MIT Women in Mathematics Lecture Series

Organizers: Shan-Yuan Ho, Mia Minnes, Gigliola Staffilani, Katrin Wehrheim

This lecture series is intended for an advanced undergraduate or beginning graduate student audience. The talks will begin in the late afternoon, usually at 5:30pm. There will be pizza available after each talk.

Mathematicians of all levels, areas, and genders are welcome!

  • The solicitation for applications and suggestions for lecture series speakers is here.
  • The "MIT: Women in Math, a Celebration" conference website is now located here.
  • For titles and abstracts from the 2008-2009 lecture series, visit the archive.

Next Lecture:

Monday May 10th 5:30pm-6:30pm 2-139
Marina Meila (University of Washington)
Statistics and Computing with Permutations

When data comes in the form of permutations, statisticians and computer scientists turn to combinatorics in order to analyze it. I will describe how combinatorics, algorithms, and statistics come together in this endeavor from defining interesting but simple distributions over permutations, to algorithms for fitting them to the data, and to predicting their properties.

At the center of the results is the code of a permutation. This natural mathematical way to represent a permutation is key both to fast computing and to better understanding the models we create.

This talk does not require previous statistical or algorithms knowledge.

Joint work with: Harr Chen, Alnur Ali, Bhushan Mandhani, Le Bao, Kapil Phadnis, Arthur Patterson, and Jeff Bilmes

Pizza after the talk.

Past Lectures:

Monday April 26th 5:30pm-6:30pm 2-139
Vera Hur (UIUC and IMA )
Symmetry or Not?

Many problems in analysis appear symmetric, yet their solutions are sometimes nonsymmetric. I will present a n\ umber of examples for this. One of them is Newton's body of minimal resistance, one of the oldest problems in the calculus of varia\ tions. I will present a collection of tricks on how to prove symmetry for solutions of PDEs or of variational problems. If time per\ mits, I will also present a research problem on symmetry of surface water waves, accessible to graduate students.

Prerequisites for the talk: none

Pizza after the talk.

CANCELLED : Wednesday April 21st 5:30pm-6:30pm 2-139
Ragni Piene (University of Oslo )
Some Counting Problems and Their Generating Functions

To a sequence of integers a1,a2,a3... is associated with a generating \ function: the formal power series ƒ(x)=∑anXn. The generating function provides a way of displaying the sequence. For example,\ if an denotes the number of ways one can write the integer n as a sum of positive integers, the generating function is the partition function p(x)=Πm≥1(1 - xm)-1, and hence one can compute an as the coefficient of xn in the power series expansion of this function.

The partition function and its generalization the MacMahon function, as well as the generating functions for the Catalan and the Bell numbers, turn up in recent problems of counting certain geometric objects (like Gromov-Witten and Donaldson-Thomas invariants). Without going into details, the talk will give some examples and some hints at explanations.

Prerequisites for the talk: none

Pizza after the talk.

Wednesday April 14th 5:30pm-6:30pm 2-139
Angelia Nedich (University of Illinois)
Random Projection Algorithms for Convex Minimization Problems

New applications in communication and control networks, as well as large data sets processing, have given rise to optimization problems with large and/or dynamically changing constraints. In this talk, we will consider such an instance where the constraint sets are unknown but they can be observed and learned over time. We will discuss algorithmic approaches for solving such problems with convex structure. The algorithms rely on random projections on the constraint sets, as the sets are learned. We investigate convergence properties of the algorithms and also provide some error bounds on the algorithms' performance both in time and asymptotic.

Pizza after the talk.



Monday April 5th 5:45pm-6:45pm 2-139
Tara Javidi (University of California, San Diego)
Foster-Lyapunov Theorem and Routing in Wireless Networks

Routing is the process of selecting paths in a network along which the network traffic is sent. Routing is performed for many kinds of networks, including the telephone network, the electronic data networks (such as the Internet), and the transportation networks. This talk is concerned primarily with routing in data networks, when information is carried in the form of bits in a data packet. In data networks, routing directs packet forwarding (the transit of logically addressed packets) from their source toward their ultimate destination through intermediate nodes; typically hardware devices called routers or switches. The manner, which routing algorithm directs packet forwarding, impacts the delivery time of packets (per packet delay) and is the topic of this talk. In particular, we will consider an abstract probabilistic model associated with routing packets in a wireless data network.

