The student seminar provides an opportunity for Wellesley College students to present mathematical research and different topics to their peers and faculty.
Presentations are usually substantial, typically lasting about fifty minutes from 12:30 to 1:20. Speakers develop both public speaking and researching skills through their participation. Talks are delivered by students in all years, from first or second year students who are eager to learn about a new subject and tell others about it to students who participated in summer research programs and are reporting on their fundings. One seminar in the fall is usually dedicated to a panel on summer research programs.
Students who are interested in speaking should be on the lookout for an email from the student seminar coordinator. Typically a call for presenters is sent out to the departmental email list at the beginnings of the spring and fall semesters. If you are not on the departmental email list, you can contact Melanie Chamberlin to find out who is coordinating the student seminar (or you can sign up for the Math Department Google group so you can stay up-to-date on departmental events).
Schedule for spring 2017
Students who are giving talks in the student seminar are asked to complete this sheet to help them prepare for their talk.
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Interested in speaking in the Student Seminar?
Any student interested in lecturing may seek faculty advice on finding a topic appropriate for her; a list of possible student talks is also available here. A PLTC public speaking tutor will be able to help in preparation. To see the seminars presented in the past, please click here. This website offers tips on giving a good presentation, as well as this document. Answers to Frequently Asked Questions are also available.
Already have a topic for the presentation in mind?
If you already have a topic in mind, please contact the student seminar coordinator (or the department administrative assistant if you do not know who the coordinator is) who can help you choose a faculty member to assist you as you develop your presentation.
Need help finding a topic for the presentation?
The key to a good presentation topic is finding a piece of mathematics that you find intriguing but would like to know more about. Then, you will have fun doing the background research for the talk and your enthusiasm for the topic will help you give a good talk. Here are some suggestions for finding a good topic:
- Has there been any math topic in class or at an internship or job that you wanted to learn more about? If so, google around about that topic for more information and ask your talk advisor for advice.
- Check out http://www.math.hmc.edu/funfacts/ to peruse some fun math facts, and see if any catch your fancy.
- Another great place to look are the magazines Math Horizons and The College Mathematics Journal. These magazines have math articles that are meant for audiences with a background in college mathematics. Math Horizons is specially meant for college students; we have (online) access through the Wellesley Library. Peruse a recent article for ideas. The College Math Journal and Mathematics Magazine (another more expository journal that can be more advanced) can be searched (but not viewed) at: http://www.math.hmc.edu/journals/journalsearch2/
In addition, here are some some ideas from our own faculty. If something sounds interesting, please feel free to contact the faculty member directly to chat about it more and try to settle on a topic.
- Topics in number theory, group theory, analysis and topology.
- Multiplicative systems without unique factorization. We all know that every integer greater than 1 can be factored uniquely into prime numbers. Though this property seems quite natural, it is actually somewhat rare in the mathematical world-at-large. There are many examples of sets (both familiar and exotic), which are closed under multiplication, but which do not enjoy a unique factorization property.
- Continued Fractions. A continued fraction is an alternate way (as opposed to the more familiar decimal expansion) of expressing a real number. As with decimals, continued fractions can be either finite or infinite, and the infinite ones can be either periodic or not, depending on the algebraic properties of the number being represented. Continued fractions are also intimately related to the theory of approximation of irrational numbers by rationals.
- Geometric mechanics. The dynamics of a ping pong ball rolling on a surface change dramatically if one changes the geometric properties of the surface (e.g., the curvature). This is just one example of the rich interplay between geometry (math) and mechanics (physics). The field of geometric mechanics seeks to study how results in one area influence the other, and is a well-established subfield of mathematical physics and applied mathematics.
- Quantizing rolling systems. Over the past two decades, international teams of scientists have been synthesizing "nanocars" - molecular machines that "roll" on atomic surfaces. The quantum mechanics of such constrained systems is not well understood. However, in some cases one can formulate a well-defined quantum theory for these systems by viewing the nanocar as a "nonholonomic system" (a mechanical system subject to velocity constraints) and applying particular results on "Hamiltonization" (a technique that seeks to embed nonholonomic systems in Hamiltonian systems, which are more straightforward to quantize).
