There are related functions called Grassmanian polylogarithms, invented by A. Goncharov, which enjoy relatively simple functional equations.
To try to relate these to webs, or to find a new more geometric approach to their functional equation, would also be interesting and potentially do-able. One of the great collaborative success stories of the past two decades has been that between complex algebraic geometers and string theorists in the mirror symmetry program. The quest to produce Calabi-Yau 3-manifolds three complex dimensions!
Moreover, they had to figure out how to resolve them -- the higher-dimensional analogue of lifting an actual string off itself. While this story has only recently been thoroughly understood, it would be well within the powers of an interested undergraduate student to provide a down-to-earth account with basic examples.
This is something I have not seen in the literature, and shouldn't be thought of as an expository project -- it would require some original thought. It would also acquaint you with toric geometry, an extremely useful tool which gives a dictionary between algebro-geometric concepts and the geometry of convex bodies like polygons and polytopes considered relative to a lattice. In working out examples, the latter boils down to some surprisingly entertaining 3-dimensional linear algebra which ultimately tells you how to draw a triangulation.
Mirror symmetry comes into this story in a number of ways. In one version, the resolutions of singularities you will construct are "mirror" to certain families of Riemann surfaces. An ambitious student might want to investigate this too. The classical theorems of Abel and Jacobi describe the divisors configurations of zeroes and poles with multiplicity of meromorphic functions on compact Riemann surfaces. Attempts to generalize these results to noncompact or singular settings, as well as to higher dimension, have motivated a lot of modern algebraic and differential geometry -- like the Bloch-Beilinson and Hodge conjectures and the theory of webs.
I don't know of a good write-up of the one-dimensional generalizations, and you could already learn a lot by trying to trying to understand the situation for unions of lines, or for multiply connected regions.
In algebraic geometry, roughly speaking, we study solution sets of algebraic equations. Replace everywhere multiplication by addition and addition by "taking the maximum," and you have an exciting new theory called tropical geometry -- which even has its own version of Abel's theorem!
Amoebas are objects which provide a connection via a limiting process between algebraic curves and tropical curves, and it would be extremely interesting though not necessary for an interesting project to devise a connection whereby one Abel's theorem becomes the limit of another. For this project, all I really ask is that a student be familiar with basic complex analysis. It would also be useful to know what a Riemann surface is, but this could be dealt with in summer reading.
I would be happy to direct a reading course and subsequent write-up as well, on any of the following topics or on appropriate student-proposed topics:. Counterexamples can be shown to exist either through probabilistic arguments i. This project would involve trying to construct more interesting families of counterexamples to the three variable von Neumann inequality in order to understand "how badly" the inequality fails.
This is a different kind of inequality for polynomials. This project would also involve looking for interesting examples to test the sharpness of known versions of this inequality. I would be happy to supervise reading projects on other topics: It is an open problem whether there will be four points that are the corners of a square.
Let G be any finite group. Can we relate the topology of U to the structure of the group? What if we allow U to live in a higher dimensional space? Does that allow more groups G to give an affirmative answer? Given a group G , can we estimate the dimension of the space in which a domain U will live that has the desired property? It is an intuitively obvious assertion that, of all planar domains, the disc has the "largest" automorphism group.
Formulate a precise version of this statement and prove it. This problem is important for the theory of partial differential equations. It is known that a convex planar U can have at most one equichordal point. But the proof is very abstract and extremely difficult. What is true in dimension three? What is true for non-convex domains? It is still of current interest to determine the maximal error of the asymptotic normal approximation to the scaled sample mean as specified by a central limit theorem.
Recent results related to the bound in the Berry-Esseen theorem, for summands of both i. This project would involve studying the methods used to obtain such bounds and investigating the accuracy using simulated data. Inferential correctness for testing hypotheses about regression coefficients after a variable selection procedure has been utilized requires a careful evaluation of the effects of the selection procedure on the final inference. This project involves studying, in real-data examples, how classical inference procedures are invalidated by the use of selection procedures.
The performance of inference procedures designed to control, respectively, selective type I error and familywise error rates FWER will be compared in theory and practice. Inference in curved exponential families, following a principled approach, requires construction of exact or approximate ancillary statistics.
Examples include the gamma hyperbola model, the bivariate normal correlation model, and ARMA p,q models used in time series analysis. This project involves motivating a principled, accurate approach to inference in such models, and real-data comparisons with conventional inference procedures which do not respect such statistical principles.
The double bootstrap was proposed as a technique to improve the coverage accuracy of confidence intervals constructed via conventional single-bootstrap methods. The warp-speed double-bootstrap attempts to achieve this using a single double-bootstrap sample for each bootstrap sample, drastically reducing the computational costs.
