Mathematics
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Item Modeling Imatinib-Treated Chronic Myelogenous Leukemia and the Immune System(2019) Peters, Cara Disa; Levy, Doron; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Chronic myelogenous leukemia can be considered as a chronic condition thanks to the development of tyrosine kinase inhibitors in the early 2000s. Most CML patients are able to manage the disease, but unending treatment can affect quality of life. The focus of much clinical research has thus transitioned to treatment cessation, where many clinical trials have demonstrated that treatment free remission is possible. While there are a lot of existing questions surrounding the criteria for cessation candidates, much evidence indicates the immune system plays a significant role. Mathematical modeling provides a complementary component to clinical research. Existing models well-describe the dynamics of CML in the first phase of treatment where most patients experience a biphasic decline in the BCR-ABL ratio. The Clapp model is one of the first to incorporate the immune system and capture the often-seen oscillations in the BCR-ABL ratio that occur later in therapy. However, these models are far from capable of being used in a predictive manner and do not fully capture the dynamics surrounding treatment cessation. Based on clinical research demonstrating the importance of immune response, we hypothesize that a mathematical model of CML should include a more detailed description of the immune system. We therefore present a new model that is an extension of the Clapp model. The model is then fit to patient data and determined to be a good qualitative description of CML dynamics. With this model it can be shown that treatment free remission is possible. However, the model introduces new parameters that must be correctly identified in order for it to have predictive power. We next consider the parameter identification problem. Since the dynamics of CML can be considered in two phases, the biphasic decline of and oscillations in the BCR-ABL ratio, we hypothesize that parameter values may differ over the course of treatment and look to identify which parameters are most variable by refitting the model to different windows of data. It is determined that parameters associated with immune response and regulation are most difficult to identify and could be key to selecting good treatment cessation candidates. To increase the predictive power of our model, we consider data assimilation techniques which are successfully used in weather forecasting. The extended Kalman filter is used to assimilate CML patient data. Although we determine that the EKF is not the ideal technique for our model, it is shown that data assimilation methods in general hold promising value to the search for a predictive model of CML. In order to have the most success, new techniques should be considered, data should be collected more frequently, and immune assay data should be made available.Item Moduli Spaces of Sheaves on Hirzebruch Orbifolds(2019) Wang, Weikun; Gholampour, Amin; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We provide a stacky fan description of the total space of certain split vector bundles, as well as their projectivization, over toric Deligne-Mumford stacks. We then specialize to the case of Hirzebruch orbifold $\mathcal{H}_{r}^{ab}$ obtained by projectivizing $\mathcal{O} \oplus \mathcal{O}(r)$ over the weighted projective line $\mathbb{P}(a,b)$. Next, we give a combinatorial description of toric sheaves on $\mathcal{H}_{r}^{ab}$ and investigate their basic properties. With fixed choice of polarization and a generating sheaf, we describe the fixed point locus of the moduli scheme of $\mu$-stable torsion free sheaves of rank $1$ and $2$ on $\mathcal{H}_{r}^{ab}$. Finally, we show that if $\mathcal{X}$ is the total space of the canonical bundle over a Hirzebruch orbifold, then we can obtain generating functions of Donaldson-Thomas invariants.Item Locally symmetric spaces and the cohomology of the Weil representation(2019) Shi, Yousheng; Millson, John; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\R) $ and $\mathrm{O}^*(2n) $. These cycles are covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson (\cite{Anderson}), we show that Poincar\'e duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G $ to vector-valued automorphic functions associated to the groups $G'=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{Sp}(m,m)$ which are members of a dual pair with $G$ in the sense of Howe. The above three groups are all the groups that show up in real reductive dual pairs of type I whose symmetric spaces are of Hermitian type with the exception of $\mathrm{O}(p,2)$.Item Mathematical Models