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!
Thursday February 12, 2009 4pm-5pm 2-190
Susan Landau
(
Sun Microsystems)
√2 + √3: Four Different Views.
How much time does it take to factor polynomials?
How can you efficiently tell if a polynomial has roots expressible in terms of
radicals? Is there a fast method to decompose a polynomial into
lower-degree components? Suppose it is claimed that
; how can you check if it is true?
You have to study the underlying algebraic structure, but often the
theorems are not conducive to efficient computation, and new
understanding --- and new results --- are needed. In the talk, I will
present four results (three mine, one not) that resulted from the
effort to find fast methods for algebraic simplification. And then I
will talk about really understanding √2+√3 sheds much
light on these seemingly unrelated results.
Tea at 3:15pm and refreshments after the talk.
Thursday February 19, 2009 5:30pm-6:30pm Room 4-145
Matilde Lalin
(
University of Alberta)
So you think you can count?
In the study of curves, zeta functions codify information about the number of solutions over finite fields, but we expect them to have more information. In particular, there is a big interest in studying the distribution of zeros of zeta functions (and L functions).
The usual approach (following the influential work of Katz and Sarnak) consists on fixing a family of curves and letting the number of points in the finite field tend to infinity. We will focus on a different point of view: fixing the number of points in the finite field and letting the genus of the curves tend to infinity. This idea was introduced in a recent work by Kurlberg and Rudnick for hyperelliptic curves (y^2=f(x)), while we are looking at the trace for the family y^3=f(x).
This is joint work with A. Bucur, A. Cojocaru, C. David, and B. Feigon.
Pizza after the talk.
Tuesday March 3, 2009 5:30pm-6:30pm Room 4-145
Rebecca Weber
(
Dartmouth)
What is computability theory?
If it is possible to write a computer program to execute a given function, we can call that function "computable". It is clear, then, how to prove that a function _is_ computable, but how do you show that it is _not_ computable? That requires a more precise definition, something not affected by advances in technology and also something to which we can apply proof techniques. Once we have such a definition, a whole world opens up -- there are more noncomputable functions than computable ones, and we can explore their properties and compare "how noncomputable" they are. Examples of noncomputable objects include the Halting Problem, also seen in computer science, and random sequences. Avoiding as many technical details as possible, we'll define computability and tour the world of the noncomputable.
Prerequisites for the talk: None.
Pizza after the talk.
Tuesday March 17, 2009 5:30pm-6:30pm Room 4-145
Rina Anno
(
University of Chicago)
The geometry of flag varieties and Springer fibers
A flag in a linear space V is a configuration of subspaces
0=V0 ⊂ V1 ⊂ ... ⊂ Vn=V of given dimensions. The flag
variety is simply the space of all flags (for a given V and a given type
of a flag). If we add a linear endomorphism Z of V into the play, we
can define a Springer fiber over Z as the space of all flags with
ZVi ⊂ Vi for all i.
These varieties lie at the crossroads of
algebraic geometry, linear algebra, and representation theory. On one
hand, it means that the Springer fibers may be studied by methods of all
three branches of mathematics; on the other hand, their study has
applications in all three. I will define these spaces and discuss their
properties (including the reasons for the terminology, namely, "fiber" and
"over Z").
Prerequisites for the talk: Linear algebra.
Pizza after the talk.
The space of diagonal harmonics has emerged as one of the key ingredients in a program initiated by Garsia and Haiman to give a representation-theoretical proof of some conjectures in the theory of Macdonald polynomials. The study of this particular space has provided a remarkable display of connections between several areas, including representation theory, symmetric function theory, and combinatorics. Over two decades since the introduction of the diagonal harmonics, the bivariate Hilbert series of the diagonal harmonics has been the object of a variety of algebraic and combinatorial conjectures.
In this lecture, we will define the diagonal harmonics and explore some of the combinatorial objects related to this space. We assume only a basic understanding of undergraduate algebra and a passing appreciation of some beautiful mathematical pictures.
Prerequisites for the talk: Undergraduate algebra.
Pizza after the talk.
Monday March 30, 2009 5:30pm-6:30pm Room 4-153
Tai Melcher
(
University of Virginia)
Brownian motion on curved spaces
Consider a perfume particle released from position x at time 0, and let Btx describe the position of the particle at any later time t>0. The dynamics of this random trajectory are well understood: Btx is Brownian motion. In particular, we know the transition probabilities of the particle, the function pt(x,y) that gives the probability of the particle transitioning to position y in time t. The function pt(x,·) is called the heat kernel of Btx.
