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Home Participants<= /A> Program = =20 Organization/= Contact=20

# Program

#### Monday 26

 9:30 - 10:00 Welcome 10:00 - 11:00 Claire T=C3=AAte Invited tutorial: Depth,=20 dimension and resolution in Commutative Algebra 30min break 11:30 - 12:30 Alain Herremann Invited talk: How=20 to define constructivity? Some few things we can learn from=20 history. lunch break (1h45) 14:15 - 15:15 Claude Quitt=C3=A9 Invited talk: Playing=20 with monomials of k[X_1, =E2=80=A6, X_n] 15:15 - 16:15 Konstantin Mischaikow Invited tutorial: Towards= =20 an Algebraic Framework for Nonlinear Dynamics 45min break 17:00 - 18:00 Timothy Gowers External event: Fully=20 automatic problem solving with human-style=20 output.

#### Tuesday 27

 9:00 - 10:00 Claire T=C3=AAte Invited tutorial: Depth,=20 dimension and resolution in Commutative Algebra 10:00 - 10:30 Francis Sergeraert Contributed talk: Spectra= l=20 Sequences downgraded to elementary Gauss = Reductions 30min break 11:00 - 12:00 Fr=C3=A9d=C3=A9ric Chyzak Invited talk: A=20 Computer-Algebra-Based Formal Proof of the Irrationality of=20 =CE=B6(3) 12:00 - 12:30 Anders M=C3=B6rtberg and Cyril Cohen Contributed talk: Formalizi= ng=20 elementary divisor rings in Coq lunch break (1h45) 14:15 - 15:15 Noam Zeilberger Invited talk: Refinin= g=20 the meaning of types 15:15 - 15:45 Xavier Caruso, David Roe and Tristan = Vaccon Contributed talk: Tracking=20 p-adic precision 15:45 - 16:15 MAP meeting Evening MAP dinner

#### Wednesday 28

 9:00 - 10:00 Konstantin Mischaikow Invited tutorial: Towards= =20 an Algebraic Framework for Nonlinear Dynamics 10:00 - 10:30 Marc Giusti and Jean-Claude Yakoubsohn Contributed talk: Multipl= icity=20 hunting and approximating multiple roots of polynomial=20 systems 30min break 11:00 - 12:00 Luca Moci Invited talk: Matroids=20 over a ring: motivations, examples, = applications. 12:00 - 12:30 Alain Giorgetti, Richard Genestier and Valerio=20 Senni Contributed talk: Software= =20 Engineering and Enumerative Combinatorics lunch break (1h45) 14:15 - 15:15 Christophe Raffalli Invited talk: Hilbert's= =20 17th problem via cut-elimination 15:15 - 16:15 Claire T=C3=AAte Invited tutorial: Depth,=20 dimension and resolution in Commutative=20 Algebra

#### Friday 30

 9:00 - 10:00 Konstantin Mischaikow Invited tutorial: Towards= =20 an Algebraic Framework for Nonlinear Dynamics 10:00 - 10:30 Peter Schuster Contributed talk: Eliminati= ng=20 Krull's Lemma for Horn Clauses 30min break 11:00 - 12:00 Ana Romero Invited talk: Spectral=20 sequences for computing persistent homology 12:00 - 12:30 Philippe Malbos Contributed talk: Linear=20 rewriting and homology of associative algebras lunch break (1h45) 14:15 - 15:15 Grant Passmore Invited talk: Exact=20 global optimization on demand 15:15 - 15:45 Bas Spitters Contributed talk: Construct= ive=20 algebra and geometric mathematics 15:45 - 16:15 Bruno Grenet Contributed talk: Computing=20 low-degree factors of lacunary polynomials: a Newton-Puiseux=20 approach

