Seminars and Colloquia by Series

Coloring graphs with no K5-subdivision: disjoint paths in graphs

Series
Dissertation Defense
Time
Tuesday, March 12, 2019 - 10:00 for 1.5 hours (actually 80 minutes)
Location
203 Classroom D.M. Smith
Speaker
Qiqin XieGeorgia Institute of Technology
The Four Color Theorem states that every planar graph is 4-colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}-subdivision is k-colorable. However, Catlin disproved Hajos' conjecture for k >= 6. It is not hard to prove that the conjecture is true for k <= 3. Hajos' conjecture remains open for k = 4 and k = 5. We consider a minimal counterexample to Hajos' conjecture for k = 4: a graph G, such that G contains no K_5-subdivision, G is not 4-colorable, and |V (G)| is minimum. We use Hajos graph to denote such counterexample. One important step to understand graphs containing K_5-subdivisions is to solve the following problem: let H represent the tree on six vertices, two of which are adjacent and of degree 3. Let G be a graph and u1, u2, a1, a2, a3, a4 be distinct vertices of G. When does G contain a topological H (i.e. an H-subdivision) in which u1, u2 are of degree 3 and a1, a2, a3, a4 are of degree 1? We characterize graphs with no topological H. This characterization is used by He, Wang, and Yu to show that graph containing no K_5-subdivision is planar or has a 4-cut, establishing conjecture of Kelmans and Seymour. Besides the topological H problem, we also obtained some further structural information of Hajos graphs.

Dynamics of Religious Group Growth and Survival

Series
Dissertation Defense
Time
Friday, June 29, 2018 - 13:00 for 2 hours
Location
Skiles 005
Speaker
Tongzhou ChenSchool of Mathematics
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.

Geometric Bijections of Graphs and Regular Matroids​

Series
Dissertation Defense
Time
Tuesday, June 26, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chi Ho YuenGeorgia Tech
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.

Topics on the longest common subsequences: Simulations, computations, and Variance

Series
Dissertation Defense
Time
Friday, June 22, 2018 - 13:30 for 2 hours
Location
Skiles 005
Speaker
Qingqing LiuGeorgia Tech
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.

Topics in percolation and sequence analysis

Series
Dissertation Defense
Time
Friday, June 22, 2018 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Chen XuGeorgia Tech
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.

The Back-and-Forth Error Compensation and Correction Method for Linear Hyperbolic Systems and a Conservative BFECC Limiter

Series
Dissertation Defense
Time
Friday, June 22, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles Building 114 (Conference Room 114)
Speaker
Xin WangSchool of Mathematics, Georgia Institute of Technology
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.

Combinatorial models for surface and free group symmetries.

Series
Dissertation Defense
Time
Tuesday, June 19, 2018 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shane ScottGeorgia Tech
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.

Local Space and Time Scaling Exponents for Diffusion on a Compact Metric Space (Thesis Defense)

Series
Dissertation Defense
Time
Monday, April 30, 2018 - 15:05 for 2 hours
Location
Skiles 006
Speaker
John DeverGeorgia Tech
We provide a new definition of a local walk dimension beta that depends only on the metric. Moreover, we study the local Hausdorff dimension and prove that any variable Ahlfors regular measure of variable dimension Q is strongly equivalent to the local Hausdorff measure with Q the local Hausdorff dimension, generalizing the constant dimensional case. Additionally, we provide constructions of several variable dimensional spaces, including a new example of a variable dimensional Sierpinski carpet. We use the local exponent beta in time-scale renormalization of discrete time random walks, that are approximate at a given scale in the sense that the expected jump size is the order of the space scale. We consider the condition that the expected time to leave a ball scales like the radius of the ball to the power beta of the center. We then study the Gamma and Mosco convergence of the resulting continuous time approximate walks as the space scale goes to zero. We prove that a non-trivial Dirichlet form with Dirichlet boundary conditions on a ball exists as a Mosco limit of approximate forms. We also prove tightness of the associated continuous time processes.

Two parameters matrix BMO by commutators and sparse domination of operators

Series
Dissertation Defense
Time
Tuesday, March 27, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dario MenaGeorgia Institute of Technology
The first part, consists on a result in the area of commutators. The classic result by Coifman, Rochber and Weiss, stablishes a relation between a BMO function, and the commutator of such a function with the Hilbert transform. The result obtained for this thesis, is in the two parameters setting (with obvious generalizations to more than two parameters) in the case where the BMO function is matrix valued. The second part of the thesis corresponds to domination of operators by using a special class called sparse operators. These operators are positive and highly localized, and therefore, allows for a very efficient way of proving weighted and unweighted estimates. Three main results in this area will be presented: The first one, is a sparse version of the celebrated $T1$ theorem of David and Journé: under some conditions on the action of a Calderón-Zygmund operator $T$ over the indicator function of a cube, we have sparse control.. The second result, is an application of the sparse techniques to dominate a discrete oscillatory version of the Hilbert transform with a quadratic phase, for which the notion of sparse operator has to be extended to functions on the integers. The last resuilt, proves that the Bochner-Riesz multipliers satisfy a range of sparse bounds, we work with the ’single scale’ version of the Bochner-Riesz Conjecture directly, and use the ‘optimal’ unweighted estimates to derive the sparse bounds.

Physical Billiards and Finite Time Predictions for First Passage Probabilities

Series
Dissertation Defense
Time
Thursday, March 15, 2018 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark BoldingGeorgia Tech
I will discuss two topics in Dynamical Systems. A uniformly hyperbolic dynamical system preserving Borel probability measure μ is called fair dice like or FDL if there exists a finite Markov partition ξ of its phase space M such that for any integers m and j(i), 1 ≤ j(i) ≤ q one has μ ( C(ξ, j(0)) ∩ T^(-1) C(ξ, j(1)) ∩ ... ∩ T^(-m+1)C(ξ, j(m-1)) ) = q^(-m) where q is the number of elements in the partition ξ and C(ξ, j) is element number j of ξ. I discuss several results about such systems concerning finite time prediction regarding the first hitting probabilities of the members of ξ. Then I will discuss a natural modification to all billiard models which is called the Physical Billiard. For some classes of billiard, this modification completely changes their dynamics. I will discuss a particular example derived from the Ehrenfests' Wind-Tree model. The Physical Wind-Tree model displays interesting new dynamical behavior that is at least as rich as some of the most well studied examples that have come before.

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