Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
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Location: Unitobler, F005 53 seats, 74m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | Room free |
| Unitobler, F005 | |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F005 | |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS147, part 1: SC-square 2019 workshop on satisfiability checking and symbolic computation |
| Unitobler, F005 | |
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10:00am - 12:00pm
SC-square 2019 workshop on satisfiability checking and symbolic computation Symbolic Computation is concerned with the algorithmic determination of exact solutions to complex mathematical problems; some recent developments in the area of Satisfiability Checking are starting to tackle similar problems, however with different algorithmic and technological solutions. The two communities share many central interests, but so far researchers from these two communities rarely interact. Furthermore, the lack of compatible interfaces for tools from the two areas is an obstacle to their fruitful combination. Bridges between the communities in the form of common platforms and road-maps are necessary to initiate a mutually beneficial exchange, and to support and direct their interaction. The aim of this workshop is to provide fertile ground to discuss, share knowledge and experience across both communities. The topics of interest include but are not limited to:
The 2016 and 2017 editions of the workshop were affiliated to conferences in Symbolic Computation. The 2018 edition was affiliated to FLoC, the international federated logic conference. More information at http://www.sc-square.org/workshops.html (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Invited Talk of SC-Square: SC-square-methods for the Detection of Hopf Bifurcations in Chemical Reaction Networks---Part I: Background and and basic methods The analytical problem of finding Hopf bifurcation fixed points for polynomial or rational vector fields (or determining that there are none) can be reduced to a purely semi-algebraic question. In the first part of the talk we explore this possibility by first giving a reduction of the parametric question on the existence of a Hopf bifurcation fixed point to a parametric first-order formula over the ordered fields of the real. We show the results of solving these with existing tools from computational logic (such as Redlog) for several standard and text book examples and compare the results of these fully automated methods to the ones of hand analyses given in textbooks. Invited Talk of SC-Square: SC-square-methods for the Detection of Hopf Bifurcations in Chemical Reaction Networks---Part II: Advanced methods for chemical reaction networks The determination of Hopf bifurcation fixed points in chemical reaction networks with symbolic rate constants yields information about the oscillatory behavior of the networks and hence is of high interest The problem is solvable in theory by the methods discussed in part I, but the generic technique leads to prohibitive large formulae even for rather small dimensions. Using the representations of chemical reaction systems in convex coordinates, which arise from the so called stoichiometric network analysis, the problem of determining the existence of Hopf bifurcation fixed points leads to first-order formulae over the ordered field of the reals that can then be solved using existing computational logic packages for somewhat larger dimensions. Using ideas from tropical geometry it is possible to formulate a more efficient method that is incomplete in theory but worked very well for the examples that we have attempted; we have shown it to be able to handle systems involving more than 20 species. Finding satisfying instances of a single (but in general very large) polynomial equation and a set of polynomial inequalities is the key challenge, which will benefit from further research in the context of SC-square-methods. Regular Paper 1 of SC-Square: Solving Constraint Systems from Traffic Scenarios for the Validation of Autonomous Driving The degree of automation in our daily life will grow rapidly. This leads to big challenges regarding the safety validation of autonomous robots which take over more and more tasks being -- as of yet -- predestinated for humans. This is in particular true for the emerging area of autonomous driving which aims at making road traffic safer, more efficient, more economic, and more comfortable. One promising approach for the safety validation of autonomous driving is the virtual simulation of traffic scenarios, i.e. conducting the majority of tests in virtual reality instead of the real world. In addition to quantity, the quality of such tests with a focus on critical traffic scenarios will be an essential ingredient for safety validation. Regular Paper 2 of SC-Square: On the proof complexity of MCSAT Satisfiability Modulo Theories (SMT) and SAT solvers are critical components in many formal software tools, primarily due to the fact that they are able to easily solve logical problem instances with millions of variables and clauses. This efficiency of solvers is in surprising contrast to the traditional complexity theory position that the problems that these solvers address are believed to be hard in the worst case. In an attempt to resolve this apparent discrepancy between theory and practice, theorists have proposed the study of these solvers as proof systems that would enable establishing appropriate lower and upper bounds on their complexity. For example, in recent years it has been shown that SAT solvers are polynomially equivalent to the general resolution proof system for propositional logic, and SMT solvers that use the CDCL(T) architecture are polynomially equivalent to the Res∗(T) proof system. In this paper, we extend this program to the MCSAT approach for SMT solving by showing that the MCSAT architecture is polynomially equivalent to the Res∗(T) proof system. Thus, we establish an equivalence between CDCL(T) and MCSAT from a proof-complexity theoretic point of view. This is a first and essential step towards a richer theory that may help (parametrically) characterize the kinds of formulas for which MCSAT-based SMT solvers can perform well. |
