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Series: SIAM Student Seminar

Fix k vertices in a graph G, say a_1,...,a_k, if there exists
a cycle that visits these vertices with this specified order,
we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown
by Thomas and Wollan that any 10k-connected graph is k-linked,
therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered
for any a_1,...,a_k. However, it is possible that we can improve
this bound when k is small. It is shown by W. Goddard that any
4-connected maximal planar graph has an (a_1,...,a_4)-ordered
cycle for any choice of 4 vertices. We will present a complete
characterization of 4-ordered cycle in planar graphs. Namely,
for any four vertices a,b,c,d in planar graph G, if there is no
(a,b,c,d)-ordered cycle in G, then one of the follows holds:
(1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3;
(2) roughly speaking, a,b,d,c "stay" in a face of G with this order.

Series: SIAM Student Seminar

Hosted also by Ben Webb

We will try to define what Khovanov homology for a link in a S^3 is. We
will then try to give a proof figuring out unknotting number of certain kinds of
knots in S^3.

Series: SIAM Student Seminar

The discussion will focus on some recent advances in improving
performance of rendering 3D scenes. First, a Monte Carlo method based upon
the Metropolis algorithm is described. Then a method of using spherical
harmonics to generate vectors and matrices which allow efficient
high-quality rendering in real time will be described. Finally, a discussion
will be made of possible future areas for improving the efficiency of such
algorithms.

Series: SIAM Student Seminar

Steinberg's Conjecture states that any planar graph without cycles of
length four or five is three colorable. Borodin, Glebov, Montassier,
and Raspaud showed that planar graphs without cycles of length four,
five, or seven are three colorable and Borodin and Glebov showed that
planar graphs without five cycles or triangles at distance at most two
apart are three colorable. We prove a statement similar to both of
these results: that any planar graph with no cycles of length four
through six or cycles of length seven with incident triangles distance
exactly two apart are three colorable. Special thanks to Robin Thomas
for substantial contributions in the development of the proof.

Series: SIAM Student Seminar

We will study a simple dynamical system with two driving vector
fields on the unit interval. The driving vector fields point to opposite
directions, and we will follow the trajectory induced by one vector field
for a random, exponentially distributed, amount of time before switching to
the regime of the other one. Thanks to the simplicity of the system, we
obtain an explicit formula for its invariant density. Basically exploiting
analytic properties of this density, we derive versions of the law of large
numbers, the central limit theorem and the large deviations principle for
our system. If time permits, we will also discuss some ideas on how to prove
existence of invariant densities, both in our one-dimensional setting and
for more general systems with random switching. The talk will rely to a
large extent on my Master's thesis I wrote last year under the guidance and
supervision of Yuri Bakhtin.

Series: SIAM Student Seminar

In this talk, I am going to give a elementary introduction of invariant manifold
theory in dynamical systems. I will start with the motivation and definition of invariant
manifolds. Then I will discuss how to construct various invariant manifolds of maps and flows.
Finally, I will discuss some applications. If time is permitted, I will also discuss a little
about invariant foliation.

Series: SIAM Student Seminar

The talk will begin with an elementary geometric discussion
of Riemann-Roch theory for sub-lattices of the integer lattice
orthogonal to some positive vector. A pair of necessary and
sufficient conditions for such a lattice to have the Riemann-Roch
property will be presented. By studying a certain chip firing game on
a directed graph related to the lattice spanned by the rows of its
Laplacian I will describe a combinatorial method for checking whether
a directed graph has the Riemann-Roch property. The talk will
conclude with a presentation of arithmetical graphs, which after the
application of a simple transformation, may be viewed as a special
class of directed graphs. Examples from this class demonstrate that
either, both or neither of the Riemann-Roch conditions may be
satisfied for a directed graph. This is joint work with Arash Asadi.

Series: SIAM Student Seminar

Almost axisymmetric flows are derived from Boussinesq equations
for incompressible fluids. They are supposed to capture special features in
tropical cyclones. We establish an unusual minimax equality as the first
step towards studying this challenging problem. I will review some basic
techniques of the calculus of variations.

Series: SIAM Student Seminar

In this talk we consider the collective dynamics of a network of
interacting dynamical systems and show that under certain conditions such dynamical
networks have a unique global attractor. This involves a combination of techniques
from dynamical systems theory as well as newly devised methods in graph theory.
However, this talk is intended to be an introduction to both areas of mathematics
with a focus on how the combination of the two is yielding new results in graph and
dynamical systems theory.

Series: SIAM Student Seminar

A constraint satisfaction problem (CSP) is an ensemble of boolean
clauses, where satisfaction is obtained by an assignment of the variables if
every clause is satisfied by such assignment. We will see that when such CSP
is arranged following certain random structure, the Fourier expansion of the
corresponding clauses allows us to understand certain properties of the
solution space, in particular getting a partial understanding of when the
'usual suspects' of the drastical failure of all known satisfiability
algorithms, namely long range correlations and clustering, appear.
Based in joint work with Prasad Tetali and Andrea Montanari.