Title Promoter Affiliations Abstract "Applied and Computational Algebraic Geometry" "Stefan Vandewalle" "Numerical Analysis and Applied Mathematics (NUMA)" "Proposed outline of the dissertation: This proposal is based on applied and computational algebraic geometry. Algebraic geometry is the study of varieties which are defined as the zero sets of systems of polynomial equations. We will focus on the polynomial systems arising from specific problems in theoretical computer science. In particular, we will develop theory and algorithms for our applications based on the following branches of algebraic geometry:[T1] Numerical Algebraic Geometry: Numerical Algebraic Geometry [SW05], as the name suggests, is the amalgamation of numerical techniques and algebraic geometry. The main objective of this theory is to develop new tools for numerically solving the system of polynomial equations. The two main techniques used in the area are homotopy continuation and eigenvalue methods. The former uses suitable homotopies to deform the given system of polynomial equations to a system that can be easily solved. Then tracking all the solutions of the deformed system to the initial system along with the chosen homotopy. This method helps find all the solutions of the system contrary to the other numerical solvers. The eigenvalue method converts the problem of solving the system of polynomial equations to solving the eigenvalue problem for corresponding Macaulay matrices [Te20]. In order to compute the Macaulay matrices, this method relies on symbolic computations. To solve the eigenvalue problem, this method relies on algorithms from computational linear algebra. Both of the methods provide numerical solutions to the system of polynomial equations. In order to use these solutions in applications, we need mathematical certificates for them, for this, we use Smale's theory [HS10] and Krawczyk's method. In recent works [Br20], [Te20], novel techniques from polyhedral geometry and toric geometry have been used to develop numerical algorithms for solving sparse polynomial systems. [T2] Real Algebraic Geometry: Real algebraic geometry [BCR13] is the branch of algebraic geometry that deals with finding real solutions to the polynomial equations. The main concepts of this area include quantifier elimination, decomposition theorems for semi-algebraic sets, the existence theory of reals, Positivstellensatz etc. In the applications, we are particularly interested in finding the real solutions for the system of polynomial equations. The classical theorems from this area (for example Putinar's Positivstellensatz) are already in use. But the full strength of this theory has not been fully utilized in the applications we are interested in. In particular, the existence theory of reals and decomposition theorems can be very useful for developing sufficient conditions for the existence of real solutions for the polynomial systems under consideration. There also have been recent advances in finding real numerical solutions to the polynomial equations using techniques from tropical geometry and polyhedral geometry. Moreover, for certain sparse polynomial systems, some new algorithms for counting real solutions have also been proposed very recently which can be helpful for the complexity analysis of our problems. [T3] Symbolic Computations: Symbolic computations deal with solving systems of polynomial equations using techniques from the theory of Gröbner basis and elimination theory [CLO13]. These methods differ from numerical algebraic geometry significantly as here the theory depends on algebraic manipulations of equations rather than relying on numerical operations. But both the methods complement each other as for certain numerical techniques like eigenvalue methods [T1], one of the crucial steps involves computation of Gröbner basis. Although the complexity of computation of Gröbner basis for a general polynomial system is doubly exponential in the size of the problem, this method is still useful for our application. As the systems that we are interested in have nice combinatorial structures (sparsity, low treewidth etc), the Gröbner basis computations for such systems are tractable. In particular, the main idea is to prove that our systems have a certain form of Gröbner basis, which in turn can help to reduce the time complexity for computing the Macaulay matrices required for the eigenvalue methods. For sparse polynomial systems, there have been recent developments for the efficient computations of Gröbner basis using properties of the graphs associated with the polynomial systems in considerations [Pa14]. There are also new results in the complexity analysis for the computation of Gröbner basis for sparse systems [Be19]. These new results can be of immense use for our proposed problems.We will use the above-mentioned branches of algebraic geometry to find the real roots of the system of polynomial equations arising in theoretical computer science. In particular in program verification, there have been recent advances and new connections from algebraic geometry [Go20]. We will focus on the three main problems from program verification: [P1] Reachability analysis, [P2] Invariant generation and [P3] Termination analysis. References:[BCR13] Bochnak J, Coste M, Roy MF. Real algebraic geometry. Springer Science & Business Media; 2013. [Be19] Bender, M. ""Algorithms for sparse polynomial systems: Gröbner bases and resultants."" PhD diss., Sorbonne université, 2019. [Br20] Brysiewicz T. Newton polytopes and numerical algebraic geometry. PhD diss., Texas A&M University, 2020[BRT20] Breiding P, Rose K, Timme S. Certifying zeros of polynomial systems using interval arithmetic. arXiv preprint arXiv:2011.05000. 