MRI holds Special Year activities from time to time.

## Past Special Years

### 2011-2012

Interactions Between Ergodic Theory, Number Theory and Noncommutative Geometry

**Organized by: Vitaly Bergelson, James Cogdell, Avner Friedman, Roman Holowinsky, Wenzhi Luo, Henri Moscovici and Nimish Shah**

The Department of Mathematics at The Ohio State University, in conjunction with the Mathematics Research Institute, ran a program during the 2011-2012 academic year entitled Interactions Between Ergodic Theory, Number Theory and Noncommutative Geometry.

Collaborative work among the fields of Ergodic Theory and Number Theory, with applications to quantum chaos in particular, has recently led to new and exciting results. At the same time, surprising links with Noncommutative Geometry have been established via an interesting connection between the operator algebra formalism of quantum statistical mechanics and arithmetic.

We hoped to raise awareness and increase collaboration between researchers in all of these fields through a series of workshops to be held during the special program. Some financial support was available for young researchers and graduate students thanks to funding from the Mathematics Research Institute, the National Science Foundation and The Ohio State University.

You can find more information about the speakers and abstracts for this conference here.

## 2010-2011: Topology and Geometric Group Theory

**Coordinator: J. F. Lafont**

### Seminars

- Basic Notions seminar - for graduate students, met weekly. Organized by J. Fowler.
- Topology seminar - met weekly. Organized by C. Ogle.
- Geometric Group Theory seminar - met weekly. Organized by I. Leary and N. Broaddus

### Workshops

- Buildings, automorphisms of RAAGs, outer space (April 25-29) - organized by M. Davis. Speakers: J. Dymara, R. Charney, B. Remy, K. Vogtmann
- Torelli group, mapping class group, and Out(F) (May 2-6) - organized by N. Broaddus and H. Min. Speakers: M. Bestvina, B. Farb, M. Feighn, L. Mosher
- CAT(0) cubical complexes and systolic spaces (May 9-13) - organized by T. Elsner. Speakers: G. Niblo, M. Sageev, J. Swiatkowski, D. Wise
- Geometrical Methods in High Dimensional Topology (May 16-20) - organized by J. Fowler, J.-F. Lafont, I. Leary. Speakers: A. Bartels, T. Farrell, C. Guilbault, G. Yu

### Weekend Conferences

- Classifying Spaces for Families of Subgroups (Dec. 4-5). Speakers: D. Farley, M. Fluch, D. Juan-Pineda, P. Kropholler, C. Martinez-Perez, Q. Khan, D. Rosenthal.
- Buildings, automorphisms of RAAGs, outer space (April 30-May 1). Speakers: T. Januszkiewicz, B. Okun, P. Przytycki, S. Witzel, and A. Thomas
- Torelli group, mapping class group, and Out(Fn) (May 7-8). Speakers: T. Brendle, D. Kielak, T. Church, C. Pfaff, F. Gultepe, M. Bridson, D. Margalit, R. Wade, M. Sato, A. Putnam, M. Bestvina, J. Birman, J. Malestin, B. Farb, M. Day, P. Przytycki
- CAT(0) cubical complexes and systolic spaces (May 14-15). Speakers: G. Niblo, D. Osajda, P. Przytycki, M. Sageev, and J. Swiatkowski
- Geometrical Methods in High Dimensional Topology (May 21-22). Speakers: J. Fowler, D. Farley, S. Hoehn, P. Nowak, N. Rounds, and Z. Su

### Additional Activities

- Workshop on "Examples of Geometries" (May 23-28) - organized by I. Chatterji
- Conference on "Geometric Group Theory" (May 31-June 4) - organized by M. Davis, J.-F. Lafont
- Proceedings of the special year will be published. Editors are M. Davis, J.-F. Lafont, and I. Leary.

### Associated Faculty at OSU

- N. Broaddus, I. Chatterji, M. Davis, T. Elsner, J. Fowler, J.-F. Lafont, I. Leary, H. Min

## 2009-2010: Noncommutative Geometry and Applications to Number Theory

**Coordinator: Henri Moscovici**

As part of the activities related to the 2009-2010 MRI program "Noncommutative Geometry and Applications to Number Theory", several series of lectures for graduate students will be held. The lectures are devoted to topics related to the general theme of the program. They do not assume expert knowledge of the subject, and are designed to be accessible to a diverse audience of graduate students. These lectures will take place twice a week, each W&F between 3:30- 5 pm in MA 240, during the Autumn and Winter quarters, according to the schedule that follows.

