## International Max Planck Research School on Moduli Spaces

The International Max Planck Research School on Moduli Spaces (IMPRS) is the graduate program of the Max Planck Institute for Mathematics in Bonn (MPIM) jointly offered with the University of Bonn. It is part of the Bonn International Graduate School in Mathematics (BIGS-Mathematics). The IMPRS is sponsored by the Max Planck Society.

**Program**

The academic training program of the IMPRS Moduli Spaces consists of courses, mini-courses, seminars and special activities, complementing the Ph.D. program of the University of Bonn.

PhD positions are available in the direction of moduli spaces and related fields for candidates with excellent Master or Bachelor degrees in the areas of research advised by one of the MPIM directors:

Further research directions are available with other IMPRS board members and with other faculty at MPIM, such as Number Theory (Blomer, Harder, Moree), Arithmetic Geometry (**Faltings**), Modular Forms (**Zagier**), Teichmueller Theory (Hamenstaedt), Complex Algebraic Geometry (Huybrechts), Symplectic Geometry (Bottman), Global Analysis (Mueller), Topology (Barthel, Ozornova, Ray, Schwede ), Geometric Topology (**Avramidi**, Stadler), Representation Theory (Stroppel) and Mathematical Physics (Klemm, Blohmann).

In addition, faculty members at the University of Bonn can also serve as advisors.

The Ph.D. should be finished within 3-4 years. The program is in English, dissertation and dissertation defense are in English too if desired. In exceptional cases, continuous supervision or graduating from the student's home university is possible.

**Application**

Students from all countries can apply, please follow the Application Instructions.

Requirements: German Diploma, Master of Science or equivalent. Depending on academic qualification, admission is for a qualifying year or directly for the Ph.D. work. Proof of proficiency in English (TOEFL-test or equivalent qualification).

**Application Deadline**

- November 30 for the summer term
- April 15 (new) for the winter term.

**IMPRS Coordinator**

Dr. Christian Kaiser

room no. B 26

kaiser[at]mpim-bonn[dot]mpg[dot]de

**Contact address for further communication**

imprs[at]mpim-bonn[dot]mpg[dot]de

**Mail Address**

Max-Planck-Institute for Mathematics

IMPRS-ModuliSpaces

P.O. Box 7280

53072 Bonn

Germany

## Application for the IMPRS Moduli Spaces

**Application**

Since 1st of December 2011 the IMPRS shares an online application

website with its partner Bonn International Graduate School Mathematics:

http://www.bigs-math.uni-bonn.de/application/online-application/

In addition to CV, statement of purpose, degrees and academic records,

proficiency in English and 2 letters of recommendation, which you should

submit at the online application website, we ask you for a copy of your

Bachelor's or Master's thesis if available (electronic version

prefered). Send this to:

**Email**

imprs$@$mpim-bonn$.$mpg$.$de

**Address**

Max-Planck-Institute for Mathematics

IMPRS-ModuliSpaces

P.O. Box 7280

D-53072 Bonn

Germany

**Important Notes**

**Visa**: Students from most countries outside the European Union will have to apply for a visa to study in Germany. For details, ask the nearst German Embassy or Consulate, advisably before you send the application. Usually you must submit a letter from us stating that you are accepted, and a proof of financial support (e.g. award of IMPRS scholarship).

- The
**deadline**for the summer term is November 30. The deadline for the winter term is April 15 (new). Applications will be considered by a Committee. Expect an answer within four weeks after the deadline.

## IMPRS Working Areas

**The topics studied in the IMPRS Moduli Spaces include:**

- Shimura Varieties, Locally Symmetric Spaces
- Moduli Spaces of Principal Bundles on Curves
- The Moduli Space of Riemann Surfaces

Combinatorics of Moduli Spaces of Curves; Homological Aspects; Teichmüller Theory and Deformations of Hyperbolic 3-Manifolds

- Modular Forms
- Bounded Geometry

Gromov Precompactness Theorem; Quasi-isometric Rigidity; Spaces of Nonpositive Curvature; Spaces of Maps; Isospectral Manifolds

- Semisimple Frobenius Manifolds and Quantum Cohomology
- Non-commutative Geometry and Moduli Spaces
- Moduli Spaces in Floer Theory
- Strings and Conformal Field Theory

More detailed information is coming soon.