The term "heat" comes from the fact that pt is also the fundamental solution to the heat equation in R3. Thus, Brownian motion is intimately related to heat flow in R3, which ultimately relies on the flat geometry of Euclidean space. In curved spaces, the dynamics of heat flow necessarily change depending on the geometry, and these changes can be characterized again in terms of Brownian motion on that space and its associated heat kernel.
Thus, the geometry of a manifold and properties of the diffusion Bt and its heat kernel measure are all intimately related. We will begin with Brownian motion in Euclidean space and use this paradigm as the basis for exploring these relationships in some simple curved examples.
Prerequisites for the talk: None.
Pizza after the talk.
Wednesday April 1, 2009 5:30pm-6:30pm Room 4-153
Ioana Dumitriu
(
University of Washington)
Linear computations: faster, bigger, better. Random?!
Why should one care that the increasing processor-memory gap
signals the end of Moore's Law? How can we build a faster matrix
multiplication algorithm, and if we can, will it be reliable? What is
"fast" linear algebra, and how can random matrices and block algorithms
help us make it faster? These are all very relevant questions in today's
numerical linear algebra, and I will attempt to answer them without
resorting to esoterica.
Prerequisites for the talk: Linear algebra.
Pizza after the talk.
In the past several decades, curvature flows, such as the Ricci flow, have proven to be remarkably useful tools for studying geometric objects. In this talk, I will discuss two equations related to the Ricci flow--the Ricci Yang-Mills flow and the cross curvature flow--and will examine some applications.
Prerequisites for the talk: Some basic knowledge of differential geometry would be useful but not required.
Pizza after the talk.
Tuesday May 5, 2009 5:45pm-6:45pm Room 4-145
Sara Billey
(
University of Washington)
How To Get A Ph.D. In Mathematics In A Timely Fashion
And What To Do From There
This will be an informal discussion about taking the next
step in your mathematical career. I will present some advice I have
collected on how to transition from a class-oriented
test-taking-machine to a full fledged mathematician. In particular,
there are two major transitions along the way: becoming a mathematical
researcher culminating in your Ph.D. and becoming an independent
researcher as a postdoc or assistant professor.
To give the flavor of the discussion, consider what is required by the
math department after you pass qualifying exams in graduate school.
Officially there are only a few hoops you are required to jump through
on your way to a Ph.D such as your thesis defense and perhaps some
language exams. The point is that you are on your own to figure out
what else needs to be done. There is very little required by the math
department. These "hoops" are just formalities. They can be used as
guidelines for progress, but they are not real achievements in and of
themselves. Your ultimate goal is to push the frontier of mathematics
(just a little bit). In order to graduate you now have to do some
original math research. You need to convert all of your class taking
skills into research skills. You will need guidance from one or more
people in making this major transition from math student to math
researcher. This is the purpose of having an advisor. In this talk I
will discuss how to go about finding an advisor. I will strongly
encourage you to make this your number one academic priority until you
find a good match. I will also try to give you some insight about
what research in mathematics really means, how you might go about
doing it, and most importantly how to have fun during the process. I
will also address the next major transition from graduate student to
postdoc/assistant professor. Finally, there will be a number of
people around who are at various stages of their mathematical career
and we will all entertain questions. Any of the more advanced
students/postdocs are welcome to come, offer advice, rebut my
opinions, and contribute funny anecdotes.
Sara Billey was formerly an undergrad and assistant/associate
professor at MIT, she got her Ph.D. from UCSD, now she is an "almost" full professor at the
University of Washington.
Prerequisites for the talk: None.
Pizza after the talk.
Friday May 8, 2009 5:30pm-6:30pm Room 4-145
Bridget Tenner
(
DePaul University)
Mathematics and Voting
Why are there so many voting procedures? How many ways can there be to tally the votes? Do different ways lead to different outcomes? What is the most "fair" option?
In recognition of the recent presidential election and Mathematics
Awareness Month 2008, we will explore some of the issues and surprising
paradoxes of mathematics and voting. This talk will touch on some of the
highlights of this vast subject, and will hopefully inspire the audience
to delve more deeply into the nuances of the field.
Prerequisites for the talk: None.
Pizza after the talk.
