## Past Talks

### Generalized Hermite reduction, creative telescoping, and definite integration of D-finite functions

Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. In this talk, we extend ...

### Walks in the three-quarter plane

Enumeration of lattice walks in cones has many applications in combinatorics and probability theory. These objects are amenable to treatment by many techniques: combinatorics, complex analysis, probability theory, computer algebra and Galois theory of difference equations. While walks restricted to ...

### Topological rewriting systems applied to standard bases and syntactic algebras

We introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence ...

### On simple semirings and related problems

Semirings are a generalization of rings, where the subtraction is not available. They appear naturally in many branches of mathematics and informatics. In cryptograhy, simple semiring were suggested as suitable candidates for so called post-quantum cryptography, i.e. for protocols ...

### Some families of completely monotonic functions

Abstract: I shall describe two families of completely monotonic functions. One family is based on the Lambert W function, and splits into two subtopics, based on the two branches which take real values. The second family is a two-parameter family ...

### I Prefer PI

I will report on my joint work with Doron Zeilberger, about how one can use experimental mathematics and symbolic algebra to measure the irrationality of famous mathematical constants; in particular, about a new world record for PI. There is still ...

### Modular Techniques for Efficient Computation of Ideal Operations

Abstract: Modular techniques are widely applied to various algebraic computations. In this talk, it is discussed how modular techniques can be efficiently applied to computation of Groebner basis of the ideal generated by a given set, and extend the techniques ...

### Explicit solutions of linear recurrence equations with polynomial coefficients

When solving functional equations, one tends to look first for an explicit representation of the solution, i.e., for an expression built from the independent variable and the constants by means of various admissible basic operations. Here we consider ...

### Symbolic computation for linear operators with matrix coefficients

For symbolic computations with systems of linear functional equations, like (integro-)differential equations or boundary problems, we need an algebraic framework that enables effective computations in corresponding rings of operators. To represent and compute with concrete linear systems usually matrices of ...

The talk will give a basic introduction to rigidity theory before we start counting rigid graphs in different spaces. We also deal with the problem of counting realizations in the respective space and give an overview on the current level ...

### Deformations of Belyi maps and dessins d’enfant

Belyi maps and their geometric representations - dessins d'enfant ("child drawings") - have appealing significance in algebraic geometry, number theory, combinatorics, transformations of modular and hypergeometric functions. Belyi maps are algebraic coverings of the Riemann sphere that branch only above ...

### How to compute volumes of semi-algebraic sets

### Reflexive Polytopes and Gröbner Bases

A reflexive polytope is a lattice polytope whose dual polytope is again a lattice polytope. In my talk, after reviewing reflexive polytopes from a viewpoint of enumeration of lattice points, current topics related to the construction of reflexive polytopes by ...

### Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals

The starting point of the talk is how to measure how big an infinite dimensional algebra is. In particular, we discuss the problem how to construct graded algebras with prescribed Hilbert series, including Hilbert series which are algebraic but not ...

### Series of Lectures and exercises: Scalling limits of random trees and random graphs

Abstract: Since the pioneering work of Aldous in the 90s about the asymptotic shape of large random trees, the connection between combinatorial structures and stochastic processes such as Brownian motion have been proven very fruitful. I will present some essential ...

### Series of Lectures and Exercises: Order Bases: Applications and Computation

Abstract: Order Bases takes as input a vector or matrix of power series F and describes all solutions (as a module) for approximation problems of the form F p = O(zω) with ω a scalar or a vector. These approximation ...

### Series of Lectures and Exercises: Contiuned fractions and Hankel-total positivity

Abstract: The expansion of power series into continued fractions goes back at least to Euler in 1746, but it gained impetus following Flajolet's seminal 1980 discovery that any Stieltjes-type (resp. Jacobi-type) continued fraction can be interpreted combinatorially as a generating ...

### Rational solutions of Painlevé equations

### Negative thinking and polynomial analogs

### What is … a Riemann-Hilbert problem?

### Applications of the Newton polygon method for differential, difference and q-difference equations

### Symbolic Description of the Boundary Curves of the 2D Projections of the Unit Ball B_n

Abstract. In classical approximation theory the unit ball of the real univariate polynomials, B_n, that is, the set of polynomials of degree at most n with supremum norm less or equal than one on the interval [-1,1] is widely ...

