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 Lfunctions, 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 ArthurSelberg 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 socalled 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 localtoglobal 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 localtoglobal 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, HitchinSteinberg basis, and Poisson summation (To appear in the proceedings of the IMS Singapore workshop on the Langlands program, 2019). (Submitted)

J. Achter, S. A. Altuğ, L. Garcia, J. Gordon (w/ Appendix by W.W. Li and T. Rüd). Counting abelian varieties over finite fields via Frobenius densities (Submitted)

S. A. Altuğ, A. Shankar, I. Varma, K. Wilson. The number of quartic D_4fields ordered by conductor (Submitted)

S. A. Altuğ, Y. Chen. A Candidate Group with Infeasible Inversion. Advances in Cryptography  ASIACRYPT 2019, Editors: S. D. Galbraith and S. Moriai, LNCS 11922, 130, 2019, doi: 10.1007/9783030346218_11.

S. A. Altuğ. Beyond Endoscopy via the Trace Formula  III: The Standard Representation. Journal of the Institute of Mathematics of Jussieu, Published online on 12 November, 139, 2018, doi: 10.1017/S1474748018000427.

S. A. Altuğ. Beyond Endoscopy via the Trace Formula  II: Fourier transforms of orbital integrals and bounds towards the Ramanujan conjecture. American Journal of Mathematics, 139 (4), 863913, 2017, doi: 10.1353/ajm.2017.0023.

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), 124, 2017, doi: 10.2140/pjm.2017.286.1.

S. A. Altuğ. Beyond Endoscopy via the Trace Formula  I: Poisson Summation and Contributions of Special Representations. Compositio Mathematica, 151 (10), 17911820, 2015, doi: 10.1112/S0010437X15007320.

S. A. Altuğ, S. Bettin, I. Petrow, Rishikesh, I. Whitehead. A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials, Quarterly Journal of Mathematics, 65,11111125, 2014, doi: 10.1093/qmath/hat054.

S. A. Altuğ, J. Tsimerman. Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms, IMRN, 13, 34653559, 2014, doi:10.1093/imrn/rnt047.

S. A. Altuğ. Beyond Endoscopy via the Trace Formula, Ph.D. Thesis, 2013, Available online at
Expository Articles

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

S. A. Altuğ, A. Arslan. BanachTarski paradoksu. Matematik Dünyası, 2006II, 5156. (in Turkish)