2020 Nov 10.[CLO13] Cox D, Little J, OShea D. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media; 2013. [Go20] Goharshady A. Parameterized and algebro-geometric advances in static program analysis PhD diss., Institute of Science and Technology Austria, 2020.[HS10] Hauenstein JD, Sottile F. alphaCertified: certifying solutions to polynomial systems. arXiv preprint arXiv:1011.1091. 2010. [Pa14] Pardo, D. ""Exploiting chordal structure in systems of polynomial equations."" PhD diss. Massachusetts Institute of Technology, 2014. [SW05] Sommese AJ, Wampler CW. The Numerical solution of systems of polynomials arising in engineering and science. World Scientific; 2005. [Te20] Simon Telen, Solving systems of polynomial equations, PhD diss., KU Leuven, 2020.[VC93] Verschelde J, Cools R. Symbolic homotopy construction. Applicable Algebra in Engineering, Communication and Computing. 1993 Sep;4(3):169-83.[VC94] Verschelde J, Cools R. Symmetric homotopy construction. Journal of Computational and Applied Mathematics. 1994 May 20;50(1-3):575-92.[VVC94] Verschelde J, Verlinden P, Cools R. Homotopies exploiting Newton polytopes for solving sparse polynomial systems. SIAM Journal on Numerical Analysis. 1994 Jun;31(3):915-30." "Derived categories and Hochschild cohomology in (noncommutative) algebraic geometry." "Wendy Lowen" "Hasselt University, Fundamental Mathematics" "Algebraic geometry is an old subject, going back to the ancient Greeks who studied the geometry of ellipses, parabolas and hyperbolas using the language of conic sections. In the 16th century, Descartes rephrased everything in terms of coordinates. The conic sections of the Greeks became solutions to quadratic polynomial equations. Finally, in the 1960s the current framework of algebraic geometry was introduced by Grothendieck, with the advent of scheme theory. An important question in the context of conic sections is their classification: how many different types are there, and how can one be related, or ""deformed"", into another? It is possible to consider this problem in the three settings introduced above, giving equivalent answers. But the high level of abstraction in the last setting allows one to really explain what is specific to the situation of conic sections, and what is true more generally. My research proposal concerns these classification and deformation problems in (non-commutative) algebraic geometry: Hochschild cohomology describes previously unknown ways of deforming objects in algebraic geometry, making them non-commutative. My goal is to study and obtain exciting and unexpected connections: Can we understand symmetries of deformations? Can we translate noncommutativity back to commutativity? How can we relate geometric objects in new ways? How are deformations similar or different?" "Singularities in algebraic geometry" "Nero Budur" Algebra "This project is in the field of algebraic geometry and is about singularities on geometrical shapes given by algebraic equations, also called algebraic varieties. We will study the effect of the presence of singularities on the geometry, the algebra, and the topology of algebraic varieties. On the geometric side, we will study contact loci of arcs associated with singularities. We aim to provide connections between contact loci and the minimal model program, to find an answer to the embedded Nash problem, and to address conjectures relating them with symplectic geometry via Floer cohomology. On the algebraic side, we will study open questions on D-modules and Bernstein-Sato ideals. On the topological side, we will study local systems and address the question of which local systems can be constructed from geometry. The algebraic and topological aspects to be treated in the project are related by generalized versions of the classical Riemann-Hilbert correspondence. Their connection with the geometric aspect to be treated in the project is the subject of the Monodromy Conjecture, one of the most interesting open problems on singularities in algebraic geometry. An underlying long-term goal is thus to provide new tools to address this conjecture." "An algebraic geometry perspective on conditional independence models." "Tom De Medts, Isabel Van Driessche, Fatemeh Mohammadi" "Department of Chemistry, Department of Mathematics: Algebra and Geometry" "The proposed research is at the interface of statistics and algebraic geometry. I will develop combinatorial, and geometric tools to study various statistical models from an algebraic viewpoint. In particular, I will focus on the study of Conditional Independence Models, Graphical Models, and Gaussoids. I will use the developed techniques to study related applications in computer vision and rigidity theory." "Applied Algebraic Geometry" "Isabel Van Driessche, Fatemeh Mohammadi" "Department of Chemistry, Department of Mathematics: Algebra and Geometry" "Algebraic Geometry is a branch of pure mathematics that deals with systems of polynomial equations and their solutions, which are called varieties. It has been extensively developed in the mathematical community, especially since the 20th century, e.g. by works of Grothendieck and Hilbert. What makes Algebraic Geometry special is that it connects many fields of mathematics, given that polynomials occur in many problems in various domains. Hence, algebraic geometry has deep conceptual connections to complex analysis, topology, and number theory, which have in turn led to the huge interest that currently exists in the mathematical community toward solving Algebraic Geometry problems. In addition to the great theoretical developments in Algebraic Geometry, there are deep and