### Abstracts

#### Fr=C3=A9d=C3=A9ric Chyzak: A Computer-Algebra-Based = Formal Proof of the=20 Irrationality of =CE=B6(3)

Slides
```We report on the formal verification  of an irrationality proof of =
the=0A=
evaluation at 3  of the Riemann zeta function.  This verification uses=0A=
the Coq  proof assistant in conjunction  with algorithmic calculations=0A=
in  Maple.  This  irrationality result  was first  proved by  Ap=C3=A9ry =
in=0A=
1978, and  our formalization  follows the proof  path of  his original=0A=
presentation.   The crux  of it  is to  establish that  some sequences=0A=
satisfy  a  common  recurrence.   We  formally  prove  this  by  an  a=0A=
posteriori  verification   of  calculations   performed  by   a  Maple=0A=
session.  This  bases   on  computer-algebra  algorithms  implementing=0A=
Zeilberger's  approach   of  creative  telescoping.   This  experience=0A=
illustrates the  limits of  the belief  that creative  telescoping can=0A=
discover recurrences for holonomic sequences that are easily checked a=0A=
posteriori.  We discuss this observation  and describe the protocol we=0A=
devised in order to produce complete formal proofs of the recurrences.=0A=
Beside   establishing  the   recurrences,  our   proof  combines   the=0A=
formalization  of  arithmetical  ingredients and  of  some  asymptotic=0A=
analysis.=0A=
=0A=
Joint work with Assia Mahboubi, Thomas Sibut-Pinote, and Enrico Tassi.=0A=
```

#### Alain Giorgetti, Richard Genestier and Valerio = Senni:=20 Software Engineering and Enumerative Combinatorics

Slides
```We  present results  on  the frontier  between  two research  =
domains,=0A=
namely enumerative  combinatorics and  software engineering,  and show=0A=
how each domain takes profit from the other one.=0A=
```

#### Timothy Gowers: Fully automatic problem solving = with=20 human-style output.

```I shall describe a program that  Mohan Ganesalingam and I created =
that=0A=
proves simple statements in elementary abstract analysis. I shall also=0A=
discuss the  broader project of  which this is  intended to be  just a=0A=
preliminary step.  Briefly, the aim is  to write a series  of programs=0A=
that solve problems  in a "fully human" way, meaning  that they do not=0A=
undertake any  search that  a human  mathematician would  not consider=0A=
undertaking. At  first, this seems  to be sacrificing  everything that=0A=
makes a computer worth using. However,  I shall attempt to explain why=0A=
we regard this view as mistaken=0A=
```

#### Bruno Grenet: Computing low-degree factors of = lacunary=20 polynomials: a Newton-Puiseux approach

Sl= ides
```I will present a new  polynomial-time algorithm for the computation =
of=0A=
the irreducible  factors of  degree at most  d, with  multiplicity, of=0A=
multivariate lacunary polynomials over  fields of characteristic zero.=0A=
The lacunary representation of a polynomial is the list of its nonzero=0A=
terms,  and  the   size  of  this  representation   is  in  particular=0A=
logarithmic in the degree of the polynomial.  Lacunary polynomials can=0A=
have   factors   of  exponential   size   which   are  therefore   not=0A=
polynomial-time computable,  whence the  restriction on the  degree of=0A=
the factors that are computed.=0A=
=0A=
The  algorithm is  a  deterministic reduction  to  the computation  of=0A=
irreducible  factors  of  degree  at most  d  of  univariate  lacunary=0A=
polynomials on  the one  hand and to  the factorization  of low-degree=0A=
multivariate polynomials  on the  other hand.   The reduction  runs in=0A=
time polynomial  in the lacunary size  of the input polynomial  and in=0A=
d. As  a result,  we obtain  a new  polynomial-time algorithm  for the=0A=
computation of low-degree factors,  with multiplicity, of multivariate=0A=
lacunary polynomials  over number  fields, but  our method  also gives=0A=
partial results for other fields, such as the fields of p-adic numbers=0A=
or for absolute or approximate factorization for instance.=0A=
=0A=
The  core of  the  algorithm  uses the  Newton  polygon  of the  input=0A=
polynomial, and its validity is  based on the Newton-Puiseux expansion=0A=
of  roots  of bivariate  polynomials.   In  particular, we  bound  the=0A=
valuation of expressions of expressions of  the form f(X,=CF=86) where f =
is=0A=
a  lacunary  polynomial  and  =CF=86   a  Puiseux  series  whose  =
vanishing=0A=
polynomial has low degree.=0A=
```

