Arbeitstreffen

On Friday, March 15, 2013 the Max Planck Institute for Mathematics will organize a meeting with some lectures to honor emeritus director Professor Günter Harder's important contributions to MPIM and Mathematics. On this occasion there will be talks by two of his former students. Professor Harder will turn 75 on March 14.

Details

The workshop "Galois representations and pencils of Calabi-Yau motives" will be held at the MPIM Bonn during the week 15-21 December, 2012 as part of the activity "Quantum cohomology, Frobenius manifolds and pencils of Calabi-Yau varieties" in 2012-2013.

Email: galois2012@mpim-bonn.mpg.de

Organisation (in short):

The workshop will consist of three parts. On each of the days 15-17 December the agenda is broken into three consecutively running seminar classes, titled as follows:

1. Congruences (moderator: Neil Dummigan)
2. Crystals (moderator: Richard Crew)
3. Congruences sheaves/Mirror symmetry (moderator: Alessio Corti)

We meet 12PM-6PM. A number of fixed talks will be given on each of these days, however, discussions and ad-lib talks are encouraged.

A short conference will run on 18 December. The Seminar on Algebra, Geometry and Physics will be part of the conference, with Jan Stienstra's talk on connections between number theory and physics. There will be a social event in the afternoon.

The days 19-21 December are dedicated to joint work in groups. However, Thursday 20 December will start with Number Theory Seminar at 10:45. Apart from that, we will start at 11 AM and finish at 5PM on these days.

The lunchtime on workdays is at 13:00, and the institute tea is 16:00, with the exception of Friday, 21 December.

Program and abstracts

 

December 14, afternoon:

Arrival

On the next three days seminars run continuously 12:00 - late afternoon, normally with one fixed talk in each seminar.

December 15, Seminars:

~12PM: Neil Dummigan: Critical values, congruences, Selmer groups

Since the seminar I am moderating is called congruences, I'll start off by explaining the possibilities for the composition factors of the reduction mod p of the type of 4-dimensional Galois representation coming from a CY 3-fold, and comparing these with the kinds of congruences observed by Anton Mellit, calling on him to report on observations about pairs of congruences and common values of p. This would set the scene. At some point later I could say something about how for at least one of the three kinds of such congruences, it should lead to an element of order p in some Selmer group, how that appears in the Bloch-Kato conjecture, which then predicts the appearance of p in an L-value, comparing and contrasting with the cases of congruences for Saito-Kurokawa lifts, and congruences connected with p-torsion on elliptic curves.

~3PM: Jan Stienstra: Formal groups associated to pencils of Calabi-Yau varieties.

The Cartier-Dieudonne module of the Artin-Mazur formal group (AMFG) equals the the unit root crystal in crystalline cohomology. A Laurent polynomial (LP) with reflexive Newton polytope defines Calabi-Yau hypersurfaces in toric varieties. There is a very concrete formula for a logarithm for a group law for the AMFG in terms of the constant terms  in powers of the LP. The AMFG is a formal group over the ring of coefficients of the LP. This ring has a natural structure of a $\lambda$-ring. For formal groups over $\lambda$-rings Cartier's theory can be reformulated in terms of Dirichlet series. This immediately leads to congruences. The natural geometric context for Laurent polynomials is toric geometry and the natural context for their variations is Gelfand-Kapranov-Zelevinsky's theory of hypergeometric functions. The aforementioned formal group law logarithm is such a GKZ hypergeometric function. This has consequences for the unit roots of L-functions.

~5PM: Piotr Achinger: A characterization of toric varieties in characteristic p

A theorem of J. F. Thomsen states that Frobenius push-forwards of line bundles on smooth toric varieties are direct sums of line bundles. Using characterization of toric varieties in terms of their Cox rings, we show that this property in fact characterizes smooth projective
toric varieties.

December 16, Seminars:

~12PM: Richard Crew: Introduction to F-isocrystals on the line

We will review basic properties of F-isocrystals on a smooth variety, with particular attention to the case of an open subset of the projective line. Topics: convergence conditions, Dwork's trick, the slope filtration, and the local monodromy theorem and its applications. If time permits we will discuss Lauder's work on the explicit computation of Frobenius matrices, and Katz's congruence formulas for the Frobenius of a curve.

~3PM: Francesco Baldassarri: Log-growth and Frobenius slope filtrations.

