H Jerome Keisler, Elementary Calculus The relation between fields, vector spaces, polynomials, and groups was exploited by Galois to give a beautiful characterization of the automorphisms of fields. From this work came the proof that a general solution for fifth degree polynomial equations does not exist. Along the way it will be possible to touch on other topics such as the impossibility of trisecting an arbitrary angle with straight edge and compass or the proof that the number e is transcendental.
Mathematicians since antiquity have tried to find order in the apparent irregular distribution of prime numbers. Let PI x be the number of primes not exceeding x. Many of the greatest mathematicians of the 19th Century attempted to prove this result and in so doing developed the theory of functions of a complex variable to a very high degree. Partial results were obtained by Chebyshev in and Riemann in , but the Prime Number Theorem as it is now called remained a conjecture until Hadamard and de la Valle' Poussin independently and simultaneously proved it in However, Hilbert's proof did not determine the numerical value of g k for any k.
Peter Schumer, Introduction to Number Theory. Primes like 3 and 5 or and are called twin primes since their difference is only 2. It is unknown whether or not there are infinitely many twin primes. In , Leonard Euler showed that the series S extended over all primes diverges; this gives an analytic proof that there are infinitely many primes.
However, in Viggo Brun proved the following: Hence most primes are not twin primes. A computer search for large twin primes could be fun too. Landau, Elementary Number Theory, Chelsea, ; pp. Do numbers like make any sense? The above are examples of infinite continued fractions in fact, x is the positive square root of 2. Moreover, their theory is intimately related to the solution of Diophantine equations, Farey fractions, and the approximation of irrationals by rational numbers. Homrighausen, "Continued Fractions", Senior Thesis, One such area originated with the work of the Russian mathematician Schnirelmann.
He proved that there is a finite number k so that all integers are the sum of at most k primes. Subsequent work has centered upon results with bases other than primes, determining effective values for k, and studying how sparse a set can be and still generate the integers -- the theory of essential components. This topic involves simply determining whether a given integer n is prime or composite, and if composite, determining its prime factorization.
Checking all trial divisors less than the square root of n suffices but it is clearly totally impractical for large n. Why did Euler initially think that 1,, was prime before rectifying his mistake? Analytic number theory involves applying calculus and complex analysis to the study of the integers. Its origins date back to Euler's proof of the infinitude of primes , Dirichlet's proof of infinitely many primes in an arithmetic progression , and Vinogradov's theorem that all sufficiently large odd integers are the sum of three primes Did you spot the arithmetic progression in the sentence above?
A finite field is, naturally, a field with finitely many elements. Are there other types of finite fields? Are there different ways of representing their elements and operations? In what sense can one say that a product of infinitely many factors converges to a number? To what does it converge?
Can one generalize the idea of n! This topic is closely related to a beautiful and powerful instrument called the Gamma Function. Infinite products have recently been used to investigate the probability of eventual nuclear war. We're also interested in investigating whether prose styles of different authors can be distinguished by the computer. Representation theory is one of the most fruitful and useful areas of mathematics. The development of the theory was carried on at the turn of the century by Frobenius as well as Shur and Burnside.
In fact there are some theorems for which only representation theoretic proofs are known. Representation theory also has wide and profound applications outside mathematics. Most notable of these are in chemistry and physics. A thesis in this area might restrict itself to linear representation of finite groups. Here one only needs background in linear and abstract algebra. Lie groups are all around us. In fact unless you had a very unusual abstract algebra course the ONLY groups you know are Lie groups.
Don't worry there are very important non-Lie groups out there. Lie group theory has had an enormous influence in all areas of mathematics and has proved to be an indispensable tool in physics and chemistry as well. A thesis in this area would study manifold theory and the theory of matrix groups.
The only prerequisites for this topic are calculus, linear and abstract algebra. One goal is the classification of some families of Lie groups. For further information, see David Dorman or Emily Proctor. The theory of quadratic forms introduced by Lagrange in the late 's and was formalized by Gauss in The ideas included are very simple yet quite profound.
One can show that any prime congruent to 1 modulo 4 can be represented but no prime congruent to 3 modulo r can. Of course, 2 can be represented as f 1,1. Let R n be the vector space of n-tuples of real numbers with the usual vector addition and scalar multiplication. For what values of n can we multiply vectors to get a new element of R n? The answer depends on what mathematical properties we want the multiplication operation to satisfy.
A thesis in this area would involve learning about the discoveries of these various "composition algebras" and studying the main theorems:. Inequalities are fundamental tools used by many practicing mathematicians on a regular basis. This topic combines ideas of algebra, analysis, geometry, and number theory. We use inequalities to compare two numbers or two functions. These are examples of the types of relationships that could be investigated in a thesis.
You could find different proofs of the inequality, research its history and find generalizations. Hardy, Littlewood, and P—lya, Inequalities, Cambridge, Ramanujan or women in mathematics , the history of mathematics in a specific region of the world e. 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.
Upon receipt of the requested dollar amount, they will send you a copy. Some constructions of irreducible representations of generic Hecke algebras of type A n.
The Mathematics of Meaning: From Triangles to Teamwork: A Mathematical Explanation for the Evolution of Cooperation. Evolutionary Games on Structured Population: Geometry in Algorithms and Complexity: Holographic Algorithms and Valiant's Conjecture. An Introduction to Contact Topology: Defining physics at imaginary time: Simplifying Complicated Simplicial Complexes: Such a student usually will have written a thesis, or pursued actuarial honors and written a mini-thesis.
