue announced a new integral in that was completely divorced from the concept of continuity and instead depended on a concept referred to as measure theory.

Interesting in their own right, the theorems of measure theory lead to facinating and paradoxical insight into the structure of sets. That is, we want a set of sets from F such that any two sets have a non-empty intersection. What is the structure of such a sub-collection? The conjecture remains open, though some particular cases have been solved. For more information see John Schmitt Snark Hunting "We have sailed many months, we have sailed many weeks, Four weeks to the month you may mark , But never as yet 'tis your Captain who speaks Have we caught the least glimpse of a Snark!

When Martin Gardner applied the name to a particular class of graphs in , a time when only four graphs including the Petersen graph of course were known to be in the class, it was an appropriate name. Snarks were hunted by Bill Tutte while writing under the pseudonym Blanche Descartes as a way to approach the then unsolved Four Color Problem. They were both an elusive and worthy prey. Now there exists several infinite classes of snarks and they have proved to be useful, though not yet in the way Tutte envisioned.

Gardner , Penguin, Gardner, Mathematical games, Scientific American, , No. For more information see John Schmitt. Two-Dimensional Orbifolds Spheres and tori are examples of closed surfaces. There is a well-known classification theorem whereby we are able to completely characterize any surface based on only two pieces of information about the surface.

A 2-dimensional orbifold is a generalization of a surface. The main difference is that in general, an orbifold may have what are known as singular points. A thesis in this area could examine Thurston's generalization of the surface classification theorem to 2-dimensional orbifolds. Another direction could be an examination of groups of transformations of the 2-dimensional plane which are used to produce flat 2-orbifolds.

This subject is full of big ideas but can be pleasantly hands-on at the same time. For classification of 2-manifolds, see Wolf, p.

Matrix Groups In linear algebra, we learn about n-by-n matrices and how they represent transformations of n-dimensional space. In abstract algebra, we learn about how certain collections of n-by-n matrices form groups. These groups are very interesting in their own rights, both in understanding what geometric properties of n-dimensional space they preserve, and because of the fact that they are examples of objects known as manifolds. There are many senior projects that could grow out of this rich subject.

See, for example, 27 above. In the late 19th century, geometry was revolutionized by the realization that if Euclid's fifth axiom, the parallel postulate, was dropped, there were a number of alternate geometries that satisfied the first four axioms but that displayed behavior quite different from traditional Euclidean geometry. These geometries are called non-Euclidean geometries, and include projective, hyperbolic, and spherical geometries.

As the theory of these geometries began to develop, one of the great mathematicians of the day, Felix Klein, proposed his Erlangen Program, a new method for studying and characterizing these geometries based on group theory and symmetries. A thesis in this area would study the various geometries, and the groups of transformations that define them. David Gans, Transformations and Geometries. For further information, see Emily Proctor. Skip to main content. Site Editor Log On. For further information, see Bruce Peterson.

The Four Color Theorem For many years, perhaps the most famous unsolved problem in mathematics asked whether every possible map on the surface of a sphere could be colored in such a way that any two adjacent countries were distinguishable using only four colors. For additional information, see Bruce Peterson. Additive Number Theory We know a good deal about the multiplicative properties of the integers -- for example, every integer has a unique prime decomposition.

For related ideas, see Waring's Problem topic Mersenne Primes and Perfect Numbers Numbers like 6 and 28 were called perfect by Greek mathematicians and numerologists since they are equal to the sum of their proper divisors e.

Kelley, Arrow Impossibility Theorems. For further information, see Mike Olinick. Mathematical Models of Conventional Warfare Most defense spending and planning is determined by assessments of the conventional ie. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has a complex root.

For further information, see Priscilla Bremser or David Dorman. Algebraic Numbers A real number r is "algebraic" if r is the root of a polynomial with integer coefficients. For further information Peter Schumer, or David Dorman. Nonstandard Analysis Would you like to see epsilons and deltas returned to Greek , where they belong?

Galois Theory The relation between fields, vector spaces, polynomials, and groups was exploited by Galois to give a beautiful characterization of the automorphisms of fields. Prime Number Theorem Mathematicians since antiquity have tried to find order in the apparent irregular distribution of prime numbers.

For further information, see Peter Schumer. Twin Primes Primes like 3 and 5 or and are called twin primes since their difference is only 2.

For further information, see Peter Schumer or David Dorman. Primality Testing and Factoring This topic involves simply determining whether a given integer n is prime or composite, and if composite, determining its prime factorization. Introduction to Analytic Number Theory Analytic number theory involves applying calculus and complex analysis to the study of the integers.