In the first part of the talk, we propose to route each individual packet along a shortest-path to its destination and provide a dynamic program whose solution enables this objective. We then identify the drawback of this shortest-path philosophy when more than one packet is to traverse the network. In particular, we discuss the impact of traffic congestion. In the second part of the talk, using simple Markov chain analysis, we specify the impact of the routing decisions on the congestion state of the network. Appealing to the ergodic theorem for Markov Chains and the Foster-Lyapunov theorem, we arrive at a routing algorithm that ensures a unique steady state distribution with finite-mean for per packet delay.

Finally, and time permitting, we propose a routing algorithm that combines aspects from both approaches and provide significantly smaller per packet delay. We first numerically contrast this solution with the solutions discussed in the earlier part of the talk before providing the theoretical results. In the process of proving theoretical guarantees for the proposed solution, we introduce a new Lyapunov function construction which gives rise to a large class of solutions with provably finite delay.

Pizza after the talk.

Wednesday March 17th 5:30pm-6:30pm 2-139
Xiaomin Ma ( Brown University)
Introduction to Discrepancy Theory from Integration Estimation

It is often impossible to calculate an exact value of the integral of an arbitrary Riemann integrable function defined, say, on the unit cube. There are a variety of numerical methods used to estimate integrals. The key issues in such methods include: 1) whether the estimate converges to the real solution, 2) the magnitude of the error, and 3) how the error term is controlled in terms of quantities that depend on properties of the functions. In this talk, we focus on quasi-Monte Carlo estimation, which utilizes the theory of discrepancy. I will introduce the theory of discrepancy of point distributions and discuss constructions of low discrepancy sets.

Prerequisites for the talk: none

Pizza after the talk.

Tuesday March 9th 5:30pm-6:30pm 2-135
Barbara Csima ( University of Waterloo)
Computable Structure Theory

An infinite set of natural numbers is computable if there is a computer program that decides membership in the set. Similarly, a function on the natural numbers is computable if there is a computer program that on every input gives the same output as the function value on the input.

In computable structure theory, one examines various countably infinite structures (such as linear orderings and graphs) for their computability theoretic properties. For example, the standard theorem that any two countable dense linear orders without endpoints are isomorphic can be carried out computably, in the sense that if the two countable dense linear orders are nicely presented, then there must be a computable isomorphism between them. However, there are many examples of computable structures thatare isomorphic but not computably isomorphic.

This talk will be an introduction to computable structure theory, explaining some standard examples, and indicating areas of current research.

Prerequisites for the talk: none

Pizza after the talk.

Monday March 1st 5:30pm-6:30pm 2-139
Megan M. Kerr (Wellesley College)
Homogeneous Spaces: Differential Geometry with Lie Groups

Which Riemannian manifolds admit a metric of positive sectional curvature? This, one of the original questions of global Riemannian geometry, has motivated deep and beautiful mathematical results. And yet this is an area of geometry that is characterized more by its open questions than by its known theorems. As an example, the Hopf conjecture is easily stated and still open:

S2 x S2 does not admit a metric of positive sectional curvature.

A weaker, but not substantially different question is "Which Riemannian manifolds admit a metric of nonnegative sectional curvature?" The first step in answering (almost) any curvature question is to try it in the case of lots of symmetries. Our symmetry groups will be Lie groups. We will see how their rich structure provides a framework for finding examples of spaces with special curvature properties.

We will focus on a particular question: whenever we have nested compact Lie groups H < K < G, we get a fibration
K/H→ G/H→ G/K. When does this fibration yield homogeneous metrics of nonnegative curvature? This work builds on a result of Schwachhofer and Tapp. In this talk, we will look closely at the concrete setting of low-dimensional examples.

Prerequisites for the talk: none

Pizza after the talk.

Special Lecture Celebrating Black History Month

Monday February 22nd 5:30pm-6:30pm 2-105
Dr. Esther M. Pearson (Lasell College)
Making the Connection: Ethnic and Cultural Effects of Mathematics

This presentation examines chronological periods in history and how African Americans affected and were affected by mathematics. The African American experience from slavery through the Information Age is examined. Evidence within each period details how mathematics has profoundly affected the progress of African Americans. From their position as slaves to the representation of African Americans as 3/5th of a human; to their liberation, right to read, right to vote, and right to fight in the armed services. From their migration from the south to the north, from inner city to suburbia, from blue collar labor to white collar, from business employee to business owner. This talk brings into perspective the need for African Americans to embrace mathematics as never before so that academic, social, and economic gains can be maintained and advanced in America. Why then, has mathematics not been held closely or embraced like a friend to those of African descent?

