Modeling HIV Drug Resistance

Despite the development of antiviral drugs and the optimization of therapies, the emergence of drug resistance remains one of the most challenging issues for successful treatments of HIV-infected patients. The availability of massive HIV drug resistance data provides us not only exciting opportunities for HIV research, but also the curse of high dimensionality. We provide several statistical learning methods in this thesis to analyze sequence data from different perspectives. We propose a hierarchical random graph approach to identify possible covariation among residue-specific mutations. Viral progression pathways were inferred using an EM-like algorithm in literature, and we present a normalization method to improve the accuracy of parameter estimations. To predict the drug resistance from genotypic data, we also build a novel regression model utilizing the information from progression pathways. Finally, we introduce a computational approach to determine viral fitness, for which our initial computational results closely agree with experimental results. Work on two other topics are presented in the Appendices. Latent class models find applications in several areas including social and biological sciences. Finding explicit maximum likelihood estimation has been elusive. We present a positive solution to a conjecture on a special latent class model proposed by Bernd Sturmfels from UC Berkeley. Monomial ideals provide ubiquitous links between combinatorics and commutative algebra. Irreducible decomposition of monomial ideals is a basic computational problem and it finds applications in several areas. We present two algorithms for finding irreducible decomposition of monomial ideals

The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree

Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged. In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Groebner Bases to move towards the intuitive bound for polynomial degree

Algorithmic Contributions to the Theory of Regular Chains

Regular chains, introduced about twenty years ago, have emerged as one of the major tools for solving polynomial systems symbolically. In this thesis, we focus on different algorithmic aspects of the theory of regular chains, from theoretical questions to high- performance implementation issues. The inclusion test for saturated ideals is a fundamental problem in this theory. By studying the primitivity of regular chains, we show that a regular chain generates its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests can be detected very efficiently. The algorithm to compute the regular GCDs of two polynomials modulo a regular chain is one of the key routines in the various triangular decomposition algorithms. By revisiting relations between subresultants and GCDs, we proposed a novel bottom-up algorithm for this task, which improves the previous algorithm in a significant manner and creates opportunities for parallel execution. This thesis also discusses the accelerations towards fast Fourier transform (FFT) over finite fields and FFT based subresultant chain constructions in the context of massively parallel GPU architectures, which speedup our algorithms by several orders of magnitude

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz