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My dissertation research is on the mathematical analysis of systems of partial differential equations that arise in modeling the equilibrium of mean filed games. I study the existence of traveling wave solutions of such systems and the large time behavior of the associated time dependent problem. Mean field models are used in optimal decision making based on stochastic games with a very large population of agents that are statistically the same. The main advantage of such models is that the overall effect of the other agents on a single one can be replaced by an averaged effect. Hence, the dynamics of a single agent can be determined as a solution of an optimal control problem that depends on the overall distribution of the agents. These models are governed by two partial differential equations, a forward Kolmogorov equation that keeps track of the distribution of agents in the model, and a backward Hamilton Jacobi Bellman (HJB) equation for the value function of the optimal stochastic control problem for each agent. An equilibrium solution to this mean field problem is a solution of the coupled system of the two equations. In my study I focus on an a mean field model for knowledge propagation and economic growth in which interaction between agents is represented by collisions and has diffusion added, in order to get a more realistic model for knowledge propagation. The equilibrium solution of such a mean field model is the solution of a Boltzmann equation for the agent distribution coupled to an HJB equation for the optimal strategy for the control problem. I have studied the existence of a BGP solution of the mean field learning model with diffusion in logarithmic variables. Such a solution can be considered as a generalization of traveling wave solutions of a system of Fisher-KPP (Fisher- Kolmogorov, Petrovsky, Piskunov) type of equation and an HJB equation. In Chapter 2 I prove that a traveling wave solution of the mean field learning model with diffusion exists and I have explored it numerically. The method that is used to construct traveling wave solutions is a variation of a well known method for the construction of traveling wave solutions of Fisher-KPP type equations. The numerical results that I have obtained in Chapter 4 support and extend the theory as many features observed numerically can be proved analytically. I have also studied analytically and numerically the forward-backward system, supplemented by an initial condition for the Fisher-KPP type of equation and a terminal condition for the HJB equation. In chapter 3 I have proved existence of solutions for a fixed terminal time T . In Chapter 4 I have tested the stability of the traveling wave numerically, by using perturbed traveling wave profiles for the initial value and I found that as time increases the solution of the time dependent problem tends to the traveling wave profiles

Kolmogrov-Petrovskii-Piskunov (KPP) equations are a class of non-linear parabolic equations which are used to model various biological, ecological, and physical phenomena. In particular it is used as a model for population dynamics. Originally it was studied by Kolmogrov, Petrovskii and Piskunov in 1937. This thesis investigates one type of integro-differential equation: the non-local KPP equation, used in population dynamics. For the non-local KPP equation, we prove estimates regarding the front location and in particular introduce logarithmic correction. We also do a non-rigorous analysis to show the profile convergence of solution to the travelling wave.

The purpose of this dissertation is to explore loop equations and string dualities in lattice gauge theories. A lattice gauge theory involves a lattice, a compact Lie group, a matrix representation of the group and a parameter which is called an inverse coupling strength. The main objects of interest in the lattice gauge theories are the Wilson loop variables. A loop equation refers to expressing the expectation of a Wilson loop variable in terms of the expectations of Wilson loop sequences obtained from the loop by various loop operations. These loop equations can be used to establish the 1/N expansion of Wilson loop expectations in a strongly coupled regime. The coefficients of this expansion are represented as absolutely convergent sums over trajectories in a string theory on the lattice, establishing one kind of gauge-string duality. Finally, we will present several applications of this expansion such as the Wilson area law upper bound, the factorization property, and the correspondence of SO(N) and SU(N) Wilson loop expectations in the large N limit.

Advection-reaction-diffusion equations (ARD) are non-linear elliptic or parabolic equations that are used to model a variety of systems in natural sciences and engineering, ranging from biological (population dynamics) to chemical and astrophysical systems (reactions in a fluid). They describe the evolution of a normalized quantity, such as a local density of a population or a temperature, as a result of three processes: transport inside the domain (advection), creation or depletion due to an energy source (reaction), and spatial movement (diffusion). These equations exhibit interesting behaviors such as growth/decay, spreading and mixing. Over the past decades there has been significant progress towards understanding and quantifying the behavior of ARD equations. A great focus was shown in equations of biological invasions, more specifically in the phenomenon of spreading, which happens as a result of the invasion of an unstable equilibrium state by a positive stable one. There are two main threads of research in this field: (1) front-like spreading, which leads to the study of traveling fronts, their speeds of propagation and stability properties, and (2) multi-directional spreading, arising from initial data that have compact support or fast decay in every direction. In this thesis we study the problem of spreading rates for the Cauchy problem in multi-dimensional periodic Fisher-KPP equations. Localized initial data give rise to an invasion that will happen typically at different speeds and profiles in each unit direction, but independent on the size and distribution of the original mass. The main result will be on precise asymptotics for the location of level sets of solutions for these data. The trajectory of the thesis will be as follows: In the first part, after a brief introduction of the problem and past results, we study the linearized Dirichlet equation in a half space moving at a constant speed derived from the slowest traveling fronts. This provides intuition for the location of the fronts in the compactly supported case, as well as concrete bounds that can be compared to the solution of the original problem. In the second part we prove the main result by controlling the propagation of the Fisher-KPP solution using sub- and super-solutions constructed from the linearized fronts.