Recent results of Chang and Hall show that, unfortunately, warp-speed double-bootstrap confidence intervals do not in general share the same coverage accuracy properties as intervals constructed by the conventional double bootstrap.
This project investigates the source of this discrepancy using large-scale simulations in different model settings of practical interest. A similar question was raised by J. Serre for polynomial rings over a field, with the a's being polynomials in several variables. This fundamental question generated an enormous amount of mathematics giving birth to some new fields and was finally settled almost simultaneously by D. Now, there are fairly elementary proofs of this which require only some knowledge of polynomials and a good background in linear algebra.
This could be an excellent project for someone who wants to learn some important and interesting mathematics. These results seem to be of great interest to people working in control theory.
Though I am not an expert, I'm willing to learn with a motivated student. This has important applications in secure transmissions over the internet and techniques like RSA cryptosystems.
Of course, the ancient method of Eratosthenes sieve method is one such algorithm, albeit a very inefficient one. All the methods availabe so far has been known to take exponential time. There are probabilistic methods to determine whether a number is prime, which take only polynomial time. The drawback is that there is a small chance of error in these methods. So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. A copy of their article can be downloaded from www.
Methylation is important to embryonic development and cancer. With the current next-generation sequencing NGS technology, people identify regions with different methylation levels under different disease status to understand the mechanism of cancer and other disorders. NGS data from methylation experiments process complicated strictures and impose challenges to statisticians.
We are developing statistical tools for the analysis of NGS data from such experiments. Anesthesiologists are still in debate about proper ways to monitor patients' anesthetic status. A recent publication on the New England Journal of Medicine vol , pages suggest that a device approved by the Food and Drug Administration FDA to reduce the risk that patients will recall their surgery does not lower the risk of the problem, known as intraoperative awareness, any more than a less expensive method.
Statistical analysis will help to find better ways of anesthetic practice. Statistical tools are crucial to understanding the transformation from the reality to the map in people's memory. We are developing statistical models to tackle these issues. At the closed end, a small quantity of gas is injected. It diffuses out the other end at a predictable rate.
Now, suppose the quantity of gas injected is increased. The flow will not scale linearly, as the effect of the pressure of the introduced gas must be considered.
I have a project with Professor Gregory Yablonksky in the Chemical Engineering department to model this flow. Imagine a large number of cameras arranged around a central object. One wants to match up the pictures, but there is some error in the measurement. Mathematically, the problem becomes approximating a large symmetric matrix by a rank 3 matrix that has 1's on the diagonal. It ties in to an active research area in systems theory: Nobody knows how to do this well.
I believe this is a statistical artifact, due principally to cultural differences in filling out death certificates. I would be willing to supervise an undergraduate who wished to hunt down the data and analyze it. The data is available at www. We would like to understand more about the tortoises's movement, for example what makes them migrate, how are they influenced by climate changes, why smaller tortoises don't migrate, how they choose routes, etc.
The Homology program of J. Welker has been used to investigate the structure of such complexes. There are many adjustments and additions which could be made to improve the program, the most ambitious of which is to make it amenable to parallel processing. There are interesting open questions and proven theorems about relating the algebraic structure of G to the combinatorial structure of this partially ordered set.
For any partially ordered set P, the set of all totally ordered subsets of P determines a simplicial complex. The topological structure of this complex is related to the combinatorial structure of P. One can hope to use this relationship productively when P is the set of subgroups of G. This area is appropriate for both reseach and expository projects. Symmetric functions appear in many areas of mathematics, including combinatorics and representation theory which involves studying a group G by understanding homomorphisms from G to various matrix groups.
There are lots of interesting open combinatorial problems involving symmetric functions many appear in the exercises after Chapter 7 of R. This area is also appropriate for expository projects. Both deal with the idea that certain variables predict whether a response is necessarily zero, and if the response is not necessarily zero, then other variables might predict its value.
A student worked on this last year, but you can extend what he did. The classic example is electro- convulsive therapy for depression.
The machine typically has about five fixed charge levels in Coulombs , and the charge is stepped up until the patient has a seizure. We thus know coarse lower and upper bounds on the seizure threshold, and the task is to estimate the exact seizure threshold.
Most large public-access data sets have this complex structure. This was covered in a course - Math I think - but that was so long ago it's not listed in the catalog. There are, in fact, two forms of complex data, the "classic" form in which each stratum has exactly two clusters, and the "certainty PSU" form.
I have a friend who has some pathological gambling data, who has extracted most of the obvious results from her data, but might be looking for help in digging out some remaining gems.
