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Basic Notions Seminar - Spring 2016
This seminar at Rutgers University is geared towards first and second year graduate students,
giving them an opportunity to learn more about the research being done in the department. Each week,
we invite a different faculty member to speak about his or her research. This also helps the students
when considering a research area and faculty mentor.
Organizer:
Chloe Urbanski
Faculty Advisers:
Dr. Doron Zeilberger and
Dr. Stephen Miller
Previous Speakers
Monday, February 29: Dr. Alex Kontorovich
Title: Some problems in number theory and ergodic theory.
Abstract: We will present [Title].
Monday, March 7: Dr. Doron Zeilberger
Title: What is Experimental Mathematics?
Abstract: Experimental Mathematics used to be an oxymoron, but the
future of mathematics is headed in that direction.
Monday, March 28: Dr. Jeffry Kahn
Title: Probabilistic Combinatorics
Abstract: There are two aspects to this:
(1) use of probabilistic ideas to attack (non-probabilistic) combinatorial problems
(2) study of the relevant random objects (e.g. random graphs) for their own sake.
We'll say a little about (1)(definitely) and (2)(maybe).
Monday, April 4: Dr. Christopher Woodward
Title: Symplectic Topology
Abstract: A symplectic manifold is a mathematical version of the "phase space"
(space of states) of a classical mechanical system.
Given a function on a symplectic manifold there is a "Hamiltonian flow" describing the evolution of the system.
Symplectic topologists study questions like, given two subsets of the manifold,
is there always a trajectory connecting one subset to another?
Which subsets have this property?
I will talk about some recent conjectures about this question
Monday, April 11: Dr. Fioralba Cakoni
Title: The Mathematical Basis of Inverse Scattering Theory
Abstract: The inverse scattering problem is central to some of the major technological advancements of the twentieth century,
in particular radar, ultrasound imaging and nondestructive testing.
Although the basic mathematical model of inverse scattering problem is deceptively simple,
this problem continues to perplex and challenge mathematicians in diverse areas of mathematics.
In this talk we will present the basic mathematical model of inverse scattering theory,
describe the stunning progress that has been made in recent years and outline a variety of profitable directions for future research.
A central theme of our talk with be the fact that the inverse scattering problem is both nonlinear and improperly posed,
thus highlighting the mathematical difficulties that are inherent to this problem.
Monday, April 25: Dr. Michael Saks
Title: P versus NP, and all that: What every mathematician should know about computational complexity
Abstract: For most mathematicians, computation and algorithms
(computing Taylor series of a given functions,
finding eigenvalues of an operator, finding the greatest common divisor of a list of nubmers) are tools that
are used in the study of objects that arise within their particular field.
The theory of computation turns the focus of mathematical inquiry
onto computation itself:
What is a computational problem, and what is an algorithm? Do all computational problems have algorithms, and if not which don't?
How do we measure efficiency of algorithms? Which problems have efficient
algorithms and which don't?
The central problem of computational complexity is the "P versus NP" problem
which is one of the 7 "million dollar" Millennial challenges posed by the
Clay Foundation. We'll discuss this as well as a number of other challenges
within the field.
Monday, April 18: Dr. Roger Nussbaum
Title: Differential-delay Equations and the question of
Analyticity versus Infinite Differentiability
Abstract: We shall give general background information about "differential-delay equations"
and why one should be interested in them.
This will be illustrated by simple examples like (*) x'(t) = f(x(t-1)),
where f is a given (usually nonlinear) function.
Even the equation (*) presents many difficult and unsolved questions.
We shall then discuss some examples for which there exist bounded,
infinitely differentiable solutions defined on the whole real line.
One might naively expect these solutions to be real analytic everywhere,
but we shall describe results which show such a solution may
sometimes be real analytic on a nonempty open set
and fail to be real analytic on an uncountable set of points.
In more complicated cases, almost nothing is known.