Research

A considerable part of my research is focused around analytic number theory and automorphic forms/representations. On the analytic number theory side I work with L-functions, analyzing their (mostly analytic) properties (zeros, poles, special values, sizes of these functions in certain domains, variation of such properties in families...), and on the automorphic side with automorphic forms/representations, trying to understand their (mostly spectral) properties (functorialty, Ramanujan conjectures, metaplectic forms, families of automorphic forms/representations..). A main goal of my research goes into trying to understand the interaction between the two areas in the spirit of beyond endoscopy (and related topics), and how the Arthur-Selberg trace formula can be combined with analytic number theory to give explicit results on the automorphic spectrum. 

I also am interested in statistical problems related to arithmetic objects (e.g. "arithmetic statistics"). A main theme in this field is to (asymptotically) count the number of certain arithmetically interesting objects in a "family" (ordered in a preferred way). A main theme in the area is the so-called Malle's conjectures (and refinements), which gives a recipe for the asymptotic order of growth of the family. Following Bhargava and Kedlaya, one could even hope to predict the leading term in the asymptotic count from the local properties of the objects in the family (a local-to-global principal). This, however, turns out not always to be the case and predictions may fail in certain cases. My research in this area focuses on understanding if and why this local-to-global phenomenon fails, and to prove it in cases when it holds.

More recently, I also got interested in certain aspects of cryptography mostly related to elliptic curves and their invariants. Many encryption schemes make use of trapdoor functions/groups, where performing an operation (like exponentiation, or multiplication) is feasible but the inverse of the operation is not. I am currently working a project to study group structures where finding the inverse of a group element is not feasible. 

 

Publications

Research Articles

  • S. A. Altuğ. Beyond endoscopy, Hitchin-Steinberg basis, and Poisson summation (To appear in the proceedings of the IMS Singapore workshop on the Langlands program, 2019). (In preparation)

 

  • S. A. Altuğ. Orbital Integrals and measure conversions (Appendix to "Elliptic curves, random matrices, and orbital integrals" by J. Achter and J. Gordon), Pacific Journal of Mathematics, 286 (1), 1-24, 2017, doi: 10.2140/pjm.2017.286.1.

 

 

 

 

 

Expository Articles

 

  • S. A. Altuğ. Nedir bu Langlands Programı? To appear in Matematik Dünyası 2020. (In preparation)