Prerequisites for the talk: none

Pizza after the talk.

Thursday January 28th 2010 1:00 PM 2-190
Jill Pipher (Brown University )
Discrepancy Theory

I intend this talk to be accessible to undergraduates with an interest in mathematical theory. I'll define everything, give some history, and indicate applications. I aim to describe some open problems and new directions of research in this field for the more advanced students in the audience.

Discrepancy theory originated with some apparently simple questions about sequences of numbers. The discrepancy of an infinite sequence is a quantitative measure of how far it is from being uniformly distributed. Precisely, an infinite sequence { a1, a2 } is said to be uniformly distributed in [0; 1] if



If a sequence {a k} is uniformly distributed, then it is also the case that for all (Riemann) integrable functions ƒ on [0; 1],



Thus, uniformly distributed sequences provide good numerical schemes for approximating integrals. For example, if α is any irrational number in [0, 1], then the fractional part { α } k :=ak is uniformly distributed. Classical Fourier analysis enters here, namely Weyl's criterion.

The discrepancy of a sequence with respect to its first n entries is



One of the original questions in this study (Van der Corput) was this: does there exist a sequence such that D ( { a k }, n) is bounded by a constant for all n? The first breakthrough was made in 1945 by Van Aardenne-Ehrenfest. Later, Roth showed that the discrepancy problem for sequences had an equivalent geometric formulation in terms of discrepancy measures of n-point distributions in the unit square, with axis-parallel rectangles taking the place of intervals. This formulation initiated an investigation of discrepancy of point distributions in Rn by Roth, Schmidt, and then Beck, Chen, and many others. The most significant contributions to this subject use ideas and techniques from Fourier analysis. Many fundamental questions remain.

There are many reasons to be interested in discrepancy theory, both pure and applied: sets of low discrepancy figure prominently in numerical applications, from engineering to finance. This talk focuses primarily on theoretical issues involving measuring discrepancy in higher dimensions and quantifying discrepancy with respect to families of sets like convex polygons and rotated rectangles.


Prerequisites for the talk: none

Refreshments after the talk.

Tuesday January 26th 2010 1:00 PM 2-190
Amanda Epping Redlich (Massachusetts Institute of Technology)
Knitting and Math

The aim of this talk is to teach mathematicians why knitting is useful, and to teach knitters why math is useful. Topics discussed include braid theory, Cartesian coordinates, Diophantine equations, Hamiltonian paths, knot theory, mobius strips, polar coordinates, the symmetric group, torii of arbitrary genus, Turing machines, cabling, circular needles, fair isle, Kitchener stitch, k2tog, M1, set-in sleeves, stockinette, tinking, and YO. No prior knowledge of any of these topics will be assumed.

Prerequisites for the talk: none

Refreshments after the talk.

Thursday January 21st 2010 1:00 PM 2-190
Lydia Bourouiba(Massachusetts Institute of Technology)
Mathematics and the Mitigation of Disease Spread

H1N1 (swine) flu, SARS, H5N1 bird flu, Tuberculosis, and Poliomyelitis is only a short list of numerous emerging or re-emerging diseases with major human and economic costs. The impact is clear when considering that the 1918-1919 (H1N1) Spanish flu killed more than 40 million people. Most common infectious diseases are preventable and their mitigation is the focus of organizations such as the Center for Disease Control and the World Health Organization. In this ongoing mitigation and prevention battle, mathematics is becoming a valuable ally, helping understand both within-host pathogen dynamics and population disease dynamics. Mathematics can help answer some basic questions about the spread and mitigation of certain diseases such as how many cases of infection should we expect in a given epidemic? How many times should certain treatments be administered given their cost and known secondary effects? Through an overview and series of examples illustrating the quantitative methods used in mathematical epidemiology, we will discuss the role of mathematics in the mitigation of disease spread; its success stories, shortcomings, and ongoing exciting areas of research.

Prerequisites for the talk: none

Refreshments after the talk.

SPECIAL PRESENTATION

Wednesday December 2nd 5:30pm-6:30pm 2-135
Debra Borkovitz(Wheelock College )
Elementary Math is Not Elementary! Thoughts on Preparing Teachers.