#### Alain Herremann: How to define constructivity? = Some few=20 things we can learn from history.

```Turing (1937) offers  to set the idea of  calculability in =
arithmetic.=0A=
Descartes  (1637)  offers  to  set the  idea  of  constructibility  in=0A=
geometry.  We will compare the two texts and show how and why they are=0A=
similar in many respects.=0A=
```

#### Philippe Malbos: Linear rewriting and homology of = associative=20 algebras

Slides
```We introduce the notion of  higher dimensional linear rewriting =
system=0A=
for   presentations   of   algebras   generalizing   the   notion   of=0A=
noncommutative  Gr=C3=B6bner bases.   The  aim is  twofold,  1/ allow  =
more=0A=
possibilities of  termination orders than those  associated to =
Gr=C3=B6bner=0A=
bases, only based  on monomial orders, 2/ give  a description obtained=0A=
by rewriting of higher syzygies for presentations of algebras.=0A=
=0A=
In homological  algebra, constructive methods based  on noncommutative=0A=
Gr=C3=B6bner  bases were  developed  to compute  projective resolutions  =
by=0A=
Anick,  Green, Berger...   In particular,  they can  be used  to study=0A=
homological properties of associative algebras such as Koszulness.  We=0A=
explain how these constructions fit into the general setting of higher=0A=
dimensional  rewriting theory  and  how linear  rewriting allows  more=0A=
flexibility  for   the  computation   of  homological   properties  of=0A=
associative algebras.=0A=
=0A=
This is a joint work with Yves Guiraud and Eric Hoffbeck.=0A=
```

#### Konstantin Mischaikow: Towards an Algebraic = Framework for=20 Nonlinear Dynamics

Sli= des first=20 tutorial
Sli= des=20 second tutorial
Sli= des third=20 tutorial
```The global dynamics  of nonlinear systems can  exhibit structures =
over=0A=
wide varieties of temporal and  spatial scales and the same phenomenon=0A=
is true with  respect to parameters. However, from  an experimental or=0A=
computational perspective only  a finite amount of  information can be=0A=
collected and measurements  can only be made up to  a certain level of=0A=
precision. With this in mind I  will argue that we need an alternative=0A=
description to the qualitative theory of dynamics, preferably based on=0A=
principles that  are easily accessed computationally.   I will present=0A=
some results that represent efforts  to develop an algebraic framework=0A=
for  dynamical systems  that is  based  on tools  and techniques  from=0A=
combinatorics and algebraic topology.=0A=
```

#### Luca Moci: Matroids over a ring: motivations, = examples,=20 applications.

Slides
```Several objects  can be associated to  a list of vectors  with =
integer=0A=
coordinates: among others, a family  of tori called toric arrangement,=0A=
a convex polytope called zonotope,  a function called vector partition=0A=
function; these  objects have been  described in  a recent book  by De=0A=
Concini and  Procesi. The  linear algebra  of the  list of  vectors is=0A=
axiomatized  by the  combinatorial notion  of a  matroid; but  several=0A=
properties of the objects above depend  also on the arithmetics of the=0A=
list.  This  can  be  encoded  by   the  notion  of  a  "matroid  over=0A=
Z".  Similarly,   applications  to   tropical  geometry   suggest  the=0A=
introduction of matroids over a  discrete valuation ring. Motivated by=0A=
the examples above, we introduce the more general notion of a "matroid=0A=
over a commutative  ring R". Such a matroid arises  for example from a=0A=
list of elements  in a R-module. When  R is a Dedekind  domain, we can=0A=
extend the usual properties and operations holding for matroids (e.g.,=0A=
duality). We can also compute  the Tutte-Grothendieck ring of matroids=0A=
over R; the class  of a matroid in such a  ring specializes to several=0A=
invariants,   such   as   the   Tutte   polynomial   and   the   Tutte=0A=
quasipolynomial. We will also  outline other possible applications and=0A=
open problems.=0A=
=0A=
(Joint work with Alex Fink).=0A=
```