Convergence polygons, log-growth polygons and Frobenius slope filtrations will be explained.

~5PM: Lucia di Vizio: Galois theory of differential equations with an action of an endomorphisms.

I'll explain how one can construct a Galois theory for differential ­equations that takes into account the action of a difference operator,i.e.,  an endomorphisms, on the solutions.The theory attaches a group scheme to a differential equation, which encodes the algebraic difference relations among the solutions of the differential equation. This is typically the case of p-adic differential equation with a Frobenius structure. This is a joint work with C. Hardouin and M. Wibmer.

December 18, Conference

10:00-10:50 Lucia Di Vizio, Introduction to Frobenius in q-difference equations.
11:00-11:50 Alexei Panchishkin, TBA
12:00-12:50 Guenter Harder Modular construction of mixed motives and congruences

I will briefly describe a construction of mixed (Anderson) motives for the symplectic group $GSp_2$. These mixed motives are labelled by  automorphic cusp forms $f$ on $Sl_2$ and they are extensions of pure Tate motives. I will give a formula for the Betti-de-Rham extension classes. Some rather speculative arguments suggest that these motives "create" congruences between elliptic and Siegel modular forms. These congruences have been checked in experiments by Berstroem, Faber, van der Geer and Ghitza, Ryan and Sulon.

Lunch

14:00-15:00 Seminar on Algebra, Geometry and Physics

Jan Stienstra (Utrecht) An amazing coincidence of formulas for the unit root part of crystalline cohomology and the density of states for physical crystals.

Certain Laurent polynomials can be viewed as discretizations of a Laplace operator with quasi-periodic boundary conditions. The eigenvalue/eigenfunction equation for this situation then describes the level sets of the Laurent polynomial. The density of states is a measure for the size of these level sets as a function of the level. It can be approximated, in yet another discretization, by counting how the points of which the coordinates are N-th roots of 1 are distributed over the various level sets. The limit as N tends to infinity is also known in number theory/algebraic geometry as the Mahler measure of the Laurent polynomial. By a simple transformation which is reminiscent of the Mellin transform the formula for the density of states becomes a formula for the formal group which describes the unit root part of the crystalline cohomology of the level sets which are then viewed as hypersurfaces in a toric variety. For real solid state physics the Laurent polynomial has to be replaced by a matrix (i.e. an endomorphism of a vector bundle) over a toric variety.

16:00 Institute tea

Social event

December 19, Joint work in groups

December 20, Joint work in groups

11.15-12.15 Number theory seminar

December 21, Joint work in groups
Departure
Official closing time of MPIM at around 14:30
 

 

Celebrating Matthias Kreck’s Birthday

 

For more details see: http://people.mpim-bonn.mpg.de/teichner/Math/Manifolds_Day.html

Organizers:

Mike Freedman, Matthias Kreck and Peter Teichner.

Further info:

http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html

Lecture Series:

• Mike Freedman, Bing Topology and Casson Handles
  (most lectures broadcast from UC Santa Barbara)
• Sergei Gukov, Link Polynomials and String Theory
• Matthias Kreck, Classification of Topological 4-manifolds
• Slava Krushkal, Open problems in 4-dimensional Topological Surgery
• Andras Stipsicz, Combinatorial Heegaard-Floer Homology
  
• Catharina Stroppel, Khovanov Homology and its Generalizations
• Peter Teichner, Whitney Towers and their Intersection Theory

The lecture series will serve two goals: One is to make accessible to a larger public those beautiful topological arguments that lead to the proof of the 4-dimensional Poincaré conjecture and the classification of many topological 4-manifolds. The other is to spell out the frontiers of knowledge and introduce some new tools that may be useful in pushing these frontiers.

All lectures series will be accompanied by homework and discussion sessions and we particularly invite young researchers to apply to this program.

Registration:

To apply for participation/funding, please submit the online application form by May 15, 2012. For any further inquiries please send an email to fourmanifolds2013@mpim-bonn.mpg.de.

Location: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn

 

We hope that bringing together specialists in different areas of mathematics would boost interesting discussions and plan to organise a few seminars and mini-courses covering, in particular, the following topics:

basics of model theory, algebraic geometry and model theory, Zariski geometries, Diophantine geometry and model theory, complex geometry and model theory, open problems in number theory and arithmetic geometry.