of Underlying Dynamics in Acute and Chronic Immunology(2019) Wyatt, Asia Alexandria; Levy, Doron; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)During an immune response, it is understood that there are key differences between the cells and cytokines that are present in a primary response versus those present in subsequent responses. Specifically, after a primary response, memory cells are present and drive the clearance of antigen in these later immune responses. When comparing acute infections to chronic infections, there are also differences in the dynamics of the immune system. In this dissertation, we develop three mathematical models to explore these differences in the immune response to acute and chronic infections through the creation, activation, regulation, and long term maintenance of T cells. We mimic this biological behavior through the use of delayed differential equation (DDE) models. The first model explores the dynamics of adaptive immunity in primary and secondary responses to acute infections. It is shown that while we observe similar amounts of antigen stimulation from both immune responses, with the incorporation of memory T cells, we see an increase in both the amount of effector T cells present and the speed of activation of the immune system in the secondary response. We conclude that our model is robust and can be applied to study different types of antigen from viral to bacterial. Extending our work to chronic infections, we develop our second and third models to explore breast cancer dormancy and T cell exhaustion. For our breast cancer dormancy model, we find that our model behaves similar to acute infections, but with constant antigen stimulation. Moreover, we observe the importance of immune protection on the long term survival of breast cancer cells. In our third model we find that while memory T cells play a major role in the effectiveness of the immune system in acute infection, in chronic infections, over long periods of time, T cell exhaustion prevents proper immune function and clearance of antigen. We also observe how the lack of long term maintenance of memory T cells plays an important role in the final outcome of the system. Finally, we propose two potential extensions to the three models developed: creating a simplified acute infection model and creating a combined breast cancer dormancy model with T cell exhaustion.Item ANALYTIC APPROACHES IN LAGRANGIAN GEOMETRY(2019) Dellatorre, Matthew; Rubinstein, Yanir A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The focus of this thesis is two equations that arise in special Lagrangian geometry: the degenerate special Lagrangian equation (DSL) and the Lagrangian mean curvature flow (LMCF). A significant part of this focus centers on Dirichlet duality, subequations, and viscosity solutions, the analytic framework which we use to formulate and study both equations. Given a Calabi--Yau manifold $(X, \omega, J, \Omega)$ and a model manifold $M$, one can construct a kind of moduli space of Lagrangians in $X$ called the space of positive Lagrangians. A Lagrangian $L\subset X$ belongs to this infinite-dimensional space if $L$ is diffeomorphic to $M$ and Re$(\Omega|_L) >0$. % i.e., the set of Lagrangians that are Hamiltonian deformations of a fixed Lagrangian, A Hamiltonian deformation class of the space of positive Lagrangians admits an $L^2$-type Riemannian metric which allows one to study this space from a geometric point of view. Geodesics in this space play a crucial role in a program initiated by Solomon \cite{S1, S2} to understand the existence and uniqueness of special Lagrangian submanifolds in Calabi–Yau manifolds. They also play a key role in a new approach to the Arnold conjecture put forth by Rubinstein–Solomon and in the development of a pluripotential theory for Lagrangian graphs \cite{RS, DR}. The DSL arises as the geodesic equation in the space of positive graph Lagrangians when $X= \mathbb{C}^n$ and $\omega$ and $\Omega$ are associated to the Euclidean structure \cite{RS}. Building on the results of Rubinstein--Solomon \cite{RS}, we show that the DSL induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth semi-flat Calabi--Yau torus fibration are governed by the Riemannian DSL on the product of the base manifold and an interval. The geodesic endpoint problem in this setting thus corresponds to solving the Dirichlet problem for the DSL. However, the DSL is a degenerate-elliptic, fully non-linear, second-order equation, and so the standard elliptic theory does not furnish solutions. Moreover, for Lagrangians with boundary the natural domains on which one would like to solve the Dirichlet problem are cylindrical and thus not smooth. These issues are resolved by Rubinstein--Solomon in the Euclidean setting by