He had written a paper called the "Trustworthy Jackknife" in which he tried to figure out when the jackknife method gave dependable variance estimates. The Jackknife was considered very mysterious.
It worked, but nobody could figure out why it worked. And sometimes it didn't work. What I thought was that the jackknife must be a differential, local kind of approximation for something else. And so when I started looking for the something else I came up with the bootstrap. In calculus, we learn that there are curves that do not have tangent lines. Such curves are usually called singular curves.
In this project, we will study geometry of singular spaces which are generalizations of singular curves. One class of such singular spaces that are interesting to us is called orbifold. It is an important object in both mathematics and physics. We will mainly focus on orbifolds and compute some useful invariants about them.
In mathematics, the shape of a donut is called torus. Noncommutative torus is the quantization of the usual torus, and appears naturally in both mathematics and physics.
We will study some interesting applications of noncommutative tori in physics. There is an interesting mathematics theory related to this answer, which is called index theory.
Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension. Combinatorics concerns the study of discrete and usually finite objects.
Aspects include "counting" the objects satisfying certain criteria enumerative combinatorics , deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria as in combinatorial designs and matroid theory , finding "largest", "smallest", or "optimal" objects extremal combinatorics and combinatorial optimization , and finding algebraic structures these objects may have algebraic combinatorics.
Logic is the foundation which underlies mathematical logic and the rest of mathematics. It tries to formalize valid reasoning. In particular, it attempts to define what constitutes a proof. Number theory studies the natural, or whole, numbers.
One of the central concepts in number theory is that of the prime number , and there are many questions about primes that appear simple but whose resolution continues to elude mathematicians.
A differential equation is an equation involving an unknown function and its derivatives. In a dynamical system , a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.
Mathematical physics is concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". The fields of mathematics and computing intersect both in computer science , the study of algorithms and data structures, and in scientific computing , the study of algorithmic methods for solving problems in mathematics, science and engineering.
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data.
Signal processing is the analysis, interpretation, and manipulation of signals. Signals of interest include sound , images , biological signals such as ECG , radar signals, and many others.
Processing of such signals includes filtering , storage and reconstruction, separation of information from noise , compression , and feature extraction. Probability theory is the formalization and study of the mathematics of uncertain events or knowledge.
The related field of mathematical statistics develops statistical theory with mathematics. Statistics , the science concerned with collecting and analyzing data, is an autonomous discipline and not a subdiscipline of applied mathematics. Game theory is a branch of mathematics that uses models to study interactions with formalized incentive structures "games".
It has applications in a variety of fields, including economics , evolutionary biology , political science , social psychology and military strategy. Operations research is the study and use of mathematical models, statistics and algorithms to aid in decision-making, typically with the goal of improving or optimizing performance of real-world systems. A mathematical statement amounts to a proposition or assertion of some mathematical fact, formula, or construction.
Such statements include axioms and the theorems that may be proved from them, conjectures that may be unproven or even unprovable, and also algorithms for computing the answers to questions that can be expressed mathematically. Among mathematical objects are numbers, functions, sets, a great variety of things called "spaces" of one kind or another, algebraic structures such as rings, groups, or fields, and many other things.
Mathematicians study and research in all the different areas of mathematics. The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied such as theoretical computer science and theoretical physics.
Good Topics for Mathematics Research Papers. A mathematics research paper is an extremely intricate task that requires immense concentration, planning and naturally clear basic knowledge of mathematics, but what is essential for a higher level research is the successful choice of a topic, matching your personal interests and level of competence.. You may be given a list of possible topics .
We have gathered a list of best research paper topics that will help you with your projects. Hire a writer to help you come up with a topic of research paper! Order Now. Research Paper Topics on Math. The influence of algorithms;.
This article itemizes the various lists of mathematics topics. Some of these lists link to hundreds of articles; some link only to a few. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. 10 Good College Research Paper Topics in Math. Math is immensely present everywhere we go. So much so, that finding a good college research topic in math should be a fun and adventurous experience. Here are some suggestions we think would make for great research topics on the subject of math.
Mathematics Research Paper Topics. Individuals Instrumental in Math: Math in World Cultures: Math in Other Disciplines: Euclid Plato Pythagoras Hypatia Thales of Miletus Alexander the reat Eratosthenes of Cyrene Archimedes Hipparchus Claudius Ptolemy Rene Descartes Johannes Kepler. Research within librarian-selected research topics on Mathematics, Chemistry, Physics, and Astronomy from the Questia online library, including full-text online books, academic journals, magazines, newspapers and more.