 

Those planning to attend include:

• M. Bays
• J-B. Bost
• E. Bouscaren
• F. Campana
• J.F. Carmona
• Z. Chatzidakis
• Delon
• J. Denef
• J. Derakhshan
• Yu.L. Ershov
• I.B. Fesenko
• H. Göral
• P. Habegger
• M. Hils
• G. Jones
• M. Kesälä
• J. Kirby
• P. Kowalski
• F. Loeser
• A. MacIntyre
• D. Marker
• A. Martin-Pizarro
• A. Medvedev
• R. Moosa
• B.Z. Moroz
• M. Morrow
• J. Pila
• A. Pillay
• F. Pop
• P. Speissegger
• S. Starchenko
• K. Tent
• I. Tomasic
• A.D. Yafaev
• U. Zannier
• B.I. Zilber

 

 

Conference on Interactions of model theory with number theory and algebraic geometry

The program will culminate in a week long research conference on "Interactions of model theory with number theory and algebraic geometry", June 11 - 15, organised by J-B. Bost, Z. Chatzidakis, and R. Moosa.

The main conference web page can be found here: http://www.math.uwaterloo.ca/~imna/.
 

Program:

Talks will be one full hour in length with an additional ten minutes for questions.

Monday, June 11, 2012
9:30 a.m.	Boris Zilber "On special sets and their geometry"
10:40 a.m.	Coffee Break I
11:10 a.m.	Françoise Delon "Pairs of algebraically closed fields"
12:20 p.m.	Lunch Break
2:50 p.m.	Martin Bays "Abelian integrals and categoricity"
4:00 p.m.	Coffee Break II
4:30 p.m.	Alice Medvedev "Geometric stability theory methods in algebraic dynamics"
Tuesday, June 12, 2012
9:30 a.m.	David Masser "Pell's Equation over Polynomial Rings"
10:40 a.m.	Coffee Break I
11:10 a.m.	Martin Orr "Families of abelian varieties with many isogenous fibres"
12:20 p.m.	Lunch Break
2:50 p.m.	Philipp Habegger "Non-Archimedean Approximations by Special Points"
4:00 p.m.	Coffee Break II
4:30 p.m.	Jonathan Kirby "Axiomatizing exponentiation"
Wednesday, June 13, 2012
9:30 a.m.	Antoine Ducros "Berkovich spaces, polytopes and model theory"
10:40 a.m.	Coffee Break I
11:10 a.m.	Immanuel Halupczok "New transfer principles between Q_p and F_p(t)"
12:20 p.m.	Lunch Break
2:50 p.m.	François Loeser "Non-archimedean Yomdin-Gromov lemma and applications to diophantine geometry"
4:00 p.m.	Coffee Break II
4:30 p.m.	Margaret Thomas "Counting algebraic points on definable sets"
Thursday, June 14, 2012
9:30 a.m.	Julia Gordon "Uniform in "p" bounds for orbital integrals"
10:40 a.m.	Coffee Break I
11:10 a.m.	Piotr Kowalski "Unlikely formal intersections"
12:20 p.m.	Lunch Break
2:50 p.m.	Tamara Servi "Quantifier Elimination and Rectilinearisation Theorem for quasi-analytic algebras"
4:00 p.m.	Coffee Break II
4:30 p.m.	Georges Comte "Grothendieck Ring of Semialgebraic Formulas and Motivic Real Milnor Fibres"
Friday, June 15, 2012
9:00 a.m.	Alexandru Buium "Differential modular forms: an overview"
10:10 a.m.	Coffee Break
10:30 a.m.	Moshe Kamensky "Tannakian formalism over fields with operators"
11:50 a.m.	Anand Pillay "Strict disintegratedness of generic Painleve equations"
1:00 p.m.	End of Conference

 

Workshop Website: http://www.aimath.org/ARCC/workshops/repnbranching.html

This workshop, sponsored by AIM, MPIM, and the NSF, will focus on branching laws for the restriction of a unitary representation to a subgroup.

The central problem of the workshop will be to find explicit branching laws. If the representations are discretely decomposable, then the explicit branching laws are often of a combinatorial nature. Here techniques from algebraic combinatorics will be extremely important and many interesting combinatorial problems should arise.

Another goal is elucidate the connection between the real and the p-adic situation. and to analyze how different methods in the real and p-adic case might lead to (conjecturally) similar phenomena. This pertains in particular to the branching laws with discrete spectrum.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for funding to particpate in this workshop has passed.