- History of Math: Parallel Postulate & the beginnings of spherical and hyperbolic geometries. Euclid's Parallel Postulate: Given a line l and a point p not on line l, there is a unique line l$'$ through p, parallel to l. For 2000 years, mathematicians attempted to prove that Euclid's fifth axiom (known as the Parallel Postulate) could be deduced from the first four axioms. Ultimately, Euclid was vindicated: by modifying the fifth axiom, one can construct completely different geometries: the sphere and hyperbolic space.
- Sphere-packing: Lattices, diamonds (carbon), the kissing number. What is the densest packing of spheres in space? The answer (to what is known as Kepler?s problem) is obvious to anyone who has seen grapefruit stacked in a grocery store, but a proof remains elusive. It is known, however, that the usual grapefruit packing is the densest packing in which the sphere centers form a lattice. The "kissing number problem" refers to the local density of packings: how many balls can touch another ball? This is a version of Kepler's problem in a spherical setting. Problems of arranging balls densely arise in many situations, particularly in coding theory (the balls are formed by the sets of inputs that the error-correction would map into a single codeword).
- Minimal surfaces. A minimal surface is a surface that locally minimizes its area. Physical models of minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film which is a minimal surface whose boundary is the wire frame.
- Double bubble theorem. A double bubble is pair of intersecting bubbles separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the figure above. Must the bubbles meet, as they do in the picture, at angles of 120 degrees? This was an open question until 2000. Its analog in four dimensions was proved by an REU group at Williams College.
- Mathematical logic. Topics such as the Continuum Hypothesis, Godel's incompleteness theorem.
- Theoretical computer science. Topics such as Computability Theory, the Recursion Theorem, P vs NP; Algebra - topics include valuations, p-adic numbers.
- Matrix theory. Topics include crazy ways to calculate/approximate eigenvalues.
- Other topics. The MAA journals (Math Horizons in particular) have lots of accessible articles that have great ideas for student seminar talks.
- Geometry of surfaces of revolution and ruled surfaces. A surface of revolution comes from taking a plane curve and rotating about a line in the plane. There are many interesting examples and it is relatively easy to find the basic geometric quantities such as sectional curvature. A surface is called ruled if each point on the surface has a line contained in the surface. There are many references; one is Alfred Gray's Modern Differential Geometry.
- Minimal surfaces in three-space. Minimal surfaces minimize area (at least locally). If you take a closed loop of wire and dip it in soapy water, the surface formed will be minimal. There are many beautiful examples of minimal surfaces, including the helicoid, Enneper's minimal surface and Hennenberg's minimal surface. There are many approaches to this topic, but I would suggest using the Weierstrass representation. Again, Alfred Gray's Modern Differential Geometry is one possible reference.
- Multiple imputation of missing data. Essentially all data sets have missing values (for example, not everyone filled out every question on a survey), and these NA values are not just an annoyance - handling missing data turns out to be an entire subfield of statistics. "Multiple imputation" is a common approach that involves predicting the missing values multiple times, based on the non-missing values, and combining these predictions in a way that takes into account the overall uncertainty in the resulting estimates. References include Little and Rubin's Statistical Analysis with Missing Data.
- Extended Monty Hall. Students often encounter the Monty Hall problem in introductory statistics courses: There are three doors, two hiding goats and one hiding a car. After you point to a door, the game show host (who knows where the car is) opens an additional door to show you a goat. We can prove that you're more likely to win a car if you switch your choice after the host shows this additional goat. This problem has been extended in a number of ways: What if there are more than three doors, with any number of goats or cars? What if the host can open more than one door? What if you can pick multiple doors or switch more than once? What if the host does not know where the car is? What if there is some randomness to the host's choice of additional door? These extensions can strengthen our conceptual understanding of the original problem and of Bayesian statistics in general.
- Error-correcting codes and check-digit schemes. When you deposit a check at the bank, you fill out a deposit slip that includes information like your account number. The teller then types your account number into a computer to begin the deposit process. Unfortunately, the various transcriptions involved in this process can lead to mistakes. For instance, a typical mistake is to transpose two numbers, writing, say, 321847 when one should have written 321487. Of course it's very important that the teller realize any transcription mistakes that have arisen (or else your deposit will end up in someone else's account). There are various mathematical schemes in place to ensure that these kinds of mistakes are "caught" before being propogating, and they all rely on some basic number theory. These codes are in place all around you: from your credit card number to the tracking code you receive at the post office.