### Thomas decomposition of nonlinear PDE systems

This talk gives an introduction to the method of Thomas decomposition for systems of nonlinear partial differential equations, which is fundamental for solving tasks like determining all power series solutions of the PDE system (around sufficiently generic points), deciding membership ...

### q-series related to weighted odd Ferrers diagrams.

### On some conjectured partition identities of Kanade and Russell

In 2014, Kanade and Russell conjectures six partition identities of the Rogers-Ramanujan type. We discuss the recent proof of the modulo 12 conjectures by Bringmann, JS, and Mahlburg, which is a combination of various q-series techniques. As time permits, we ...

### The Gamma function and its functional inverse

### Sum-Product Identities, Partitions, and Affine Lie Algebras

### Statistics for unimodal sequences of integers

### Asymptotic behavior of Linear and Integer Programming and the stability of the Castelnuovo-Mumford regularity

In this talk I will explain a Connection between Commutative Algebra and Linear and Integer Programming. In the first part, it is explained how one can translate the Problem of bounding the index of stability of the Castelnuovo-Mumford regularities of ...

### The Algebro-Geometric Solution Method for Algebraic Differential Equations – Theory and Software

### Effective Localization

We propose a new method for localization of polynomial ideals, which we call `Local Primary Algorithm.' For an ideal I and a prime ideal P, our method computes a P-primary component of I after checking if P is associated ...

### Radical Varieties

In many applications, where geometric constructions appear, parametrizations of the geometric objects are used. In practice, when the geometric entities are algebraic, the varieties are assumed to be rational, and hence representable by means of rational functions. This is a ...

The recent work of George Andrews and Mircea Merca on the truncated version of Euler’s pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. In collaboration with ...

### Solving a linear system symbolically – how and why

### A Polynomial Version of the Rogers-Ramanujan Identities and Ramanujan Type Functions.

### Reduction of matrices of difference operators and inverse matrix computation

### Combinatorics on polynomial equations: do they describe nice varieties?

Combinatorics on polynomial equations: do they describe nice varieties? Joachim von zur Gathen Abstract: We consider natural combinatorial questions about systems of multivariate polynomials over a finite field and the variety V that they define over an algebraic closure. Fixing ...

Chain Partition Analysis Matthias Beck Abstract: We introduce and study a hybrid of (restricted) partition functions from combinatorial number theory and zeta polynomials from the theory of partially ordered sets (posets), giving rise to a concept that in a sense ...

### Quantum Field Theory and a Hierarchy of Special Function Spaces

### Eigenvalues, invariant factors and random integer matrices

Integer matrices are often characterized by the lattice of combinations of their rows or columns. This is captured nicely by the Smith canonical form, a diagonal matrix of invariant factors, to which any integer matrix can be transformed through left ...

### Ramanujan’s Lost Notebook in Five Volumes__Reflections

Ramanujan's Lost Notebook in Five Volumes__Reflections George E. Andrews Abstract: Bruce Berndt and I have recently completed the fifth and final volume on Ramanujan's Lost Notebook. All of Ramanujan's assertions (with perhaps one of two exceptions) have been proved or, ...

### Definite sums as solutions of linear recurrence equations

### Some Hankel determinants with nice evaluations

Some Hankel determinants with nice evaluations Johann Cigler Abstract: We give some results and conjectures about Hankel determinants with nice evaluations: Hankel determinants of convolution powers of Catalan numbers Cn, different approaches to Hankel determinants of sequences such as (1, ...

### Modeling Sensor Networks by using Finite-Source Retrial Queueing Systems Tamas Berczes

A new finite source queueing model is introduced in order to calculate the most important system performance characteristics (e.g. mean waiting time, mean number of requests waiting for transmission). The sensors are classified according to their working purposes: The ...

### Some simple observations on Ehrhart Polynomials, Vector Partition Function and Minkowski Decomposition

In this talk we will see some basic facts about Ehrhart polynomials after introducing the necessary notions from polyhedral geometry. The Vector Partition Function, which can be thought of as a generalization of Ehrhart polynomials, will be then explored from ...