An outstanding student who writes a mini-thesis, or pursues actuarial honors and writes a paper, might also be considered. In all cases, the award of honors and highest honors is the decision of the Department. Here is a list of possible colloquium topics that different faculty are willing and eager to advise. You can talk to several faculty about any colloquium topic, the sooner the better, at least a month or two before your talk.
For various reasons faculty may or may not be willing or able to advise your colloquium, which is another reason to start early. Here is a list of faculty interests and possible thesis topics.
You may use this list to select a thesis topic or you can use the list below to get a general idea of the mathematical interests of our faculty. I work in low-dimensional topology.
Specifically, I work in the two fields of knot theory and hyperbolic 3-manifold theory and develop the connections between the two. Knot theory is the study of knotted circles in 3-space, and it has applications to chemistry, biology and physics. Hyperbolic 3-manifold theory utilizes hyperbolic geometry to understand 3-manifolds, which can be thought of as possible models of the spatial universe.
An n -crossing is a crossing with n strands of the knot passing through it. Every knot can be drawn in a picture with only n -crossings in it. The least number of n -crossings is called the n -crossing number. Determine the n -crossing number for various n and various families of knots. In other words, the projection looks like a daisy. The petal number of a knot is the least n for such a projection.
Determine petal number for various knots, and see how it relates to other traditional knot invariants. Still unknown for two twisted strands. Can also consider lattice stick knots, where all sticks are parallel to the x,y,z axes.
When are they hyperbolic? Quasi-Fuchsian surfaces generalize totally geodesic surfaces. Show that many surfaces in knot complements are quasi-Fuchsian. Particularly interested in topology, knot theory, graph theory, tiling theory and geometry but will consider other topics as well.
Mathematical modeling, theoretical ecology, population biology, differential equations, dynamical systems. My research uses mathematical models to uncover the complex mechanisms generating ecological dynamics, and when applicable emphasis is placed on evaluating intervention programs.
My research is in various ecological areas including I invasive species management by using mathematical and economic models to evaluate the costs and benefits of control strategies, and II disease ecology by evaluating competing mathematical models of the transmission dynamics for both human and wildlife diseases.
Any topics in applied mathematics, such as: Statistical methodology and applications. One of my research topics is variable selection for high dimensional data. I am interested in traditional and modern approaches to select a smaller number of variables from a large candidate set. I am especially interested in selecting variables for survival models, which study how risk factors influence the time until the occurrence of a certain event of interest.
Additionally, I am also interested in analyzing the rhetorical styles in English text data using statistical methods. The traditional survival models usually study the relationship between several risk factors and time to the occurrence or recurrence of a certain disease.
We may evaluate whether the treatment is effective, identify important risk factors and compare the survival features among different demographic groups.
On the other hand, survival models can be extended to be used in other settings, e. We could answer questions like how current network structures predict future development of the network. For example, for survival models in the previous topic, we could include all potentially relevant risk factors or network features initially, then select the most important ones for a simpler model with easy interpretations.
Other examples are not restricted to models previously mentioned. We could also study the time-varying effects of predictor variables on the response variable. The data have hierarchical structure and contain very rich information about of the rhetorical styles being used.
We could apply statistical models to reduce dimensions and have a more insightful understanding of the text. Open to any problems in statistical methodology and applications, not limited to my research interests and the possible thesis topics above.
My research interests are in both statistical methodology and in statistical applications. For the first, I look at different methods and try to understand why some methods work well in particular settings, or more creatively, to try to come up with new methods.
For the second, I work in collaboration with an investigator e. I have been especially interested in problems dealing with large data sets and the associated modeling tools that work for these problems.
I have been working on models for assessing the effect of age on performance in running and swimming events. There is still much work to do. Classical statistics deals with quantitative and categorical variables, but what happens when the variables have even less structure? Using text mining and other recent work from statistics and machine learning can we figure out how people feel about a topic by analyzing what they say as well as their actions?
How do we choose variables when we have dozens, hundreds or even thousands of potential predictors? Various model selection strategies exist, but there is still a lot of work to be done to find out which ones work under what assumptions and conditions.
Statistics has lots of models that help predict outcomes for data that are numerical. But what if the data are text? What can we say about documents based only on the words they contain? Can we use comments in surveys to help answer questions traditionally modeled only by quantitative variables?
In this era of Big Data, not all methods of classical statistics can be applied in practice. What methods scale up well, and what advances in computer science give insights into the statistical methods that are best suited to large data sets? In collaboration with a scientist or social scientist, find a problem for which statistical analysis plays a key role.
Geometry and Number Theory.
If you are looking for information and advice from students and faculty about writing a senior thesis, look at this document. It was compiled from comments of students and .
Math Thesis Archive This is a list of all dissertations that have been submitted in partial satisfaction for the degree of Doctorate of Philosophy (Ph.D) in Mathematics at UCSD. Dissertations are sorted by thesis topic, author's last name, and year of graduation.
Undergraduate Senior Thesis in Mathematics The Department of Mathematics at UC Davis offers undergraduate students the opportunity to participate in original mathematical research that will culminate in a senior undergraduate thesis. A full year of either Senior Thesis (Math ) or Mathematics Clinic (Math ) is required for the mathematics major. Senior thesis offers the student, guided by the faculty advisor, a chance to experience a taste of the life of a professional research mathematician. The work is largely independent with guidance from the research advisor.
Available in PDF format (linked below) and in hard copy from the Mathematics Library (through , more recent ones available soon). For more information, see Senior Thesis Guidelines. Mathematics is not a static, immutable body of knowledge. New approaches to old problems are constantly being investigated and, if found promising, developed. Nonstandard analysis is a good and exciting example of this mathematical fact of life.