| 3:00pm - 5:30pm | MS147, part 2: SC-square 2019 workshop on satisfiability checking and symbolic computation |
| Unitobler, F005 | |
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SC-square 2019 workshop on satisfiability checking and symbolic computation Symbolic Computation is concerned with the algorithmic determination of exact solutions to complex mathematical problems; some recent developments in the area of Satisfiability Checking are starting to tackle similar problems, however with different algorithmic and technological solutions. The two communities share many central interests, but so far researchers from these two communities rarely interact. Furthermore, the lack of compatible interfaces for tools from the two areas is an obstacle to their fruitful combination. Bridges between the communities in the form of common platforms and road-maps are necessary to initiate a mutually beneficial exchange, and to support and direct their interaction. The aim of this workshop is to provide fertile ground to discuss, share knowledge and experience across both communities. The topics of interest include but are not limited to:
The 2016 and 2017 editions of the workshop were affiliated to conferences in Symbolic Computation. The 2018 edition was affiliated to FLoC, the international federated logic conference. More information at http://www.sc-square.org/workshops.html (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Regular Paper 3 of SC-Square: Algorithmically generating new algebraic features of polynomial systems for machine learning There are a variety of choices to be made in both computer algebra systems and satisfiability modulo theory (SMT) solvers which can impact performance without affecting mathematical correctness of the end result. Such choices are candidates for machine learning (ML) approaches, however, there are difficulties in applying standard ML techniques, such as the efficient identification of ML features from input data which is typically a polynomial system. Our focus is selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation which is also now used and adapted for SMT-solvers. We studied prior ML work here and recognised a framework around the features used. Enumerating all options in this framework led to the automatic generation of many additional features. We validate the usefulness of these with an experiment which shows that an ML choice for CAD variable ordering is superior to those made by human created heuristics, and further improved with these additional features. We expect that this technique of feature generation could be useful for other choices related to CAD, or even choices for other algorithms with polynomial systems for input. Extended Abstract 1 of SC-Square: On variable orderings in MCSAT for non-linear real arithmetic Satisfiability-modulo-theories (SMT) is a technique for checking the satisfiability of logical formulas. In this context, a framework called model-constructing satisfiability calculus (MCSAT) was introduced which allows the simultaneous construction of the Boolean and theory model enabling more freedom for decision on the models variables. In this paper we report on implementation issues for non-linear real arithmetic and our work in progress on heuristics for decision orderings on variables. Extended Abstract 2 of SC-Square: On Benefits of Equality Constraints in Lex-Least Invariant CAD McCallum was the first to show that it was possible to reduce the projection set for quantifier elimination problems which have equality constraints. Lazard provided a projection operator that reduces the projection set as compared to Collins' original algorithm. In this paper, we aim to extend Lazard's work and provide a modification to his projection operator that reduces the projection set even further when there is an equality constraint in the quantifier elimination problem. This is similar to McCallum's modification and outputs a sign-invariant CAD: consequently, it cannot be used inductively, but only in the first step of the projection phase. In the further steps of the projection phase, we use Lazard's original projection operator. Nonetheless, reducing the output in the first step has a domino effect throughout the remaining steps, which significantly reduces the complexity. Extended Abstract 3 of SC-Square: Evolutionary Virtual Term Substitution in a Quantifier Elimination System Quantifier Elimination over real closed fields (QE) is a topic on the borderline of both the Satisfiability Checking and Symbolic Computation communities, where quantified statements of polynomial constraints may be relevant to solving a Satisfiability Modulo Theory (SMT) problem. Feasible algorithms for QE date as far back as 1975 with Cylindrical Algebraic Decomposition (CAD), and Virtual Term Substitution (VTS) in 1988. While implementations of these can be found in software such as QEPCAD and Redlog, they