### African research institute for Mathematics

AIMS is the main African research institute for Mathematics. The research area of the chairman Professor Foupouagniagni is computational mathematics, in particular symbolic computation and mathematical software. He will speak about the structure and the far-reaching goals of AIMS for ...

### Laurent Series Solutions of Algebraic Ordinary Differential Equations

### Ramanujan’s Lost Notebook: History and Content

In the spring of 1976, George Andrews discovered Ramanujan's "Lost Notebook" in the library of Trinity College, Cambridge. The "Lost Notebook" is not a notebook, but a sheaf of over 100 handwritten pages made by Ramanujan during the last year ...

Generally regarded as India's greatest mathematician, Srinivasa Ramanujan was born in the southern Indian town of Kumbakonam on December 22, 1887 and died in Madras at the age of 32 in 1920. Before going to England in 1914 at the ...

### Walks, Difference Equations and Elliptic Curves

In the recent years, the nature of the generating series of the walks in the quarter plane have attracted the attention of many authors. The main questions are: are they algebraic, holonomic (solutions of linear differential equations) or at least ...

In 1887, Hölder proved that the Gamma function, defined by the difference equation y(x+1) = x y(x), satisfies no nonzero polynomial differential equation with complex coefficients. In this talk I will describe a Galois theory that allows one to reprove ...

### Historische Streiflichter zum RISC Jubiläum

### 4-Shadows in q-Series: Gupta, Kimberling, the Garden of Eden and the OEIS.

TALK ANNOUNCEMENT: Special RISC Algorithmic Combinatorics Seminar Abstract: This talk is devoted to discussing the implications of a very elementary technique for proving mod 4 congruences in the theory of partitions. It starts with a tribute to the late Hans ...

### The Man Who Knew Infinity: the Movie, the Man, and the Mathematics

*RISC Colloquium Announcement with corrected date* Abstract: In the spring of last year, the motion picture, The Man Who Knew Infinity, was released. It is now available on DVD. The movie tells the life story of the Indian genius, Ramanujan. ...

### A new approach to categorical semantics for procedural language

The semantics of programs written in some languages is concerned with the interpretation in various types of models. We present a new approach to semantics: behavior of programs, i.e. changes of states is modeled in the category of states. ...

### Approximate polynomial GCD by approximate syzygies, redux

I will show in brief how one can compute the GCD of a pair of multivariate polynomials by finding a syzygy. I will then show how we can weaken this and create an "approximate syzygy" to find an approximate GCD. ...

### talk on some of his recent research in easy terms

Most algebraists believe they know Linear Algebra. The purpose of this talk is to indicate that this is not necessarily true. We show a substantial amount of "new" Linear Algebra and its connection to Algebraic Geometry, in particular to the ...

### A new companion to Göllnitz’ (Big) partition theorem

Title: A new companion to Göllnitz' (Big) partition theorem Speaker: Professor Krishnaswami Alladi University of Florida, USA Time and Location: Tuesday, March 3, 2016 Seminar room castle, RISC, Hagenberg Abstract: One of the deepest results in the theory of partitions ...

### The number of prime factors of integers – a selection of charming results both old and new

Note: This is a (blackboard) talk for a general audience. Abstract: The first systematic study of the number of prime factors was due to Hardy and Ramanujan in 1917. This eventually gave rise to probabilistic number theory with the work ...

### Partitions with non-repeating odd parts – q-hypergeometric and combinatorial identities

ABSTRACT: By studying partitions with non-repeating odd parts using representations in terms of 2-modular graphs, we first derive a Lebesgue type q-series identity and use this to give a unified treatment of several fundamental identities in the theory of q-hypergeometric ...

### Inequalities on Ranks and Cranks of Partitions

### Paul Erdoes – one of the most influential mathematicians of our times

Note: This is a talk for a general audience and for students. Abstract: Paul Erdoes (1913-1996) was one of the most influential mathematicians of the twentieth century. 2013 was his 100-th birthday year. A Hungarian by birth, Erdoes had no ...

Title: Twisting the quantum Speaker: Prof. Charles W. Clark Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland College Park, Maryland, USA Time and Location: Thursday, March 3, 2016 2:00 pm, Seminar room pond, RISC, ...