**Alexander Gorokhovsky, University of Colorado****Theme**: Formal deformations and the algebraic index theorem**Period**: November 16th - December 11th, 2009**Topics covered:**

- Generalities on the deformation theory of associative algebras (Hochschild complex, statement of Kontsevich's formality theorem)
- Deformations of functions on symplectic manifolds (Fedosov quantization)
- Cyclic cohomology
- Lie algebra cohomology and Gelfand-Fuchs construction
- The algebraic index theorem of Nest and Tsygan
- Relation to the Atiyah-Singer index theorem

**Bahram Rangipour, University of New Brunswick, Canada****Theme**: Hopf cyclic cohomology and geometric applications**Period**: January 25th - February 12th, 2010**Topics covered:**

- Cyclic category
- Hopf algebras
- Symmetry in noncommutative geometry
- Hopf cyclic cohomology, examples and computational tools
- Spectral sequences
- Cup products in Hopf cyclic cohomology

**Matthias Lesch, University of Bonn, Germany****Theme**: Pseudo--differential calculus and geometric applications**Period**: February 15th - February 26th, 2010**Topics covered:**

- Parameter dependent pseudo--differential calculus, resolvent and heat expansions
- Noncommutative residue, Connes' Trace theorem on the relation between the Dixmier trace and the noncommutative residue
- Local Index Theorem
- Local Index Theory techniques on manifolds with singularities (conic singularities, cylindrical ends)
- Chern-Connes character and its limiting behavior

**Raphael Ponge, University of Tokyo, Japan****Title**: The Local Index Formula in Noncommutative Geometry and Applications in Transverse Geometry**Period**: March 1st - March 12th, 2010**Topics covered:**

This series of 4 lectures will be devoted to presenting the framework for the local index formula in noncommutative geometry, which was derived by Alain Connes and Henri Moscovici in the mid-90s. In the setting of noncommutative geometry the role of manifolds is played by spectral triples (A,H,D), where H is a Hilbert space, A is an algebra represented in a Hilbert space H and D is an unbounded selfadjoint operator on H commuting with A modulo bounded operators. Given a spectral triple (A,H,D) the operator D gives rise to a Fredholm index with coefficients in the K-theory of A. This index map can be computed by pairing the K-theory of A with a cocyle in the (periodic) cyclic cohomology of A, the celebrated Chern-Connes character, which was the main motivation of Alain Connes for inventing cyclic cohomology. The Chern-Connes character is hard to compute. However, Connes-Moscovici exhibited a representative, the so-called CM cocycle, which is given by formulas which are local in the sense of noncommutative geometry, i.e., they involve an analogue for spectral triple of the noncommutative residue trace of Guillemin and Wodzicki. This yields the local index formula in noncommutative geometry. It allows us to easily recover the local index formula of Atiyah-Singer for Dirac operators, but it also allows us to deal with new geometric settings. This will be illustrated by explaining how this local index formula together with the use of Hopf cyclic cohomology enables us to compute the index of transversally elliptic operators on foliations.

## 2008-2009: Analytic and Algebraic Geometry: Multiplier Ideals

**Coordinators: Herbert Clemens and Jeffery McNeal**

A particular technique from PDE's was brought to bear on problems in algebraic geometry, with remarkable success.

The basic version of this technique consists of establishing certain crucial inequalities, that do not universally hold, by studying the set of functions or operators which "tame'' the inequality, i.e. the set of multipliers for a particular inequality. In many situations, it turns out that the set of multipliers possesses a remarkable number of unexpected properties, for example they form an ideal. These properties then allow one to formulate conditions, of both algebraic and geometric character, which give situations where the original inequality holds.

In the hands of algebraic geometers, this basic idea has been expanded upon and greatly abstracted. And it has been successfully applied to many questions of current interest, including the invariance of plurigenera, the construction of Kähler-Einstein manifolds on algebraic manifolds, and Fujita's conjecture, among others. These successes, however, point to even more possible uses of the multiplier ideal notion. By bringing together the analysts and algebraic geometers who work in directions close to multiplier ideals -- but who speak about the concept in different languages and who rarely interact with one another -- we expect that the range of application for this general idea will be greatly broadened and enrich both the algebraic and analytic sides in the process. More math details

Mechanics of the program.