are not often found together, and especially not used concurrently in terms of one poly-algorithm. This paper briefly explores the implications of such a poly-algorithm between CAD and VTS, which the author is presently developing as part of a package in collaboration with Maplesoft, intended to make its way into a future version of Maple. One such implication of the system requires incremental CAD to be effective, which has already had some attention in this workshop series via. This paper in particular focuses on proof of concept new methods for incremental and decremental VTS, that works for any multivariate problem previously solved by VTS. This may not only be desirable for QE when used in an SMT system, but we also discuss its potential ramifications when used in the author’s work in progress poly-algorithmic QE system, where users may be interested in incrementality and decrementality for stock QE. Extended Abstract 4 of SC-Square: Lemmas for Satisfiability Modulo Transcendental Functions via Incremental Linearization Incremental linearization is a conceptually simple, yet effective, technique that we have recently proposed for solving satisfiability problems over nonlinear real arithmetic constraints, including transcendental functions. A central step in the approach is the generation of linearization lemmas, constraints that are added during search to the SMT problem and that form a piecewise-linear approximation of the nonlinear functions in the input problem. It is crucial for both the soundness and the effectiveness of the technique that these constraints are valid (to not remove solutions) and as general as possible (to improve their pruning power). In this paper, we provide more details about how linearization lemmas are generated for transcendental functions, including proofs of their soundness. Such details, which were missing in previous publications, are necessary for an independent reimplementation of the method. |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS137, part 1: Symbolic Combinatorics |
| Unitobler, F005 | |
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10:00am - 12:00pm
Symbolic Combinatorics In recent years algorithms and software have been developed that allow researchers to discover and verify combinatorial identities as well as understand analytic and algebraic properties of generating functions. The interaction of combinatorics and symbolic computation has had a beneficial impact on both fields. This minisymposium will feature 12 speakers describing recent research combining these areas. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Enumeration of walks in three quarters of the plane Walks in the quarter plane with a prescribed set of steps have, in the past decades, attracted a lot of attention. One of the main question is to find the nature of the generating function (algebraicity, D-finiteness, etc.) as a function of the step set. Since these walks are now rather well understood, we turn our attention to a related problem, walks in three quarters of the plane, for which we present the latest results. Work in common with Alin Bostan and Killian Raschel. On the growth of algebras The growth function of an algebra is a combinatorial invariant that gives insight into the underlying algebraic structure. There are several well-known combinatorial constraints that growth functions must have. We show that in fact all constraints are now known: given a function satisfying the previously known constraints, we show that there is an algebra having this function as its growth function. This is joint work with Efim Zelmanov. A Gessel way to the diagonal theorem on D-finite power series Special functions that satisfy linear differential equations with polynomial coefficients appear ubiquitously in combinatorics and mathematical physics. Such special functions are said to be D-finite by Stanley. In the early 1980's, Gessel and Zeilberger independently proved that the diagonal of D-finite power series in several variables is D-finite. However their proofs were not complete and later Lipshitz gave a complete proof using an elimination lemma. Zeilberger completed his proof with the theory of holonomic D-modules. We follow the spirit of Gessel's proof strategy and fix the gap in his proof directly. This is a joint work with Chaochao Zhu. Inhomogeneous Lattice Walks We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating functions are always algebraic. For the models in the quarter plane, we have carried out an experimental classification of all models with small steps. We discovered many (apparently) D-finite cases for most of which we have no explanation yet. Joint work with Manuel Kauers. To appear in the proceedings of FPSAC 2019. |
| 3:00pm - 5:00pm | MS188: Probability and randomness in commutative algebra and algebraic geometry |
| Unitobler, F005 | |
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3:00pm - 5:00pm
