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!
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.
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.