- Primality testing and pseudoprimes. A large part of internet security relies on the fact that if a number is a product of two "large" prime factors, then it's very difficult to actual compute those factors. These cryptosystems therefore rely on knowing some prime numbers. But this produces an obvious conundrum: if large numbers are hard to factor, how do we know if a given large number is prime? A careful investigation into the properties of prime numbers allow some very clever techniques for answering this question.
- The nature of pi. Mathematics students are exposed to the concept of $\pi$ early in their careers, and it's clear from the beginning that this constant is ubiquitous in mathematics. Yet there is much about $\pi$ that is elusive. For instance, what can we say about the decimal expansion of $\pi$. Much is known about this question, and yet there is a lot that is still unknown. This topic has the benefit that it deals with an object that many are familiar with while still providing some unfamiliar (but easily understood) results.
- Linear algebra has many applications. One particularly good source of applications is the text Linear Algebra: A Modern Introduction, by David Poole. The applications there are in some cases self contained, and in other cases the text givea a brief introduction to whet your appetite, with additional references. I've listed some topics below; there are many more in this text.
- Error detecting codes and Error correcting codes. Examples are UPC codes on items you buy, ISBN codes for books, codabar for credit cards and ATM cards. The goal is to take a potential code, say a UPC code, that someone has entered and catch simple errors such as a transposition or to correct errors due to noise in transmissions from space vehicles.
- GPS system (Global Positioning System). This gives a brief introduction explaining how GPS works, with references given to sources with more details.
- Robotics. This shows how linear algebra can be used to describe motions a robot might make with its arms or hands.
- Markov Chains. If each year10% of iPhone users replace their iphone with an android model, and 20\% of android users switch to using an iphone, what will the long run distribution of iphone/android users be if the total number of users is fixed? Here annual transitions can be described by a matrix, multiplying by the matrix describes transitions from one year to the next, and the long run trends can be analyzed by considering eigenvalues and eigenvectors of the matrix.
- Leslie population model. This uses a matrix to describe the proportion of net births (births minus deaths) by age group in a population, and uses that matrix to predict long run population growth. This application is related to Markov chains, but since the population size isn?t fixed this leads to different mathematical issues.
- Leontief input/output models. These use matrices to describe the evolution of national economies by analyzing input and output of different. Leontief won the Nobel Prize in economics for his contributions involving such models.
- Predator-prey models. Foxes are a predator, and hares their prey. The populations of such predators and prey tend to go in cycles. Matrices can be used to predict such cyclic patterns.
- Fibonacci sequence. 1, 1, 2, 3, 5, 8, ? One can use linear algebra to find a formula for the nth Fibonacci number.
- Digital image compression. A photograph can be described by a matrix which gives the color of each pixel. That matrix can be represented in a systematic way (called the singular value decomposition) , and then approximated by much simpler matrices that still describe a picture quite similar to the original. These simpler approximations take up a lot less space (and less time to transmit.) This is one idea involved in image compression, which plays a crucial role in sending images and movies electronically.
- Cake cutting and fair division. How can a cake be divided fairly between n people so that each person feels they have a fair share? Are there good algorithms for dividing up chores between roommates, or the proceeds of an estate between family members? There are interesting mathematical papers on this subject, for example by Francis Su at Harvey Mudd College
- The pigeon-hole principle and double counting (Combinatorics). A good source for this topic is Chapter 22 of the book Proofs from THE BOOK by Martin Aigner and Gunter Ziegler.
- Topology. This is a place where various seemingly disparate areas of mathematics come together - algebra, analysis, geometry, combinatorics, and others - which makes it a fertile ground for student presentation. Some accessible topics are the classification of surfaces in 3-dimensional space, knot theory and its applications, algebraic mathods for telling topological spaces apart, or the topological combinatorics or graphs and their generalizations.
- Number theory. This field abounds with beautiful theorems, such as Euler's Theorem or Quadratic Recipricity, that are easy to state and whose proofs can be explained in a student presentation. In addition, some of the coolest applications of number theory in cryptography are not too hard to understand and explain. Some potential presentation topics in this area are RSA or elliptic curve cryptosystems, cryptography in our daily lives, and the issue of the regulation of cryptography and privacy.