The program is designed to follow up on the Park City Mathematics Institute (PCMI) which took place in the summer of 2008 on the same mathematical topic, and it will incorporate extended interaction with the FRG in Algebraic Geometry at the University of Michigan.

We first outline the research and graduate education aspects of the Park City program of 2008. The mathematicians listed below gave a week-long course for graduate students (and their colleagues) at PCMI2008:

**Bo Berndtsson**, Chalmers University of Technology:*Introduction to L2-methods in complex geometry***John D'Angelo**, University of Illinois at Urbana-Champaign:*Real and complex varieties and orders of contact***Jean-Pierre Demailly**, Université de Grenoble**Christopher Hacon**, University of Utah:*Finite generation of the canonical ring***János Kollár**, Princeton University**Robert Lazarsfeld**, University of Michigan:*Introduction to multiplier ideals***Mircea Mustata**, University of Michigan:*Resolution of singularities.***Dror Varolin**, SUNY at Stony Brook:*Invariance of plurigenera. Skoda's theorems.*

Each PCMI course was accompanied by a problem session for less advanced graduate students and research working groups (RWG) for more advanced ones. During the three weeks of PCMI2008, senior mathematicians (possibly including the ones above) also gave seminar-style lectures on topics connected to the courses above which are on the research frontier.

The Special-Topic Cross-disciplinary Research Year in Analytic and Algebraic Geometry at Ohio State University sent an 8-person delegation of researchers and graduate students as full-program participants at the IAS Park City Summer Institute in Analytic and Algebraic Geometry (July 6-26, 2008), organized five one-week follow-up workshops at Ohio State on research areas highlighted by the PCMI Summer Institute, and, in early 2010, will sponsor a subsequent capstone lecture series "Finite generation of the canonical ring" by Christopher Hacon reviewing recent progress on that central problem of common interest of analytic and algebraic geometers.

The focus of the five workshops was to develop a future generation of researchers who dominate both analytic and algebraic methods, a cross-disciplinary competence currently enjoyed by only a very few of those working in either field. These candidates for the future are advanced graduate students--our workshop participants were roughly evenly split between those on the analytic and algebraic side.

In addition, as a special initiative for 6 of our own advanced graduate students working in the area, Clemens and McNeal led a year-long, 4-hour-per-week seminar centered on deepening their understanding of multiplier ideals as a tool in both the analytic and algebraic theory, as well as applications to vanishing, basepoint-free and non-vanishing theorems, and finally to the results of Demailly and his school giving remarkable analytic 'approximations' to the answers to classical algebraic questions such as the unirationality of projective manifolds of Kodaira-dimension minus infinity. The five week-long workshops for graduate students were:

- Oct. 6-10, 2008:
*Introduction to the minimal-model program*(J. Kollár) - Nov. 17-21, 2008:
*Del-bar methods*(B. Berndtsson) - Jan. 5-9, 2009:
*L2-methods in complex geometry*(D. Varolin) - Feb. 23-27, 2009:
*Multiplier Ideals and their Applications*(R. Lazarsfeld. M. Mustata) - May 11-15, 2009:
*Analytic aspects of multiplier ideals*(M. Paun, D. Kim, D. Varolin)

All 76 graduate student participants at PCMI2008 were invited to the 5 workshops. Average graduate-student participation in the workshops was about 15. The last two workshops were the largest, each with participation of over 20 graduate students from around the U.S. (and even a couple of graduate students from abroad).

## Workshops and Conferences

### Dynamics on Homogeneous Spaces and Number Theory

September 12 - 16, 2011

Speakers: Tim Austin, Don Blasius, S.G. Dani, Manfred Einsiedler, Alex Gamburd, Alex Gorodnik, Patrick Ingram, Dmitry Jakobson, Florent Jouve, Dubi Kelmer, Alex Kontorovich, Par Kurlberg, Philippe Michel, Amir Mohammadi, Shahar Mozes, Paul Nelson, Barak Weiss, Peng Zhao, Tamar Ziegler

### Noncommutative Geometry: Multiple Connections

May 7 - 18, 2012

The first week of the workshop will emphasize the links between Noncommutative Geometry and Number Theory while the second week will emphasize the links between Noncommutative Geometry and Ergodic Theory. Please send any questions related to the conference to Roman Holowinsky via e-mail.