#### Anders M=C3=B6rtberg and Cyril Cohen: Formalizing = elementary=20 divisor rings in Coq

Slides=
```An elementary divisor ring is a commutative ring where every matrix =
is=0A=
equivalent to  a matrix in  Smith normal form.  It is well  known that=0A=
principal ideal  domains are  elementary divisor rings,  but it  is an=0A=
open problem whether the generalization to B=C3=A9zout domains, rings =
where=0A=
every finitely  generated ideal  is principal, are  elementary divisor=0A=
rings  or not.   We have  formalized,  using the  Coq proof  assistant=0A=
together with the SSReflect extension, different extension that we can=0A=
extend B=C3=A9zout domains  with in order to show that  they are =
elementary=0A=
divisor rings. The  extensions we have considered are:  existence of a=0A=
gdco operation, adequacy, Krull dimension less  than or equal to 1 and=0A=
rings where strict divisibility is well-founded.=0A=
```

#### Grant Passmore: Exact global optimization on=20 demand

```Joint work with Leonardo de Moura of Microsoft Research, Redmond.=0A=
=0A=
We present a method for exact global nonlinear optimization based on a=0A=
real  algebraic  adaptation  of the  conflict-driven  clause  learning=0A=
(CDCL) approach of modern SAT  solving.  This method allows polynomial=0A=
objective  functions to  be constrained  by real  algebraic constraint=0A=
systems with arbitrary boolean  structure.  Moreover, it can correctly=0A=
determine when  an objective  function is  unbounded, and  can compute=0A=
exact  infima  and  suprema  when they  exist.   The  method  requires=0A=
computations  over   real  closed  fields   containing  infinitesimals=0A=
(cf. ).=0A=
=0A=
References:=0A=
1. Leonardo  de Moura  and Grant Olney  Passmore. Computation  in Real=0A=
Closed Infinitesimal  and Transcendental Extensions of  the Rationals.=0A=
In  Proceedings  of the  24th  International  Conference on  Automated=0A=
```

#### Claude Quitt=C3=A9: Playing with monomials of = k[X_1, =E2=80=A6,=20 X_n]

```The context is  the graded resolution of monomial ideals  of k[X_1, =
=E2=80=A6,=0A=
X_n], where k is a field. The speaker wishes to explain carefully some=0A=
experiments which  led him to  understand parts of  the correspondance=0A=
between the monomial world and  the simplicial world.  In particular :=0A=
formulas  (Hochster,  Bayer-Charalambous-Popescu), Alexander  duality,=0A=
etc. Tools: algebraic and simplicial Koszul complexes.=0A=
```

#### Christophe Raffalli: Hilbert's 17th problem via=20 cut-elimination

Slides=
```Hilbert's seventeen problem, solved by Artin, says that every =
positive=0A=
polynomial  can   be  written  as   a  sum  of  squares   of  rational=0A=
fraction. Since Artin,  effective proof have been given  by Lombardi -=0A=
Roy - Perrucci for the latest one  which give a bound to the degree as=0A=
a tower of five exponentials.=0A=
=0A=
We will  see how  such a  proof can be  presented as  a result  of cut=0A=
elimination, showing that replacing model  theory by proof theory is a=0A=
possible method to make a result effective.=0A=
=0A=
The main contribution of our work  is the fact that we implemented the=0A=
procedure and  we introduced  a notion of  PBDD that  requires smaller=0A=
degrees, making it possible to  extract the wanted equalities yet only=0A=
for the simplest  positive polynomials (which was  not possible before=0A=
from such an effective proof).=0A=
```

#### Ana Romero: Spectral sequences for computing = persistent=20 homology

Slides
```A filtration of the object to be studied is the heart of both =
subjects=0A=
persistent homology and spectral sequences; in the talk I will present=0A=
the  complete  relation between  them,  focusing  on constructive  and=0A=
algorithmic  points of  views.  By  using some  previous programs  for=0A=
computing spectral  sequences, we obtain persistent  homology programs=0A=
applicable to spaces not of finite type (provided they are spaces with=0A=
effective   homology)    and   with    Z-coefficients   (significantly=0A=
generalizing  the usual  presentation  of persistent  homology over  a=0A=
field).=0A=
```