adapting the Dirichlet duality framework of Harvey--Lawson to domains with corners \cite{RS}. We further develop these analytic techniques, specifically modifications of the Dirichlet duality theory in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence, in certain settings, continuous geodesics in the space of positive Lagrangians. The uniqueness of solutions to the Dirichlet problem in the Euclidean formulation of Dirichlet duality theory relies on an important convex-analytic theorem of Slodkowski \cite{Slod}. Motivated by the significance of this result and the technical, geometric nature of its proof, we provide a detailed exposition of the proof. We then study some of the quantities involved using the Legendre transform, offering a dual perspective on this theorem. Given a Lagrangian submanifold in a Calabi--Yau, a fundamental and still open question is whether or not there is a special Lagrangian representative in its homology or Hamiltonian isotopy class. A natural approach to this problem is the Lagrangian mean curvature flow, which preserves not only the Lagrangian condition but also the homology and isotopy class. Assuming the flow exists for all time and converges, it will converge to a minimal (i.e., zero mean curvature) Lagrangian. In the Calabi--Yau setting these are precisely the special Lagrangian submanifolds. A major conjecture in this area is the Thomas--Yau conjecture \cite{TY}, which posits certain stability conditions on the initial Lagrangian under which the LMCF will exist for all time and converge to the unique special Lagrangian in that isotopy class. Thomas--Yau stated a variant of their conjecture for a related, more tractable flow, called the almost Lagrangian mean curvature flow (ALMCF). In the setting of highly symmetric Lagrangian spheres in Milnor fibers, and under some additional technical assumptions, they make significant progress towards a proof of this variant of the conjecture \cite{TY}. We study the flow of $2$-spheres from a slightly different perspective and provide a relatively short proof of the longtime existence of viscosity solutions under certain stability conditions, and their convergence to a special Lagrangian sphere.Item Branching diffusion processes in periodic media(2019) Hebbar, Pratima; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In the first part of this manuscript, we investigate the asymptotic behavior of solutions to parabolic partial differential equations (PDEs) in $\real^d$ with space-periodic diffusion matrix, drift, and potential. The asymptotics is obtained up to linear in time distances from the support of the initial function. Using this asymptotics, we describe the behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the $k-$th moment dominates the $k-$th power of the first moment for some $k$), while, at distances that grow sub-linearly in time, we show that all the moments converge. In the second part of the manuscript, we obtain asymptotic expansions for the distribution functions of continuous time stochastic processes with weakly dependent increments in the domain of large deviations. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying H\"ormander condition on a $d$--dimensional compact manifold admit asymptotic expansions of all orders in the domain of large deviations.Item Universal Deformations and p-adic L-functions(2019) Sehanobish, Arijit; Washington, Lawrence C; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis we study deformations of certain $2$-dimensional reducible representations whose image is in the Borel subgroup of $GL_2(\F)$. Our method of understanding the universal deformation ring is via the Jordan-H\"older factors of the residual representation. Using the vanishing of cup products of appropriate cohomology classes we can compute the tangent space of the universal deformation ring and some obstruction classes to lifting representations. In this process, we can also explicitly construct certain big meta-abelian extensions inside the fixed field of the kernel of the universal representation. We give an explicit example of our construction of an unramified extension in the case of elliptic curves of conductor $11$. We also give an Iwasawa theoretic description of various fields that are cut out by the universal representation. The Galois theoretic description of the constructed meta-abelian unramified extension is then later used as an ingredient for the isomorphism criterion in the modularity lifting results. When the isomorphism criterion is satisfied, we could prove some modularity lifting results allowing us to recover some results of Skinner-Wiles and prove a conjecture of Wake in this special case. We also show that the representations considered by Skinner-Wiles have big image inside the universal