Probability and randomness in commutative algebra and algebraic geometry Randomness has long been used to study polynomials. Several classical instances include Lit- tlewood and Offord’s examination of the expected number of real roots of an algebraic equation defined by random coefficients, as well as work of Kac and Kouchnirenko on varieties defined by random coefficients on a fixed Newton polytope support. Additionally, the use of smooth analysis, which measures the expected performance of an algorithm under slight random perturbations of worst-case inputs, has been used in the context of algebraic geometry. The aim of this minisymposium is to highlight a recent surge of interactions between the fields of probability and commutative algebra/algebraic geometry, in which questions of expected (average, typical) or unlikely (rare, non-generic) behavior of ideals and varieties are studied formally using probability distributions. Recent work has seen the successful application of techniques from statistics and probabilistic combinatorics in this setting. Our goal is to bring researchers working in this intersection together to share their work and form potential new collaborations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) What can be predicted in algebraic geometry? This talk explores how supervised machine learning can be used to predict the algebraic and combinatorial properties of polynomial ideals prior to doing a computation. This is practical for fast approximations of algebraic invariants, and also for the problem of "algorithm selection." When several algorithms exist for performing an exact computation, we train a neural network to automatically select the fastest algorithm, on a case-by-case basis, by learning features of the input that are predictive of algorithm performance. Joint work with Jesus De Loera, Robert Krone, and Zekai Zhao. Degree of Random Monomial Ideals In joint work with Lily Silverstein and Dane Wilburne, we investigate the behavior of the standard pairs of a random monomial ideal. We then use this to explore the degree and arithmetic degree of random monomial ideals. Stochastic Exploration of Real Varieties Nonlinear systems of polynomial equations arise naturally in many applied settings. The solution sets to these systems over the reals are often positive dimensional spaces that in general may be very complicated yet have very nice local behavior almost everywhere. Standard methods in real algebraic geometry for describing positive dimensional real solution sets include cylindrical algebraic decomposition and numerical cell decomposition, both of which can be costly to compute in many practical applications. In this talk, we communicate recent progress towards a Monte Carlo framework that provides a probabilistic method for exploring such real solution sets. After describing how to construct probability distributions whose mass focuses on a variety of interest, we show how state-of-the-art Hamiltonian Monte Carlo methods can be used to sample points near the variety that may then be magnetized to the variety using endgames. We conclude by showcasing trial experiments using practical implementations of the method in the probabilistic programming language Stan. Random numerical semigroups A numerical semigroup is a subset of the natural numbers that is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M, and select a generating set for S from the integers 1,...,M where each potential generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep geometric and combinatorial structures that arise naturally in the process. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS137, part 2: Symbolic Combinatorics |
| Unitobler, F005 | |
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10:00am - 12:00pm
Symbolic Combinatorics In recent years algorithms and software have been developed that allow researchers to discover and verify combinatorial identities as well as understand analytic and algebraic properties of generating functions. The interaction of combinatorics and symbolic computation has had a beneficial impact on both fields. This minisymposium will feature 12 speakers describing recent research combining these areas. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Mahlerian analogues of Riccati equations and proofs of hypertranscendence A Mahler function~$f(x)$ is a function whose evaluations at iterated $b$th powers of the variable, $f(x)$, $f(x^b)$, $f(x^{b^2})$, $f(x^{b^3})$, etc, are linearly dependent. The interest in them was recently renewed, when a difference-Galois-theoretic approach was developed to determine whether a Mahler function is hypertranscendental, that is, whether $f(x)$~and all its derivatives are algebraically independent. For Mahler functions defined by a linear equation of order~2, a criterion was given by Dreyfus, Hardouin, and Roques: $f(x)$~is hypertranscendental if and only if two auxiliary non-linear equations have no rational solutions. These equations are Mahlerian analogues of Riccati equations, as they encode the right-hand factors of corresponding linear Mahler equations. I will present preliminary results of a joint work with Dreyfus, Dumas, and Mezzarobba in which we develop an algorithm to solve Riccati equations for their rational solutions. Walk in the quarter plain and differential Galois theory The determination of the nature of the generating series of walks in the quarter plain is a very vivid topic. Recently, differential Galois theory gave tools to understand what are the algebraic and differential equations satisfied by the latter. In practice, we are reduced to determine whether a telescoper relation exists or not. The latter problem may be treated with computer algebra in many situation. Systems of equations for sets of permutations and limit shapes Enumerating and analyzing sets of permutations avoiding some patterns (permutation classes) is a standard question in enumerative combinatorics. One method is to use the so-called substitution operation to decompose recursively the objects: in particular, when the permutation class contains finitely many indecomposable permutations (called simple permutations), one can obtain in an automatic way a system of combinatorial equations describing the class. This system