|
Videos of SUMS 2007 are now posted. Mathematics - Princeton University
The Symmetries of Things Mathematics - Brown University
Shapes of Polygons AbstractTwo polygons have the same shape if they are related by a
Euclidean similarity. We will discuss the (surprisingly rich) geometry
of spaces of shapes of polygons, focusing on some explicit examples such
as pentagons and hexagons with fixed angles or side lengths.Mathematics - Harvard University
Canonical forms: A mathematician's view of musical canons AbstractMusical canons, from simple rounds like "Three Blind Mice"
to the compendium of canons Bach compiled in his Musical Offering,
have a history almost as long as that of Western music itself,
and continue to fascinate musical composers, performers and listeners.
In a canon the same melody is played or sung in two or more parts
at once; this melody must therefore make musical sense both as a tune
and in harmony with a delayed or otherwise modified copy of itself.
How does one go about constructing such a melody? This challenge
has a mathematical flavor. It turns out that some kinds of canons
are so easy to create that they can be improvised in real time,
while other kinds are more demanding, and in some cases only
a handful of examples are known. The talk will be illustrated
with both abstract diagrams and specific musical examples,
and may also digress into generalizations of canons (the forms
known collectively as "invertible counterpoint") and the reasons
-- besides showing off -- that so many composers incorporate canons
into their music.
Operations Research/Management Science
- University of Massachusetts
Operations Research and the Captivating Study of Networks and Complex Decision-Making
AbstractIn this talk, I will overview some of the major early and recent contributions
to the formal mathematical study of networks and associated decision-making,
from the perspective of an operations researcher. I will highlight novel
mathematical tools, such as nonlinear optimization, game theory, variational
inequalities, and projected dynamical systems, that have been utilized for the
rigorous formulation of numerous network-based problems, and their effective
and efficient solution. Some of the operation-research applications that I
will discuss are: congested transportation networks and the Internet,
including the Braess paradox (with fixed and time-varying demands), supply
chains, financial and social networks, and energy/environmental networks.
The mathematical network-based discoveries continue to impact numerous
disciplines, including: engineering, computer science, physics, economics, and
biology, where the formalism of networks brings new, refreshing, and unifying
insights. Interestingly, and, not surprisingly, several of the fundamental
discoveries in the network-application domain have been made by Brown faculty
and Brown graduates!
Links: SlidesCombinatorial Optimization - Brown University
Structure and Symmetry in Constraint Programming AbstractConstraint Programming is one of the key techniques to solve real-world
applications. Many problems exhibit a lot of symmetry which complicates
the solution process considerably. Consequently, symmetry breaking was
found to be an important method to speed up the search in constraint
satisfaction problems that contain symmetry. When breaking symmetry
by dominance detection, a computationally efficient symmetry breaking
scheme can be achieved if we can solve the dominance detection problem in
polynomial time. We study the complexity of dominance detection when value
and variable symmetry appear simultaneously in constraint satisfaction
problems (CSPs). Particularly, we devise an efficient dominance detection
algorithm for CSPs with interchangeable variables and values. Our method
yields symmetry-free search trees and is based on the abstraction to the
actual, intuitive structure of a symmetric CSP.Mathematics - University of Connecticut
Fractals and fixed points AbstractA fixed point of a function  is a solution of the
equation  . For example, the function  has one fixed
point, approximately
.73908, which can be found graphically by intersecting the graphs of
 and  ; at the intersection point  we have  . The fixed point of cosine can also be found numerically by
hitting
the cosine button (in radians, please) on your calculator repeatedly
starting from any initial value you wish: ![$x_0,\ \cos x_0,\
\cos(\cos x_0),
\cos(\cos(\cos x_0)),\ldots$](sums/latexrender/pictures/cdea0c73db658fa582ce4e3cfa4e77c0.gif "$x_0,\ \cos x_0,\
\cos(\cos x_0),
\cos(\cos(\cos x_0)),\ldots$") will always tend to the fixed point
.73908... (try
it!). We will indicate why the concept of a fixed point is important in