For many years, it was commonly believed in the U.S. that future elementary school teachers learned all the math content they needed to know by the end of high school. Now it is widely recognized that for teachers to take school mathematics beyond calculation without understanding, they need to develop a much deeper understanding of elementary mathematics. There is a specialized body of math content knowledge for elementary teaching, just as there are such bodies of knowledge for engineering and other professions (and even people with Ph.D's in math have not necessarily mastered this math content).

In this talk we will look at some not so elementary examples of K-8 math to get a sense of some of the deeper issues involved for both elementary teachers and for those who prepare them. We will also look at some efforts to improve mathematics education in Massachusetts and beyond. I will also share a bit of my own path from a PhD in math from MIT to working primarily with future elementary teachers.

The talk will be participatory, so come prepared to think, to share experiences, and to take a fresh look at third-grade math.

Prerequisites for the talk: none

Pizza after the talk.

Tuesday January 12th 2010 1:00 PM 2-190
Natasa Pavlovic (University of Texas, Austin)
The Enigma of the Equations of Fluid Motion: a Survey of Existence and Regularity Results

The partial differential equations that describe the most crucial properties of the fluid motion are the Euler equations. They are derived for an incompressible, inviscid fluid with constant density. Some basic questions concerning Euler equations in 3 dimensions are still unanswered. For example, it is an outstanding problem to find out if solutions of the 3D Euler equations form singularities in finite time.

The equations that describe the most fundamental properties of viscous fluids are the Navier-Stokes equations. As with the Euler equations the theory of the Navier-Stokes equations in 3D is far from being complete. The major open problems are global existence, uniqueness and regularity of smooth solutions of the Navier-Stokes equations in 3D.

In this talk we will give a survey of some known results addressing existence and regularity of solutions to these equations.

Prerequisites for the talk: none

Refreshments after the talk.

Thursday January 14th 2010 1:00 PM 2-190
Alison Malcolm (Massachusetts Institute of Technology)
Using Math to See Inside the Earth

Seismic waves (similar to those excited by Earthquakes) are used all the time to make images of structures beneath the Earth. This is done for oil exploration, carbon sequestration, geothermal energy, contaminant monioring and slope stability studies to name just a few examples. Constructing images from these waves requires an understanding of how to compute the paths on which waves travel through the Earth. Through a series of examples, we'll see how simple techniques for solving equations can be applied to approximately compute these paths, quickly and accurately, and how the questions we ask in imaging the Earth also lead to new mathematical questions.

Prerequisites for the talk: none

Refreshments after the talk.

Monday October 26th 5:30pm-6:30pm 4-145
Valentina Harizanov (George Washington University)
Priority methods.

In computable mathematics the existence of certain objects is often demonstrated by actually building them. We will present an example of a construction which will in a very simple setting illustrate the main ideas and give the flavor of a computability theoretic technique called the finite injury priority method. This method, which was invented in the 1950's and revolutionized computability theory, represents the first level in the hierarchy of priority methods. This intricate and powerful technique allows us to satisfy mutually conflicting requirements by fitting together opposite strategies.

Prerequisites for the talk: none

Pizza after the talk.

Thursday October 22nd, 2009 5:30pm-6:30pm 2-135
Julia Wolf (Rutgers University)
What the Fourier transform can and cannot tell us about the integers.

It is surprisingly straightforward to count the number of solutions to simple equations such as x+y=2z (representing a 3-term arithmetic progression) or x-y=z2 (a square difference) in a "random-looking" subset of the integers. The discrete Fourier transform provides a natural way of quantifying what we mean by random-looking, but fails us once we start to consider longer arithmetic progressions and other more intricate structures. This failure opens the door to a rich and as yet largely unexplored theory of higher-degree Fourier analysis, which we shall try and catch a glimpse of in this talk.

Prerequisites for the talk: modular arithmetic, roots of unity and the Cauchy-Schwarz inequality; some familiarity with the Fourier transform is desirable but not essential.

Pizza after the talk.

Wednesday September 16, 2009 5:30pm-6:30pm 2-135
Johanna Franklin (Fields Institute)
Defining Randomness.

What does it mean for an infinite binary sequence to be random? What does it mean for an infinite binary sequence to be nowhere near random? I will present three different approaches to defining randomness, show how they can be made to be equivalent, and describe some of the other properties that random binary sequences can have. Then I will present ways in which a real can be said to be "far from random," discuss whether or not these definitions are equivalent, and explore some of the other properties of these binary sequences.

Prerequisites for the talk: None.

Pizza after the talk.