WIN3: Women in Numbers 3 (14w5009)

Arriving Sunday, April 20 and departing Friday April 25, 2014

Organizers

Ling Long (Iowa State University)
Rachel Pries (Colorado State University)
Katherine Stange (University of Colorado, Boulder)

Objectives

Applications

There are a dozen remaining slots available for participation in WIN3. Women in number theory who are interested in joining the conference should complete the short application process at the link below by October 15.

https://docs.google.com/forms/d/1UyioLacI3oudz9ZfzBOvMCx2YaWRdtMnKnqZANj5ouo/viewform

Please be aware that we expect to have more applicants than available slots. Priority will be given to advanced graduate students, recent PhDs, and applicants whose research interests align best with project topics.


Number theory is a fundamental subject with connections to a broad spectrum of mathematical areas including algebra, arithmetic, analysis, topology, cryptography, and geometry. The vitality of this field depends on the full participation of all number theorists. The number of female number theorists is steadily growing, especially at the graduate student and postdoctoral level; however, there are still relatively few women reaching high profile positions. Visibility of women at international workshops and conferences tends to be low, with percentages of female speakers rarely exceeding 20%. For instance, in the last three cycles of CNTA and ANTS conferences, only 4 of the 41 plenary speakers were women. The lack of female leaders in the area is an issue that perpetuates itself, since it has repercussions in attracting and training the next generation of female mathematicians.

The proposed workshop aims to promote research and leadership among female number theorists within a supportive environment. Many participants told us that the previous WIN conferences ignited their careers. For the graduate students, it was eye-opening to see the level of intensity of the projects. Junior faculty, both postdocs and assistant professors, seemed to gain the most from the WIN experience. Several mentioned that the WIN projects introduced them to completely new directions for research. Faculty with high teaching loads appreciated the chance to focus on research. For group leaders, it was a formative experience to find new research problems and direct a research group.

The specific goals of the workshop are:

  • to generate research in significant topics in number theory;
  • to broaden the research programs of female number theorists, especially pre-tenure;
  • to train female graduate students in number theory, by providing experience with collaborative research and the publication process;
  • to strengthen and extend a research network of potential collaborators in number theory and related fields;
  • to enable female faculty at small colleges to participate actively in research activities including the training of graduate students; and
  • to highlight research activities of women in number theory.

The focus of the workshop is on supporting new research collaboration within small groups. Before the workshop, each participant will be assigned to a working group according to her research interests. Each group will have one or two leaders chosen for their skill in both research and communication. Prior to the conference, these leaders will design projects and provide background reading and references for their groups. At the conference, the project leaders will give morning lectures, but afternoons and evenings will be dedicated to the working groups. Project leaders will direct their group's research effort and provide mentorship. At the end of the week, members of each research group (typically postdocs or graduate students) will describe their group's progress on the research problems as well as future directions for the work.

The scientific themes of the workshop will include L-functions, automorphic forms, Shimura varieties, abelian varieties, and quadratic forms. These intertwined focal areas are all central themes in modern number theory, ranging from topics in analytic and combinatorial number theory to topics centered on algebraic number theory and arithmetic geometry. The broad scope of the WIN conferences encourages the leaders to find connections between their topics. For example, at WIN2, unexpected synergy emerged between several groups focused on surfaces (reduction of abelian surfaces, modular forms on K3 surfaces, zeta functions of surfaces, etc).

The topic of L-functions originated in the study of the Riemann zeta function and includes a Millenium Problem, namely, the Riemann hypothesis. Recent progress in this area includes a new connection with random matrix theory. Analytic properties of L-functions have already led to profound applications in areas such as arithmetic statistics, which was the main topic of the spring semester at MSRI in 2011. Under the influential Langlands' program, the L-functions of Galois representations arising from algebraic varieties or motives are related to L-functions of modular forms and more generally automorphic forms. The most renowned example is the proof of the Taniyama-Shimura-Weil conjecture which relates the L-functions of elliptic curves over the field of rationals to L-functions of Hecke modular forms. A distinguished application of this profound relation is the proof of the Sato-Tate conjecture which predicts a certain statistical distribution of arithmetic data arising from an elliptic curve. In the area of automorphic forms, there are exciting breakthroughs in the theory of partition functions, a classical topic in combinatorial number theory, using mock modular forms.

Underlying many of the algebraic topics is the concept of a moduli space which parametrizes structures of a given type. The recent discoveries of Bhargava about parametrization of quadratic forms are transforming the study of number fields and have led to a proof of another Millenium Problem, the Birch and Swinnerton-Dyer conjecture, for a positive proportion of elliptic curves. Shimura varieties are higher dimensional analogues of modular curves, which parametrize isomorphism classes of elliptic curves with additional structure. Some Shimura varieties can be realized as moduli spaces of certain abelian varieties. Due to their special properties, Shimura varieties form a natural realm of examples for testing open conjectures like the Langlands' programs. Likewise, abelian varieties provide an important testing ground for the Langlands' program and Tate's conjecture. In addition, the arithmetic properties of Jacobians, Galois representations, and torsion points of abelian varieties continue to fascinate number theorists. The structure of lattices plays an important role in studying ranks of abelian varieties and also emerges in the study of lattice-based cryptography, sphere packing, coding theory, and a class of modular forms called theta series. Finally, concrete problems about rational points on abelian varieties over number fields and finite fields are useful for applications to cryptography.



We aim to make a few improvements based on participant feedback from the previous WIN conferences. The schedule will include fewer lectures, in order to give more time for group work. We will increase the background requirements for graduate student participants, to ensure that they can be fully involved in the research projects. More social activities will be planned to spark more interaction between the research groups. We will also include discussions on job searches, balance of work and personal life, and grant applications.