**The Theme "Moduli Spaces"**

Already Riemann was able to count the "number of free parameters" defining a Riemann surface of fixed genus. In the formally precise terminology of modern algebraic geometry, he was counting the dimension of the moduli space of Riemann surfaces. Traditionally, one studies these spaces via deformation theory (degenerations and compactifications - this often involves interesting combinatorics) or by uniformization theory (Teichmüller space).

Very surprisingly, these and other more complicated moduli spaces (e.g., of vector bundles, of stable maps, etc.) were discovered in recent years to play an important role in mathematical physics, especially in the theory of quantum strings, which strives to the unification of quantum field theory and the theory of gravity.

Currently, moduli spaces are studied from three perspectives, which we will now describe in more detail.

Initially, moduli spaces were introduced and investigated in order to understand better the objects they parametrize. It turned out that moduli spaces can be used as important tools in proofs of classical results. Often it requires a deep insight to see how a moduli space can be employed to obtain a result that only deals with individual objects. The 1983 proof of the Mordell and Shafarevich conjectures by G. Faltings is a famous example; here the moduli space of abelian varieties plays a decisive role. As it was thus realized that moduli spaces are significant mathematical objects, mathematicians started to study them in their own right. Their intrinsic beauty also contributed to the flourishing of this subject.

Another application of moduli spaces is to consider them as a tool to construct interesting varieties. There are various arithmetic conjectures about varieties that are inaccessible in general. By making use of the fact that a moduli space is parametrizing certain structures it becomes possible to prove some of these conjectures for these special varieties. For instance we can attach an $L$-function to an algebraic variety over $\mathbb Q$ and certain conjectures about its analytic properties can be stated, but we can prove them only in very few cases, and these cases are usually modular varieties. The most spectacular example is the theorem of Wiles that an elliptic curve over $\mathbb Q$ in some sense occurs in a suitable modular curve. These modular curves are the simplest examples of Shimura varieties.

In global Riemannian geometry one considers spaces of isometry classes of complete Riemannian manifolds, defined by specific bounds on the geometry. The questions of interest here include the triviality of such spaces (rigidity) and the structure of their boundaries (e.g., Gromov-Hausdorff convergence), among others. One also studies minimal submanifold immersions or isospectral metrics by looking at their moduli spaces.

The second perspective is to view moduli spaces as a way of generating new geometries. As an example from differential geometry, we mention the Weil-Petersson metric which is a natural Kähler metric on moduli spaces of Calabi-Yau manifolds. In the case of three-dimensional Calabi-Yau manifolds this metric satisfies additional properties which lead to the notion of a special Kähler metric. This geometry was discovered first by physicists in an entirely different context. There it arose as a constraint of extended supersymmetry in four-dimensional supergravity Yang-Mills theories. The link between these two appearances of special Kähler geometry is provided by string theory. The low-energy limit of a ten-dimensional superstring theory compactified down to four dimensions with a Calabi-Yau threefold is identical to the supergravity Yang-Mills theory.

Another example is furnished by the so-called Frobenius manifolds. Physicists discovered that moduli spaces of topological and conformal field theories come up together with a new structure: their tangent vectors can be multiplied as elements of an algebra. After a suitable axiomatization, it was realized that several other constructions lead to the same structure. In particular it emerges on the unfolding spaces of isolated singularities (Kyoji Saito et al.), and the cohomology spaces of certain differential graded algebras (S. Barannikov and M. Kontsevich).

The most surprising and much studied phenomenon, again predicted by physicists, is the famous mirror symmetry. It is expressed in the existence of its morphisms between Frobenius manifolds given by totally different constructions, for example, quantum cohomology (genus zero Gromov-Witten invariants) and extended moduli spaces of Calabi-Yau manifolds.

Mirror symmetry is closely connected with studying degenerations which lie at the boundary of a moduli space. It was recognized recently that moduli spaces may have "invisible" boundary strata which parametrize non-commutative varieties in the sense of Alain Connes. This is a very promising new direction of research.

Finally, the third perspective originates from a fundamental idea about constructing invariants of geometric spaces. The idea is to assign to a geometric space a moduli space and to prove that standard invariants of the moduli space are actually invariants of the geometric space.