#### Peter Schuster: Eliminating Krull's Lemma for = Horn=20 Clauses

```Scarpellini   has  characterised  algebraically  when in  =
classical=0A=
logic a  certain kind  of propositional  geometric implication  in the=0A=
language of rings can be proved for commutative rings and for integral=0A=
domains,   respectively.     By   interpolation,   we    extend   this=0A=
characterisation to  the provability of such  implications for reduced=0A=
rings: that  is, commutative rings  in which  0 is the  only nilpotent=0A=
element.=0A=
=0A=
The characterisation above implies that the theory of integral domains=0A=
is conservative  for Horn  clauses over the  theory of  reduced rings.=0A=
This allows to eliminate Krull's Lemma for commutative rings from some=0A=
of the so-  called short and elegant proofs in  commutative algebra in=0A=
which the  general case is  reduced to  the special case  by reduction=0A=
modulo a generic  prime ideal.  The method applies,  in particular, to=0A=
the example studied  in [1, 2, 4]: that the  non-constant coecients of=0A=
an invertible polynomial are all nilpotent.=0A=
=0A=
1.  Henrik Persson.  An application  of the constructive spectrum of a=0A=
ring. In Type  Theory and the Integrated Logic  of Programs.  Chalmers=0A=
University and University of Goeteborg, 1999. PhD thesis.=0A=
=0A=
2.      Fred     Richman.       Nontrivial     uses     of     trivial=0A=
rings. Proc. Amer. Math. Soc.,103(4):1012-1014, 1988.=0A=
=0A=
3. Bruno  Scarpellini. On  the metamathematics  of rings  and integral=0A=
domains.  Trans. Amer. Math. Soc., 138:71-96, 1969.=0A=
=0A=
4.  Peter Schuster.  Induction in algebra: a first case study. In 2012=0A=
27th Annual  ACM/IEEE Symposium  on Logic  in Computer  Science, pages=0A=
581-585.  IEEE Computer Society Publications, 2012.  Proceedings, LICS=0A=
2012, Dubrovnik,  Croatia, June 2012.  Journal  version: Log.  Methods=0A=
Comput. Sci. (3:20) 9 (2013).=0A=
```

#### Francis Sergeraert: Spectral Sequences = downgraded to=20 elementary Gauss Reductions

Slides
```The  standard  Spectral  Sequences  of  Algebraic  Topology  had  =
been=0A=
downgraded  25 years  ago by  Julio  Rubio to  the Basic  Perturbation=0A=
Lemma,  an  "elementary"  result   of  homological  algebra,  in  fact=0A=
relatively  complex.  The  talk  is  devoted to  a  reduction of  this=0A=
"lemma"  to   elementary  Gauss  reductions,  already   known  by  the=0A=
Babylonians, combined with the invertibility of 1-x when |x| < 1.=0A=
```

#### Bas Spitters: Constructive algebra and geometric=20 mathematics

```Vickers observed that predicative  techniques from formal topology =
are=0A=
often preserved under pullbacks  of geometric morphisms. This provides=0A=
a  clear  technical  advantage to  such  constructions.   Constructive=0A=
algebra,  as  developed  by  Coquand  and  Lombardi  provides  another=0A=
important application.=0A=
=0A=
I  will provide  a  short  overview of  some  recent  such results  in=0A=
analysis, mostly based on our recent article:=0A=
=0A=
Bas Spitters, Steven Vickers, and  Sander Wolters - Gelfand spectra in=0A=
Grothendieck  toposes using  geometric  mathematics, 1310.0705,  2013.=0A=
under revision for QPL 2012 post-proceedings in EPTCS.=0A=
=0A=
http://www.cs.ru.nl/~spitters/geoBS.pdf=0A=
```