allows to sample large uniform permutation in the class and to try and describe their limit shape. We will explain how this limit shape is related to combinatorial and analytic properties of the system. Joint work with Frédérique Bassino (Paris-Nord), Mathilde Bouvel (Zurich), Lucas Gerin (École Polytechnique), Mickaël Maazoun (ENS Lyon) and Adeline Pierrot (Orsay). The location of variables in lambda-terms with bounded De Bruijn levels We consider lambda-terms with a bounded number of De Bruijn levels, say bounded by $k$, and are interested in the shape of such a term, if it is chosen uniformly at random from all such terms of a given size. The number of such terms of given is known asymptotically, and moreover that the asymptotic expression behave in a very strange way: The subexponential term in the asymptotics is different if $k$ belongs to a certain doubly exponentially growing sequence. It is conjectured that the reason lies in the distribution of the variables within a term. Under some technical condition we first show that the number of variables is asymptotically normally distributed with mean and variance asymptotically proportional to the size. Then, we investigate the number of variables in the different De Bruijn levels and thereby exhibit a so-called ``unary profile'' of a random lambda term. It turns out that almost all the variables are located in the last De Bruijn levels and the number of these levels is proportional to $loglog k$. This is joint work with Isabella Larcher. |
| 3:00pm - 5:00pm | MS131, part 1: Computations in algebraic geometry |
| Unitobler, F005 | |
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3:00pm - 5:00pm
Computations in algebraic geometry This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some applications. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Regularity of S_n-invariant monomial ideals Consider a polynomial ring R in n variables with the action of the symmetric group Sn by coordinate permutations. I will describe an explicit recipe for computing the graded components of the modules Ext(I,R), when I is an arbitrary Sn-invariant monomial ideal, as well as the maps induced by inclusions of such ideals. As a consequence, this gives explicit formulas for the the regularity of Sn-invariant monomial ideals. A homological approach to numerical Godeaux surfaces Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. By work of Miyaoka and Reid it is known that the torsion group of such a surface is cyclic of order at most 5, a full classification has been given for the cases where this order is 3,4, or 5. In my talk, I will discuss recent progress by Isabel Stenger towards the classification of numerical Godeaux surfaces with a trivial torsion group. Following a suggestion by Frank-Olaf Schreyer, the starting point of Stenger's work is a syzygy-type approach to the study of the canonical ring of such a surface. Particular attention is paid to the hyperelliptic curves arising in the fibration induced by the bicanonical system. Asymptotic syzygies for products of projective space We will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.
Where can toric syzygies live? Syzygies of toric varieties admit a natural grading by the character lattice of the corresponding torus. I will give some results on the the regions in the character lattice in which toric syzygies can be supported. This is joint work with Castryck and Lemmens. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | Room reserved (unless you reserved it, please don't enter) |
| Unitobler, F005 | |
| 3:00pm - 5:00pm | MS131, part 2: Computations in algebraic geometry |
| Unitobler, F005 | |
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3:00pm - 5:00pm
Computations in algebraic geometry This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some applications. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) The semigroup and cone of effective divisor classes on a hypersurface in a toric variety The computation of the semigroup (or even the cone) of all effective divisor classes on a Calabi–Yau hypersurface of a toric variety is an important open problem, whose solution would have a number of applications in theoretical physics. This is a difficult computation, with no known effective algorithms. We present computational tools and algorithms for investigating this and related problems, and describe some results in this direction. On subring counting and simultaneous monomialization The task of determining the order zeta function for certain number rings (which is just a sophisticated form of counting subrings) gives rise to a particular kind of p-adic integrals. The domain of integration of these stubbornly withstands standard techniques, including even an out-of-the-box Hironaka-style resolution of singularities. However, choosing centers of blow-ups using the structural properties of the problem, a simultaneous monomialization of the conditions can be achieved, making the problem again accessible to usual methods. This talk is based on joint work with Josh Maglione, Bernd Schober, and Christopher Voll. Fröberg-Macaulay conjectures for algebras In a joint work with Aldo Conca, we look at what should correspond to Macaulay’s Theorem and Fröberg’s Conjecture for the Hilbert function of subalgebras of standard graded polynomial rings. Upper bounds correspond to generic forms and lower bounds correspond to strongly stable monomial ideals. Singular value decomposition for complexes In this talk, the concept of singular value decomposition of complexes will introduced and applied to the computation of syzygies. |