The year 2016 marks 20 years since the publication of the paper "On the Lambert W function". As will be pointed out, the function was studied before 1996, but this publication has proved to be the most cited reference. The ...

### Effective analytic combinatorics in several variables, with applications to lattice path enumeration

### Burnside Rings (D. Patil, IlSc Bangalore); Border Basis Schemes (M. Kreuzer)

### Problems I cannot do – that people at RISC hopefully could do

I will discuss open problems in enumeration, including partitions, and special functions. This will include the Rogers-Ramanujan identities, Gessel paths, the Omega operator, generalized super Catalan numbers, very well poised series, finite difference calculus, finite fields, and homology questions. ...

A well studied (q,t)-analogue of symmetric functions are the Macdonald polynomials. In this talk I will survey another (q,t)-analogue, where q is a prime power from a finite field and t is an indeterminate. Analogues of facts about ...

The Fishburn numbers, originally considered by Peter C. Fishburn, have been shown to enumerate a variety of combinatorial objects. These include unlabelled interval orders on n elements, (2+2)--avoiding posets with n elements, upper triangular matrices with nonnegative integer entries and ...

### Buchberger Theory for Effective Associative Rings

The extension of Buchberger Theory and Algorithm from the classical case of polynomial rings over a field[1, 2, 3] to the case of (non necessarily commutative) monoid rings over a (non necessarily free) monoid and a principal ideal ring was ...

### The linearisation of the lambda-calculus: a tale of two conjectures.

### Induced term order unter some ring homomorphism and related topics

### On the classification of singularities and report on a novel database for mathematical software

First I report on new results on the classification of isolated hypersurface singularities in positive characteristic. The corresponding classification over the real and complex numbers was achieved by V.I. Arnold in his pioneering work in the mid sixties. A ...

### Report on the Computation of Extremal Polynomials and the Teaching of Hermite Interpolation

The author in his PhD thesis and in the talk in the Theroma Seminar last year showed how exact explicit formulas for low degree extremal polynomials can be given with aid of quantifier elimination. In the first part of the ...

### A New Finite-Source Queueing Model for Spectrum Renting in Mobile Cellular Networks

This presentation proposes a new finite-source retrial queueing model to consider spectrum renting in mobile cellular networks, in which service providers may rent each other’s unutilized frequency bands. We present a novel way to take into account the renting fee, ...

### Generating network topologies with clustering similar to online social networks

We propose a novel method to model real online social networks where our growing scale-free networks have tunable clustering coefficient. Models which based on purely preferential attachment are not able to describe high clustering coefficient of social networks. Beside the ...

### Program Extraction from Coinductive Proofs

Refinements of the celebrated tangent and secant numbers give rise to bivariate statistical distributions that can be expressed, either by a finite difference equation system, or by a three-variate exponential generating function. The underlying combinatorial sets, counted by tangent and ...

### Telescopers for compatible rational functions and the inverse problem in parameterized Picard–Vessiot theory

### Computing Galois Groups of Linear Differential Equations

### On the ubiquity of modular forms and Apery-like numbers

In the first part of this talk, we give examples from the theories of short random walks, binomial congruences, positivity of rational functions and series for $1/\pi$, in which modular forms and Apery-like numbers appear naturally (though not necessarily obviously). ...

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and stationary bisecants. We express the degree of this ...

### Teaching Semantics with a Proof Assistant or No more LSD trip proofs

The gulf between many computer science students and rigorous proofs is well known and much lamented. Teachers are frequently confronted with student ``proofs'' that look more like LSD trips than coherent chains of logic. In this talk I will present ...

### Old and New Results for Generalized Frobenius Partition Functions

In his 1984 AMS Memoir, George Andrews defined two families of generalized Frobenius partition functions which he denoted $\phi_k(n)$ and $c\phi_k(n)$ where $k\geq 1.$ Both of these functions "naturally" generalize the unrestricted partition function $p(n)$ since $p(n) = \phi_1(n) ...

### A unified method to prove Rogers-Ramanujan generalizations

The first of the famous Rogers-Ramanujan identities states that the number of partitions of a positive integer n into distinct non-consecutive parts equals the number of partitions of n into parts that are 1 or 4 mod 5. Gordon later ...