Finite Fields A finite field is, naturally, a field with finitely many elements. Representation Theory Representation theory is one of the most fruitful and useful areas of mathematics. Serre, Linear Representations of Finite Groups. For further information, see David Dorman.

Lie Groups Lie groups are all around us. Quadratic Forms and Class Numbers The theory of quadratic forms introduced by Lagrange in the late 's and was formalized by Gauss in Davenport, The Higher Arithmetic. Generalizations of the Real Numbers Let R n be the vector space of n-tuples of real numbers with the usual vector addition and scalar multiplication.

A thesis in this area would involve learning about the discoveries of these various "composition algebras" and studying the main theorems: Frobenius' theorem on division algebras. The Arithmetic-Geometric Inequality and Other Famous Inequalities Inequalities are fundamental tools used by many practicing mathematicians on a regular basis.

For further information, see Bill Peterson. For further information, see Michael Olinick or Peter Schumer. Decision-Theoretic Analysis and Simulation Medical researchers and policy makers often face difficult decisions which require them to choose the best among two or more alternatives using whatever data are available. Theory and Decision Making The power of modern computers has made possible the analysis of complex data set using Bayesian models and hierarchical models.

MA , with Ma a plus. Ma , with Ma a plus. For further information, see John Emerson. Pseudo-Random Number Generation Because a computer is deterministic, it cannot generate truly random numbers. Fall is also a good time to continue your reading and literature review, and begin your writing of the methods and introduction to the thesis; again this is up to your advisor.

At the end of the fall semester, a decision will be made by the faculty supervisor in consultation with the student as to whether good progress is being made, and whether the student should continue working on the thesis over the course of the year, or drop the project and convert Psychology to Psychology Independent Study with a grade.

Data collection and writing continue; data analysis may also be starting. Winter study and spring semester are usually especially busy times for thesis work. You will register for WS Thesis no other course is required , and the Thesis Coordinator will offer group workshops and support. Lots of data analysis and lots of writing, with multiple revisions.

Final copies of the thesis are due to the department a week before the second day of reading period. On the second day of reading period, all students present their theses in talks to the psychology department faculty, other thesis students, and interested others. You can take 8 regular psychology courses plus the thesis to fulfill your major requirements.

If you would like to check out psychology research on a limited basis, you can volunteer to help out on ongoing research, perhaps helping to collect data, prepare materials, assist a thesis student, etc.

As a paid work-study research assistant. Sometimes professors hire research assistants to help with research tasks such as data collection, data coding, entry, library research, etc.

As an Independent Study Psych fall or spring , Independent Study, allows you to get involved in research more intensively and independently, but is not as involved as a thesis. Students work with a professor to design, carry out, and write up a project.

Occasionally, students arrange an Independent Study that is library research and discussion based, studying a topic that is not offered in the regular Psych curriculum. In Independent Studies, students and profs meet weekly and the work is equivalent to a level psych course. This must be approved by the professor and the Psychology Department Chair before the end of the drop-add period so that if it is not approved you still have time to find another course.

You can take an Independent Study for one, but not more, of your level psych courses. This can be a good alternative to a thesis, if you are interested in the research but not the major writing and year-long commitment that a thesis involves. Summer research assistantships A limited number of summer research assistantships are available in Psychology and Neuroscience, through the Summer Science Research program.

These are paid positions for up to 10 weeks, doing research of various kinds. Depending on the nature of their work, professors may take anywhere from no summer students to several, and often but not always, give preference to thesis students.

If you are interested, contact the professor during Winter Study or as early in February as possible. Senior Thesis What is a senior thesis? Who should consider doing a senior thesis in psychology? What is the timetable for a senior thesis? Summer before senior year: Fall Semester All projects have their own timetables, and yours will be determined by you and your advisor and the nature of the project.

Winter Study Data collection and writing continue; data analysis may also be starting. Spring Semester Lots of data analysis and lots of writing, with multiple revisions.

Does the thesis count toward a course in the major? What are some other ways to get involved in research other than doing a thesis?

## Main Topics

Many high school seniors pick topics like "What is the bravest moment you had?," "Why do we sleep?" and "What Olympic events were practiced in ancient Greece?" There are many thousands of topics from which to choose. There are thousands of topics to choose from as a high school senior, but it is.

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It provides an essential experience for those planning to do graduate work, especially in history. A senior thesis means "doing" history, not just learning it; it helps you to discover how the historian conducts research and transforms that raw information into a coherent story and analysis. Previous Senior Thesis Topics. Toggle.