important connections between Algebraic Geometry on the one hand, and problems in physics, biology and neuroscience on the other. The goal is to make Algebraic Geometry tools applicable to real-world problems, especially in Engineering and Neuroscience, and to translate the techniques and results that have been developed in the Algebraic Geometry community over the past two centuries to programs, tools, and methods that are easy-to-use and accessible for other scientists. The research in Applied Algebraic Geometry has interfaces with mathematical branches such as Algebra, Combinatorics and Convex Geometry, and other sciences such as phylogenetics, physics, statistics, computer science, neuroscience and reliability engineering." "Non-commutative algebraic geometry" "Michel VAN DEN BERGH" Algebra "This project involves a number of very promising applications of non-commutative algebraic geometry. Non-commutative algebraic geometry is a fairly new discipline which reinterprets geometric concepts in terms of algebraic or categorical language. This makes it possible to employ geometric intuition in cases where it is not normally applicable." "Applied and Computational Algebraic Geometry" "Fatemeh Mohammadi" Algebra "This PhD proposal centers on systems of polynomial equations that mathematically model several problems in network reliability theory, rigidity theory, and statistics. The main idea is to exploit combinatorial and geometric structures in these systems and use them to efficiently study their solutions spaces. Solving systems of polynomials in general is extremely difficult, however, in the aforementioned applications, the main problem is reduced to certifying the existence of a positive or real solution. In particular, such systems have a canonical underlying graph that captures the geometric information of polynomials. We will restrict our attention to such families of equations to develop new tools in real algebraic geometry. We will exploit the combinatorial structures of these graphs and use them to find efficient algorithms. In particular, we will use problem-specific insights along with tools in numerical and computational algebraic geometry to decompose the solution spaces (varieties) into more manageable components, hence speeding up the solution procedure." "Combinatorial and Computational Algebraic Geometry" "Stefan Vandewalle" "Numerical Analysis and Applied Mathematics (NUMA), Algebra" "My project lies in the area of Commutative Algebra and its interactions with Algebraic Geometry, Tropical Geometry, Combinatorics, and Convex Geometry. The main goal is to associate convex polytopes to algebraic varieties such that significant geometric properties of the variety can be read off from their polytopes. A toric variety is a certain algebraic variety modeled on a convex polytope. My main goal is to develop new and unifying tools to extend the tools from toric varieties to general varieties via toric degenerations. I will first develop combinatorial tools to construct points in tropical varieties and study their associated Gröbner degenerations. Then, I find explicit characterization for such points leading to toric degenerations, that is their corresponding polynomial ideal is binomial and prime. Furthermore, I will study the relations among toric degenerations of a given variety by studying their associated polytopes. In particular, I will study the combinatorial mutations of these polytopes. My goal is to determine isomorphic degenerations and characterize them. Finally, I will provide algorithms to compute tropicalization and toric degenerations for specific families of varieties. As a particular case, I plan to study the Grassmannians and flag varieties." "Topology, birational geometry and vanishing theorem for complex algebraic varieties" "Nero Budur" Algebra "In this proposal, we focus on three aspects of algebraic varieties. Firstly, we want to study two algebro-geometric properties of smooth algebraic varieties: the linearity of  the set of holomorphic 1-forms with zeros on smooth complex projective varieties, which reflects deep topological and birational nature of algebraic varieties; the surjectivity of quasi-Albanese map for smooth quasiprojective varieties, which is a crucial property for quasiprojective varieties and involves both topological nature and birational aspects of them. Both of these two projects (Project A and D in this proposal) are based on generic vanishing theorems for line bundles and local systems. Secondly, we want to study the crucial topological invariant involving first and second Chern classes of good minimal models motivated by a conjecture of Kollar, for which we want to understand the index of good minimal models using birational geometry techniques.  Thirdly, we want to establish Kodaira-Akizuki-Nakano type vanishing theorems for Higgs bundles over projective V-manifolds, which admit mild singularties in birational geometry, based on the vanishing theorem for Higgs bundles on smooth projective varieties by D. Arapura, Y. Deng, H. Li, and the applicant. " "Topics in singularity theory and algebraic geometry" "Nero Budur" Algebra "We will work on selected topics in singularity theory and algebraic geometry. We will focus on uncovering the geometric details of the contact loci of polynomials inside jet spaces. There are two possible directions for applications. One is in arithmetic, where contact loci play a prominent role in the monodromy conjecture. Another one is in symplectic geometry, where contact loci are conjectured to provide an algebraic formulation of the Floer cohomology of the iterations of the monodromy map. In this thesis both directions will be explored."