## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, August 21, 2018 - 15:00 , Location: Skiles 006 , Professor Veli Shakhmurov , Okan University , , Organizer: Ronghua Pan
The boundary value and mixed value problems for linear and nonlinear degenerate abstract elliptic and parabolic equations are studied. Linear problems involve some parameters. The uniform L_{p}-separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems.     In this talk, the classes of boundary value problems (BVPs) and mixed value problems (MVPs) for regular and singular degenerate differential operator equations (DOEs) are considered. The main objective of the present talk is to discuss the maximal regularity properties of the BVP for the degenerate abstract elliptic and parabolic equation     We prove that for f∈L_{p} the elliptic problem has a unique solution u∈ W_{p,α}² satisfying the uniform coercive estimate ∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)} where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be defined later. We also prove that the differential operator generated by this elliptic problem is R-positive and also is a generator of an analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for the corresponding degenerate parabolic problem. This fact is used to obtain the existence and uniqueness of  maximal regular solution of the MVP for the nonlinear parabolic equation.
Monday, August 20, 2018 - 15:05 , Location: Skiles 005 , Esther Ezra , Georgia Tech , Organizer: Prasad Tetali
A recent extension by Guth (2015) of the basic polynomial partitioning technique of Guth and Katz (2015) shows the existence of a partitioning polynomial for a given set of k-dimensional varieties in R^d, such that its zero set subdivides space into open cells, each meeting only a small fraction of the given varieties.  For k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently.  This, in particular, applies to the setting of n algebraic curves, or, in fact, just lines, in 3-space.  In this work we present an efficient algorithmic construction for this setting almost matching the bounds of Guth (2015); For any D > 0, we efficiently construct a decomposition of space into O(D^3\log^3{D}) open cells, each of which meets at most O(n/D^2) curves from the input.  The construction time is O(n^2), where the constant of proportionality depends on the maximum degree of the polynomials defining the input curves.  For the case of lines in 3-space we present an improved implementation using a range search machinery. As a main application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently been studied by Aronov et al.  Joint work with Boris Aronov and Josh Zahl.
Series: Other Talks
Tuesday, August 14, 2018 - 14:00 , Location: Skiles 154 , , Georgia Tech , , Organizer: Justin Lanier
Take a branched covering map of the sphere over itself so that the forward orbit of each critical point is finite. Such maps are called Thurston maps. Examples include polynomials with well-chosen coefficients acting on the complex plane, as well as twists of these by mapping classes. Two basic problems are classifying Thurston maps up to equivalence and finding the equivalence class of a Thurston map that has been twisted. We will discuss ongoing joint work with Belk, Margalit, and Winarski that provides a new, combinatorial approach to the twisted polynomial problem. We will also propose several new research directions regarding Thurston maps. This is an oral comprehensive exam. All are welcome to attend.
Friday, July 20, 2018 - 13:00 , Location: Skiles 006 , Kashyap Rajeevsarathy , IISER Bhopal , Organizer: Dan Margalit
Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2. Given a finite subgroup H < Mod(Sg), let Fix(H) denote the set of fixed points induced by the action of H on the Teichmuller space Teich(Sg). In this talk, we give an explicit description of Fix(H), when H is cyclic, thereby providing a complete solution to the Modular Nielsen Realization Problem for this case. Among other applications of these realizations, we derive an intriguing correlation between finite order maps and the filling systems of surfaces. Finally, we will briefly discuss some examples of realizations of two-generator finite abelian actions.
Monday, July 2, 2018 - 01:55 , Location: Skiles 005 , Isabelle Kemajou-Brown , Morgan State University , , Organizer: Luca Dieci
We assume the stock is modeled by a Markov regime-switching diffusion process and that, the benchmark depends on the economic factor. Then, we solve a risk-sensitive benchmarked asset management problem of a firm. Our method consists of finding the portfolio strategy that minimizes the risk sensitivity of an investor in such environment, using the general maximum principle.After the above presentation, the speaker will discuss some of her ongoing research.
Friday, June 29, 2018 - 13:00 , Location: Skiles 005 , Tongzhou Chen , School of Mathematics , , Organizer: Tongzhou Chen
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.
Tuesday, June 26, 2018 - 14:00 , Location: Skiles 005 , Chi Ho Yuen , Georgia Tech , Organizer: Chi Ho Yuen
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.
Friday, June 22, 2018 - 13:30 , Location: Skiles 005 , Qingqing Liu , Georgia Tech , Organizer: Qingqing Liu
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.
Friday, June 22, 2018 - 11:00 , Location: Skiles 005 , Chen Xu , Georgia Tech , , Organizer: Chen Xu
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.
Friday, June 22, 2018 - 11:00 , Location: Skiles Building 114 (Conference Room 114) , Xin Wang , School of Mathematics, Georgia Institute of Technology , , Organizer: Xin Wang
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.