deformation ring.Item POSITIVE TUPLES OF FLAGS, PIECEWISE CIRCULAR WAVEFRONTS, AND THE 3-DIMENSIONAL EINSTEIN UNIVERSE(2019) Kirk, Ryan Timothy; Zickert, Christian K; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Fock and Goncharov defined the notion of positive subsets of a complete flag manifold G/B in order to study higher Teichmüller spaces. In this dissertation, we study the finite positive subsets when G = PSp(4,R) ∼= SO0(3,2). The main tool is the fact that the 3-dimensional Einstein universe, or Lie quadric, is one of the parabolic homogeneous spaces of G and it parametrizes oriented circles in the 2-sphere. We interpret complete flags in this setting as pointed oriented circles in the 2-sphere and the action of G as contactomorphisms of the unit tangent bundle of S2. This leads to an interpretation of positive subsets in G/B in terms of oriented piecewise circular curves in the 2-sphere, or equivalently piecewise linear Legendrian curves in RP3. We parametrize positive triples of flags by a pair of real-valued cross ratios. We explicitly describe a homeomorphism between the configurations space of positive triples of flags and the moduli space of 6-sided, labeled, positive, oriented piecewise circular wavefronts in S2.Item Special Unipotent Arthur Packets for Real Reductive Groups(2019) Fernandes, Jonathan Francis; Adams, Jeffrey D; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Let $\GR$ be a real reductive group. In this thesis we study the unitary representations of $\GR$. In particular, we study the special Arthur unipotent parameters and the associated packets of irreducible representations of $\GR$. It is conjectured that these unipotent representations form the building blocks for all unitary representations of $\GR$. \\ To understand unipotent representations, we will need to compute the following invariants of irreducible representations of $\GR$: complex associated variety and the theta associated variety. Even though these invariants are theoretically understood, there are no known (at least to this author) results/algorithms to compute them explicitly. \\ The primary results of this thesis provide algorithms to compute these invariants explicitly in many cases. We then use these invariants to compute information about unipotent Arthur packets, and in favorable cases, their entire contents explicitly. In unfavorable cases, we show how to extract more information from our results by using the stable sum formula. \\ We have implemented these algorithms into the Atlas of Lie Groups software, available at www.liegroups.org. We also provide some tables of data compiled using the output from Atlas.Item Center of pro-p-Iwahori-Hecke algebra(2019) Gao, Yijie; He, Xuhua; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Let G be a connected reductive group over a p-adic eld F. The study of representations of G(F) naturally involves the pro-p-Iwahori-Heche algebra of G(F). The pro-p-Iwahori-Hecke algebra is a deformation of the group algebra of the pro-p-Iwahori Weyl group of G(F) with generic parameters. The pro- p-Iwahori-Hecke algebra with zero parameters plays an important role in the study of mod-p representations of G(F). In a series of paper, Vigneras introduced a generic algebra HR(q~s; c~s) which generalizes the pro-p-Iwahori-Hecke algebra of a reductive p-adic group. Vign- eras also gave a basis of the center of HR(q~s; c~s) when HR(q~s; c~s) is associated with a pro-p-Iwahori Weyl group. This basis is dened by using the Bernstein presentation of HR(q~s; c~s) and the alcove walk. In this article, we restrict to the case where q~s = 0 and give an explicit description of the center of HR(0; c~s) using the Iwahori-Matsumoto presentation. First, we introduce the generic algebra. Let W be the semidirect product of a Coxeter group and a group acting on the Coxeter group and stabilizing the generating set of the Coxeter group. Let W(1) be an extension of W with a commutative group. Let R be a commutative ring. We give the denition of the R-algebra HR(q~s; c~s) of W(1) with parameters (q~s; c~s). Then for any pair (v;w) in W W with v w, we dene a linear operator rv;w between R-submodules of HR(q~s; c~s). It takes some work to show that rv;w is well dened. Next, we restrict W to be an IwahoriWeyl group. We show that the maximal length terms of a central element in HR(q~s; c~s) is given by a union of nite conjugacy classes in W(1). Then we prove some techical results regarding rv;w acting on the maximal length terms of a central element in HR(q~s; c~s). In the last part, we restrict to the case when q~s = 0 and give a explicit basis of the center of HR(0; c~s) in the Iwahori-Matsumoto presentation by using the operator rv;w. Two examples are given to help understand how this basis looks like.