The School of Mathematics is an international center of research and postdoctoral training in many diverse aspects of mathematics including pure mathematics, theoretical computer science, mathematical physics and applied mathematics.

Fifty to sixty mathematicians are invited to the School each year to study with the Faculty and to pursue research projects of their own. A small number of memberships for a longer period of time are also available. Funding for candidates comes from a variety of sources. Some mathematicians are funded by the Institute, others receive financial aid from their home institutions, and a portion receive grants from governments or foundations.

Term dates for the 2014-2015, 2015-2106, 2016-2017 and 2017-2018 years are as follows: **2014-2015** - term I, Monday, September 22 to Friday, December 19, term II, Monday, January 12 to Friday, April 10, 2015; **2015-2016** - term I, Monday September 21 to Friday, December 18, term II, Monday, January 11 to Friday, April 8, 2016; **2016-2017** - term I, Monday September 19 to Friday, December 16, term II, Monday, January 16 to Friday, April 14, 2017; **2017-2018** - term I, Monday September 25 to Friday, December 22, term II, Monday, January 15 to Friday, April 13, 2018.

Please note that the School's term II begins and ends one week later than the rest of the Institute.

The School frequently sponsors special programs. However, these programs comprise no more than one-third of the membership so that each year a wide range of mathematics is supported.

Several years ago the School established the **von Neumann Fellowships**. Up to eight of these fellowships will be available for each academic year. To be eligible for the von Neumann Fellowships, applicants should be at least five but no more than fifteen years following the receipt of their Ph.D.

The **Veblen Research Instructorship** is a three-year position which was established in partnership with the Department of Mathematics at Princeton University in 1998. Three-year instructorships will be offered each year to candidates in pure and applied mathematics who have received their Ph.D. within the last three years. The first and third year of the instructorship will be spent at Princeton University and will carry regular teaching responsibilities. The second year will be spent at the Institute and dedicated to independent research of the instructor's choice.

Claire Voisin, Institut de Mathématiques de Jussieu, will be the School's Distinguished Visiting Professor during the **2014-15** academic year. Professor Voisin will lead a special program on** The Topology of ****Algebraic Varieties**.

The topology of algebraic varieties is traditionally understood in two different senses: either one considers real or complex algebraic varieties, and then one studies (mostly via Hodge theory) the topology of (the set of real or complex points of) this algebraic variety, endowed with the Euclidean topology, or one studies this algebraic variety endowed with the Zariski or étale topology with the help of various cohomology theories (étale, de Rham...)

The second viewpoint is closer in spirit to the theory of motives, as it includes the various comparison isomorphisms, and because it does not use non-algebraic data. The complex point of view is, in fact, also very well-adapted to the study of algebraic cycles, in view of the Bloch-Beilinson conjectures.

This program intends to bring a mix of people interested in various aspects of the subject: Motives, K-theory, Chow groups, periods, fundamental groups.

During the **2015-16** academic year, the School will have a special program on **Geometric Structures on ****3-manifolds**, and Ian Agol of the University of California, Berkeley, will be the Distinguished Visiting Professor.

Thurston proposed the classification of geometric structures on 3-manifolds. While the spectacular Geometrization Theorem classified the geometric structures on 3- manifolds with compact isotropy group, i.e. locally homogeneous Riemannian metrics, there is a cornucopia of other fascinating structures such as contact structures, foliations, conformally flat metrics and locally homogeneous (pseudo-) Riemannian metrics.

The goal of this program is to investigate these other geometric structures on 3-manifolds and to discover connections between them. Additionally, it is important to forge connections between geometric structures on 3-manifolds and other geometric constructs, such as gauge theory, PD(3) groups, minimal surfaces, cube complexes, geometric structures on bundles over 3-manifolds, and strengthened structures such as taut foliations, tight contact structures, pA flows, convex projective structures and quasi-geodesic foliations. Many of these do not even have a conjectural classification (in terms of topological restrictions and moduli), and specific examples are still being constructed.

Two special program workshops will be held during term I. The first workshop, "Geometric structures on 3-manifolds", will take place during the week of October 5. The second workshop on "Flows, foliations and contact structures", will be during the week of December 7-11, 2015.

The goal of the October workshop will be to explore the classification of geometric structures on 3-manifolds, broadly interpreted. Constraints and consequences for the topology of manifolds with a given geometric structure, and connections between different geometric structures will be investigated.

During the **2016-17** academic year, the School will have a special program on **Homological Mirror Symmetry**. Paul Seidel of MIT will be the Distinguished Visiting Professor. Maxim Kontsevich, IHES, will be attending the program for one month during each of the fall and spring terms (from mid-October to mid-November, and for the month of February). Denis Auroux, UC Berkeley, will be attending for the spring term.

Homological Mirror Symmetry (HMS) was initiated by Kontsevich. It benefits from a close relationship with string theory, and has developed into a powerful and versatile idea. During the program, we will consider the core conjectures of HMS, and its role as a framework within which wider questions from mirror symmetry and other parts of mathematics can be studied. This is still a developing subject, and the program is open to a variety of approaches and viewpoints.

The intention is that the fall term will have a greater focus on the core building blocks of HMS as currently understood: the A-model theory (Lagrangian submanifolds, holomorphic curves and their generalizations), the B-model theory (derived categories in algebraic geometry), and mathematical interpretations of the Strominger-Yau-Zaslow approach, including the Gross-Siebert program. Specific questions of interest include: the role of singular Lagrangian submanifolds (such as Lagrangian skeleta); the effect of instanton corrections on the construction of mirror manifolds; and the structure of wrapped Fukaya categories. We will also consider the interplay between the various algebraic notions that appear in HMS.

The spring term would widen the focus, allowing space for emerging interactions between HMS and other areas. Examples are the theory of Special Lagrangian submanifolds, tropical geometry, and non-archimedean analytic geometry, as well as sheaf-theoretic methods. We also intend to look at applications of ideas from homological mirror symmetry to specific classes of manifolds, such as complex symplectic manifolds and cluster varieties.

There will be two workshops during the special program. The term I workshop, "homological mirror symmetry: methods and structures", will be held November 7-11, 2016. The term II workshop, "homological mirror symmetry: emerging developments and applications", will be held March 13-17, 2017.

For the **2017-2018** academic year, the School will have a special program on **Locally Symmetric Spaces: Ananlytical and Topological Aspects**. Akshay Venkatesh of Stanford University will be the Distinguished Visiting Professor.

The topology of locally symmetric spaces interacts richly with number theory via the theory of automorphic forms (Langlands Program). Many new phenomena seem to appear in the non-Hermitian case (e.g., torsion cohomology classes, relations with mixed motives and algebraic K-theory, derived nature of deformation rings). One focus of the program will be to try to better understand some of these phenomena.

>The Institute for Advanced Study is committed to diversity and strongly encourages applications from women and minorities.

The School is grateful for the continued support of its programs by the National Science Foundation.

APPLICATION PROCEDURE: Apply Online

APPLICATION DEADLINE: December 1.