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Series: Dissertation Defense

We model and analyze the dynamics of religious group membership and size. A groups
is distinguished by its strictness, which determines how much time group members are
expected to spend contributing to the group. Individuals differ in their rate of return for
time spent outside of their religious group. We construct a utility function that individ-
uals attempt to maximize, then find a Nash Equilibrium for religious group participation
with a heterogeneous population. We then model dynamics of group size by including
birth, death, and switching of individuals between groups. Group switching depends on
the strictness preferences of individuals and their probability of encountering members of
other groups. We show that in the case of only two groups one with finite strictness and
the other with zero there is a clear parameter combination that determines whether the
non-zero strictness group can survive over time, which is more difficult at higher strictness
levels. At the same time, we show that a higher than average birthrate can allow even the
highest strictness groups to survive. We also study the dynamics of several groups, gaining
insight into strategic choices of strictness values and displaying the rich behavior of the
model. We then move to the simultaneous-move two-group game where groups can set up
their strictnesses strategically to optimize the goals of the group. Affiliations are assumed
to have three types and each type of group has its own group utility function. Analysis
on the utility functions and Nash equilibria presents different behaviors of various types
of groups. Finally, we numerically simulated the process of new groups entering the reli-
gious marketplace which can be viewed as a sequence of Stackelberg games. Simulation
results show how the different types of religious groups distinguish themselves with regard
to strictness.

Series: Dissertation Defense

The Jacobian of a graph, also known as the sandpile group or the critical group, is a finite group abelian group associated to the graph; it has been independently discovered and studied by researchers from various areas. By the Matrix-Tree Theorem, the cardinality of the Jacobian is equal to the number of spanning trees of a graph. In this dissertation, we study several topics centered on a new family of bijections, named the geometric bijections, between the Jacobian and the set of spanning trees. An important feature of geometric bijections is that they are closely related to polyhedral geometry and the theory of oriented matroids despite their combinatorial description; in particular, they can be generalized to Jacobians of regular matroids, in which many previous works on Jacobians failed to generalize due to the lack of the notion of vertices.

Series: Dissertation Defense

The study of the longest common subsequences (LCSs) of two random words is a classical problem in computer science and bioinformatics. A problem of particular probabilistic interest is to determine the limiting behavior of the expectation and variance of the length of the LCS as the length of the random words grows without bounds. This dissertation studies the problem using both Monte-Carlo simulation and theoretical analysis. The specific problems studied include estimating the growth order of the variance, LCS based hypothesis testing method for sequences similarity, theoretical upper bounds for the Chv\'atal-Sankoff constant of multiple sequences, and theoretical growth order of the variance when the two random words have asymmetric distributions.

Series: Dissertation Defense

We will present three results in percolation and sequence analysis. In the first part, we will briefly show an exponential concentration inequality for transversal fluctuation of directed last passage site percolation. In the the second part, we will dive into the power lower bounds for all the r-th central moments ($r\ge1$) of the last passage time of directed site perolcation on a thin box. In the last part, we will partially answer a conjecture raised by Bukh and Zhou that the minimal expected length of the longest common subsequences between two i.i.d. random permutations with arbitrary distribution on the symmetric group is obtained when the distribution is uniform and thus lower bounded by $c\sqrt{n}$ by showing that some distribution can be iteratively constructed such that it gives strictly smaller expectation than uniform distribution and a quick cubic root of $n$ lower bound will also be shown.

Series: Dissertation Defense

In this dissertation, we studied the Back and Forth Error Compensation and Correction (BFECC) method for linear hyperbolic PDE systems and nonlinear scalar conservation laws. We extend the BFECC method from scalar hyperbolic PDEs to linear hyperbolic PDE systems, and showed similar stability and accuracy improvement are still valid under modest assumptions on the systems. Motivated by this theoretical result, we propose BFECC schemes for the Maxwell's equations. On uniform orthogonal grids, the BFECC schemes are guaranteed to be second order accurate and have larger CFL numbers than that of the classical Yee scheme. On non-orthogonal and unstructured grids, we propose to use a simple least square local linear approximation scheme as the underlying scheme for the BFECC method. Numerical results showed the proposed schemes are stable and are second order accurate on non-orthogonal grids and for systems with variable coefficients. We also studied a conservative BFECC limiter that reduces spurious oscillations for numerical solutions of nonlinear scalar conservation laws. Numerical examples with the Burgers' equation and KdV equations are studied to demonstrate effectiveness of this limiter.

Series: AMS Club Seminar

Inkscape is an powerful open-source drawing program suitable for making
figures for your math papers and lectures. In this talk I will discuss
some of the useful tricks and features that you can take advantage of in
this
software, as well as some things to avoid. This will be a live demonstration talk, please bring a laptop if you can.

Series: Dissertation Defense

The curve complex of Harvey allows combinatorial representation of a surface mappingclass group by describing its action on simple closed curves. Similar complexes of spheres,free factors, and free splittings allow combinatorial representation of the automorphisms ofa free group. We consider a Birman exact sequence for combinatorial models of mappingclass groups and free group automorphisms. We apply this and other extension techniquesto compute the automorphism groups of several simplicial complexes associated with map-ping class groups and automorphisms of free groups.

Series: Stochastics Seminar

Random balls models are collections of Euclidean balls whose centers and radii are generated by a Poisson point process. Such collections model various contexts ranging from imaging to communication network. When the distributions driving the centers and the radii are heavy-tailed, interesting interference phenomena occurs when the model is properly zoomed-out. The talk aims to illustrate such phenomena and to give an overview of the asymptotic behavior of functionals of interest. The limits obtained include in particular stable fields, (fractional) Gaussian fields and Poissonian bridges. Related questions will also be discussed.

Series: School of Mathematics Colloquium

Given two complex polynomials, we can try to mathematically paste them
together to obtain a rational function through a procedure known as
mating the polynomials. In this talk, we will begin by trying to
understand the "shape" of complex polynomials in general. We will then
discuss the mating of two quadratic polynomials: we explore examples
where the mating does exist, and examples where it does not. There will
be lots of movies and exploration in this talk.

Wednesday, May 30, 2018 - 14:00 ,
Location: Skiles 006 ,
Tongzhou Chen ,
GT Math ,
Organizer: Jiaqi Yang

We model and analyze the dynamics of religious group membership
and size. A groups is distinguished by its strictness, which determines
how much time group members are expected to spend contributing to the
group. Individuals differ in their rate of return for time spent outside
of their religious group. We construct a utility function that individ-
uals attempt to maximize, then find a Nash Equilibrium for religious
group participation with a heterogeneous population. We then model
dynamics of group size by including birth, death, and switching of
individuals between groups. Group switching depends on the strictness
preferences of individuals and their probability of encountering members
of other groups.