mathematics, and in particular see how a fractal like the Sierpinski
gasket
is a "fixed point" which can be approximated by iteration starting from
any
initial set in the plane.Mathematics - Brown University
The Birch and Swinnerton-Dyer Conjecture AbstractIn the early 1960s, Bryan Birch and Peter Swinnerton-Dyer conjectured a
relation between the group of rational points on an elliptic curve and a
special value of the Hasse-Weil L-function attached to the curve. I will
present an exposition of their conjecture, some numerical evidence in
support of it, and an overview of progress made in the last 40 years. No
prior knowledge of elliptic curves required.Mathematics - City University of New York, Queens College
The Halting Problem and Computability Theory AbstractA brief introduction to computability theory via "The Halting Problem." Discussion will include the relationship between diagonalization and the noncomputable nature of both the halting problem and the set of functions whose domain is all the natural numbers. This will segue to a brief introduction to the Turing Degrees, via oracles and a partial order "<" on noncomputable sets, where A < B means A is "easier" to compute than B.Mathematics - Harvard University
Knots, Polynomials, and Khovanov Homology AbstractMy goal is to give an introduction to Khovanov's homology theory of knots
and links, as a categorification of the Jones polynomial. This new knot
invariant has connections to many older invariants, including of course
the Jones polynomial, as well as the knot determinant. I will discuss
these connections, make some sample calculations, and describe some
conjectured combinatorial patterns. If there's time, I'll discuss a
newly-proved result relating Khovanov homology to maps from the knot group
into SU2. Note: No prior knowledge of knot theory required.Mathematics - Brown University
Generalized Isoperimetric Problems for Space Polygons AbstractIt is well-known that for a quadrilateral of given edge lengths, the
largest area is achieved when the vertices lie on a circle. What if we
twist the quadrilateral into three-space to form the vertices of a
tetrahedron? Which such tetrahedron will have the largest volume? More
generally, what is the largest hypervolume in n-space of the smallest
convex set determined by the vertices of a k-gon with given
edge lengths
(the convex hull of the polygon)? We solve the problem for an equilateral
pentagon in 4-space and show its relationship with equiangular pentagons
investigated by others. We solve the problem for the equilateral n-gon in
(n-1)-space and the equilateral pentagon in 3-space.Mathematics and Economics - Emmanuel College
What Are the Determinants of Free Agents' Salaries in Major League
Baseball? AbstractThis paper uses ordinary least squares to determine which statistics are
most important in determining free agent's salary in Major League Baseball. Offense, defense, experience, and position
variables are all examined. The primary determinants of a free agent's
salary are his offensive statistics. Teams are more interested in
attracting good offensive players and fans are more interested in watching
offensive ball games.Mathematics - City University of New York, Queens College
Measures of Growth AbstractA thorough discussion of orderings for functions is presented with particular emphasis on two of them. A proof that smooth functions are dense in C(R, R) is given and then used to prove the existence of large smooth functions. Integrals of certain large functions are shown to be less than the initial function, but not significantly less. A generalization of these orderings is given and then used for a more refined discussion of the system of derivatives at a point.Mathematics - Tufts University
Variational Principle for Locating Periodic Orbits of Dynamical Systems AbstractI will introduce a variational principle for finding periodic orbits of
dynamical systems. The method may be applied to both discrete-time and
continuous-time dynamical systems, and may be used to locate either stable
or unstable periodic orbits. I will describe the variational principle,
and present examples of its application, including the determination of
various unstable periodic orbits of the Lorenz attractor.Mathematics - Harvard University
Kummer and Fermat's Last Theorem AbstractI will outline a 19th century partial proof of Fermat's Last Theorem in
the case where the exponents are regular primes. The formulation of these
ideas is due to Kummer. The talk will touch on the theory of cyclotomic
fields and Bernoulli numbers; it may hint at class field theory as well.
Disclaimer: these ideas have little to do with the work of Andrew Wiles,
and in no way suggest a complete proof of Fermat's Last Theorem. |