### Combinatorial Applications of Ehrhart Theory, Hypergraph Coloring Complexes and the Ehrhart f*-vector

The Ehrhart function of a set X in Euclidean space counts the number of integer points in the k-th dilate of X. If X is a polytope with integral vertices, the Ehrhart function of X coincides with a polynomial at ...

ROGUE WAVES: A LINEAR APPROXIMATION Sergei K. Suslov School of Mathematical and Statistical Sciences, Arizona State University Abstract We elaborate on the phenomenon of Giant, Freak, or Rogue Waves in the ocean. A simplest possible explanation, in a linear approximation, ...

### Around the Numerical Evaluation of D-Finite Functions

### Completing the Classification of Walks with Small Steps in the Quarter Plane

In joint work with Amanda Folsom, we resolve a question of Kac, and explain the automorphic properties of certain characters due to Kac and Wakimoto. We prove that they are essentially holomorphic parts of certain generalizations of harmonic weak Maass ...

### Improvements and Extensions to CAD-Based Quantifier Elimination

The talk will informally describe some relatively recent improvements and extensions to cylindrical algebraic decomposition (CAD) based quantifier elimination (QE). The improvements include improved projection operators and the use of equational constraints (where present) to further reduce the size of ...

For a prime number l, the classical modular polynomial Phi_l (sometimes called the modular equation of level l) is a polynomial with integer coefficients such that Phi_l(j(t),j(lt))=0 where j(t) is the j-invariant well known from the theory of elliptic ...

### A bijection for switching all crossing and nesting numbers for a subclass of matchings, set partitions, and permutations.

For a subclass of matchings, set partitions, and permutations, we describe a direct bijection involving only arc annotated diagrams that not only interchanges maximum nesting and crossing numbers, but also all refinements of crossing and nesting numbers. Furthermore, we show ...

### Set theoretical aspects of Game Theory

At the beginning mathematics enabled a systematic approach to popular games giving advantage to the perceptive player, however nowadays it is also common to reformulate conjectures and theorems in term of games, in a way that if one were able ...

### Tree transformations: The equational point of view

### Developing Design Tools for Critical Embedded Systems: A Model Transformation Playground

The design of critical embedded systems necessitates a thorough quality assurance process to guarantee that the target software meets all its requirements for safe operation. Therefore, development and verification tools used for designing such systems also need to undergo a ...

### A generalized queueing network model to study Proxy Cache Servers

In this talk we treat a modification of the performance model of Proxy Cache Servers to a more powerful case when the inter-arrival times and the service times are generally distributed. First we describe the original Proxy Cache Server model ...

### Algebraic& Statistic Computation for Model Checking in BioInformatics

Temporal model checking is an algorithmic and formal approach for automatically verifying whether a finite-state concurrent system such as a sequential circuit design functions correctly. Typically, computation is carried over Boolean algebras using binary decision diagrams (BDDs) or satisfiability (SAT) ...

### Combinatorial proofs of (some of) Aek’s identities

### Schur functions and generating functions for Kostka, Littlewood-Richardson and Kronecker coefficients, and their stretched versions.

Some basic properties of the symmetric functions known as Schur functions will be presented including the ubiquitous Cauchy identity. The expansions of Schur functions as sums of monomials define Kostka coefficients, The evaluation of outer and inner products of Schur ...

### Lattice walks restricted to the positive quarter plane (and octant)

The study of partitions and compositions (i.e., ordered partitions) of integers goes back centuries and has applications in various areas within and outside of mathematics. Partition analysis is full of beautiful--and sometimes surprising--identities, starting with Euler's classic theorem ...

The study of partitions and compositions (i.e., ordered partitions) of integers goes back centuries and has applications in various areas within and outside of mathematics. Partition analysis is full of beautiful--and sometimes surprising--identities, starting with Euler's classic theorem ...

We use generating functions and complex-analytic methods to count integer lattice points in polytopes with rational vertices. More precisely, we study the number of lattice points as the polytope gets dilated by an integer factor. This expression is known as ...

The precise definition of functions such as inverse sine is generally thought of as a tedious business, settled by Abramowitz and Stegun. However, the concept of branch cuts, and more generally the fact that functions such as sine do not ...