#### Claire T=C3=AAte: Depth, dimension and resolution in = Commutative=20 Algebra

```First we introduce  two fondamental tools in the theory  of depth: =
the=0A=
Koszul complex and the Cech complex for a finite sequence (a_1,=E2=80=A6 =
,a_n)=0A=
in a commutative ring.  We give some cohomological properties of these=0A=
complexes.  By using some basic examples,  we try to make explicit the=0A=
links between depth,  Krull dimension and resolutions.  At  the end we=0A=
illustrate a celebrated Serre  theorem characterizing regular rings by=0A=
existence of finite projective  resolutions: we explain THE resolution=0A=
of the ideal of a point of an hypersurface.=0A=
```

#### Xavier Caruso, David Roe and Tristan Vaccon: = Tracking p-adic=20 precision

Slides
```We present a new method to propagate p-adic precision in =
computations,=0A=
which also applies to other  ultrametric fields. We illustrate it with=0A=
many examples and give a toy  application to the stable computation of=0A=
the SOMOS  4 sequence.  Our  methods relies on  elementary ultrametric=0A=
calculus and  can either  be applied  on the fly  or as  a preparatory=0A=
step.=0A=
```

#### Marc Giusti and Jean-Claude Yakoubsohn: = Multiplicity=20 hunting and approximating multiple roots of polynomial systems

Slides
```The computation of the multiplicity  and the approximation of =
isolated=0A=
multiple roots of polynomial systems is a difficult problem.=0A=
=0A=
In recent years, there has been  an increase of activity in this area.=0A=
Our goal is  to translate the theoretical background  developed in the=0A=
last century  on the theory  of singularities in terms  of computation=0A=
and complexity.=0A=
=0A=
This  paper presents  several  different views  that  are relevant  to=0A=
address  the following  issues :  predict the  multiplicity of  a root=0A=
and/or determine  the number of  roots in  a ball, approximate  fast a=0A=
multiple  root and  give  complexity results  for  such problems.   We=0A=
propose a new method to  determine a regular system, called equivalent=0A=
but deflated,  i.e., admitting the  same root as the  initial singular=0A=
one.=0A=
=0A=
Finally, we perform a numerical analysis of our approach.=0A=
```

#### Noam Zeilberger: Refining the meaning of = types

Slides
```Complete the following sequence:=0A=
=0A=
* number theorists study numbers=0A=
* group theorists study groups=0A=
* knot theorists study knots=0A=
* type theorists study ___=0A=
=0A=
Many  mathematicians  would hesitate  at  giving  the obvious  answer,=0A=
because of their  inability to explain just what  "types" are. Instead=0A=
they might point  to a bunch of different papers  where various formal=0A=
systems  are studied,  and conclude  that "type  theorists study  type=0A=
theories".=0A=
=0A=
I believe that an important reason  for this lack of consensus is that=0A=
in  practice,  types  are  actually   used  with  two  very  different=0A=
intuitions in  mind, corresponding  to the  distinction the  late John=0A=
Reynolds  termed  "intrinsic"  versus "extrinsic"  .   For  several=0A=
years,   Paul-Andr=C3=A9   Melli=C3=A8s   and   I   have   been   =
developing   a=0A=
category-theoretic  approach to  type  theory that  aims to  reconcile=0A=
these two opposing  intuitions, the essence of which  may be explained=0A=
by the following simple idea: just as any category may be viewed (=C3=A0 =
la=0A=
Lambek) as a deductive  system (supporting composition of deductions),=0A=
any *functor* may  be viewed as a type  system (supporting composition=0A=
of typing derivations). Our approach is closely related to (and partly=0A=
inspired by)  an idea which  appears in the  type-theoretic literature=0A=
under  the  heading of  "refinement",  developed  especially by  Frank=0A=
Pfenning  and his  students. In  the talk,  I will  begin by  giving a=0A=
partial  survey  of the  work  on  type  refinement  and some  of  the=0A=
practical motivations  behind it. Then  I will introduce  our abstract=0A=
definition of a type (refinement) system, and describe various ways it=0A=
can be used.=0A=
=0A=
=0A=
1.  John C.  Reynolds,  "The Meaning  of Types  --  From Intrinsic  to=0A=
Extrinsic   Semantics",  BRICS   Report   Series  RS-00-32,   December=0A=
2000. Available from http://www.brics.dk/RS/00/32/=0A=
```
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