As an example of this we mention Donaldson and Seiberg-Witten invariants of 4-manifolds. The gauge-theoretic moduli spaces arising in these theories are spaces of solutions of certain partial differential equations defined in terms of geometric objects such as connections and spinors, modulo a large group of gauge symmetries. These moduli spaces provide invariants of smooth 4-manifolds, and by dimensional reduction and gluing formulae, invariants of 3-manifolds in the form of Floer homology theories. An intensive study of gauge theoretic moduli spaces was initiated in the early eighties, with Donaldson's famous result on obstructions to the existence of smooth structures on certain classes of 4-manifolds. More recent results have uncovered deep connections between moduli spaces of Donaldson and Seiberg-Witten invariants of 4-manifolds, and between Seiberg-Witten and Gromov-Witten invariants of symplectic 4-manifolds.

In the study of all kinds of moduli spaces from all of these perspectives, explicit formulas are often given by identities involving modular functions (i.e., functions on moduli spaces). The appearance of modular forms is ubiquitous in the theory of $L$-functions, but very surprisingly, they are also involved in many identities arising from mirror symmetry, vertex algebras and statistical mechanics.

## Financial and other support for IMPRS students

A few doctoral positions are available. Ph.D. students receive a Ph.D. support contract of the Max Planck Society for three

years, based on the German TVöD 13 scale (75%). Extension to a 4th year is possible.

The Max Planck Institute will assist in finding housing (housing[at]mpim-bonn[dot]mpg[dot]de). All information about Visa/Residence permit Ph.D. students will obtain by the administration (personnel[at]mpim-bonn[dot]mpg[dot]de).

For a sketch of the approximate living expenses, have a look at some information provided by BIGS.

The IMPRS provides support for students to visit specialists in their fields at German or other European institutions with whom the IMPRS is connected.

There is also a limited budget for participation in conferences or workshops at these partner institutions, or for short invitations of specialists from these institutions to Bonn.

## IMPRS Courses

**University of Bonn**

- Algebra

- Number Theory

- Algebraic Groups

- Algebraic and Arithmetic Geometry

- Modular Forms

- Differential Geometry

- Algebraic Topology

- Complex Analysis

- Global Analysis and

- Mathematical Physics

For a complete list see the "Vorlesungsverzeichnis" of Bonn University.

**Max Planck Institute for Mathematics**

- Oberseminar Topology

- AG Homotopy Theory (jointly with the Universities Wuppertal, Düsseldorf and Bochum)

- AG Complex Analysis (jointly with Wuppertal University)

- Number Theory Seminar

- Algebraic and Complex Geometry Seminar

- Seminar on Algebra, Geometry and Physics

- Gauge Theory Seminar

as well as further activities and seminars on timely topics.

For a complete list have a look at MPIM's Web-Site

**IMPRS**

The IMPRS will install two new seminars specifically for the graduate students:

- a reading course, where the students work with some advisor through material necessary for, or supplementary to, their research work;

- a seminar where the students present their own work and/or discuss ideas with the other graduate students (who might be working in other fields).

For those students who first do their qualifying year, a crash course will be organized.

The IMPRS and BIGS will also organize a series of lectures by invited speakers on recent research in the fields of the graduate students.

Each graduate student is expected to participate in the weekly "Mathematical Colloquium", where invited speakers give talks about their latest work.

Apart from the scientific program, the IMPRS students can participate in the language courses and the social program the BIGS will offer.

## IMPRS Board Members

Prof. Dr. W. Ballmann (U. Bonn and MPIM):

- Differential Geometry

Prof. Dr. V. Blomer (U. Bonn):

- Number Theory

- Arithmetic Algebraic Geometry

Prof. Dr. D. Gaitsgory (MPIM):

- Geometric Langlands Theory

Prof. Dr. U. Hamenstädt (U. Bonn):

- Differential Geometry, Teichmüller Theory

Prof. Dr. G. Harder (U. Bonn and MPIM):

- Number Theory

Prof. Dr. D. Huybrechts (U. Bonn):

- Complex Geometry

- Mathematical Physics: String Theory, Algebraic Geometry

Prof. Dr. Werner Müller (U. Bonn):

- Global Analysis, Locally Symmetric Spaces

- Topology

Prof. Dr. P. Scholze (U. Bonn and MPIM):

- Arithmetic Algebraic Geometry

Prof. Dr. S. Schwede (U. Bonn):

- Topology

Prof. Dr. Catharina Stroppel (U. Bonn):

- Representation Theory, Topology, Category Theory

Prof. Dr. P. Teichner (UC Berkeley and MPIM):

- Topology

Prof. Dr. D. Zagier (MPIM and Collège de France):

- Modular Forms, Number Theory

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