Research interests: number theory, analytic number theory, Diophantine geometry

Here is a list of my recent preprints and publications.

• Points of bounded height on certain subvarieties of toric varieties (with M. Pieropan), arXiv:2403.19397, 22 pages
• Rational points near manifolds, homogeneous dynamics, and oscillatory integrals (with R. Srivastava and N. Technau), submitted
• Generalised quadratic forms over totally real number fields (with T.  D. Browning and L. B. Pierce), accepted for publication in Journal of the Institute of Mathematics of Jussieu
• Hyperbola method on toric varieties (with M. Pieropan), Journal de l'École polytechnique - Mathématiques 11 (2024), 107 – 157
• Density of rational points near/on compact manifolds with certain curvature conditions (with S. Yamagishi), Adv. Math 403 (2022), Paper No. 108358, 36pp
• On prime values of binary quadratic forms with a thin variable (with P. Lam and S. Xiao), J. London Math. Soc. 102 (2) (2020), 749 - 772
• Diophantine inequalities for generic ternary diagonal forms, Int. Math. Res. Not., no.11 (2020), 3396–3416
• On the algebraic Brauer classes on open degree four del Pezzo surfaces (with J. Jahnel), J. Number Theory 203 (2019), 376 - 427
• On the frequency of algebraic Brauer classes (with J. Jahnel), Canad. Math. Bull. 62 (2019), 551 - 563
• Sarnak's saturation problem for complete intersections (with E. Sofos), Mathematika 65 (2019), no. 1, 1 - 56
• Strong approximation and a conjecture of Harpaz and Wittenberg (with T. D. Browning), Int. Math. Res. Not. 14 (2019), 4340 - 4369
• On integral points on degree four del Pezzo surfaces (with J. Jahnel), Israel J. Math. 222 (2017), no. 1, 21 - 62
• Counting rational points on hypersurfaces and higher order expansions, J. Number Theory 173 (2017), 332-370
• Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme (with J. Jahnel), Annales de l'Institut Fourier 67, no.4 (2017), 1783-1807
• Representations of integers by systems of three quadratic forms (with L. B. Pierce and M. M. Wood), J. London Math. Soc. (3) 113, no.3 (2016), 289-344
• On the number of certain Del Pezzo surfaces of degree four violating the Hasse principle (with J. Jahnel), Journal of Number Theory 162 (2016), 224-254
• On the Brauer-Manin obstruction for degree four del Pezzo surfaces (with J. Jahnel), Acta Arithmetica 176 (2016), 301-319
• Manin's conjecture for certain biprojective hypersurfaces, J. reine angew. Math. 714 (2016), 209-250
• A variant of Weyl's inequality for systems of forms and applications, Advances in the theory of numbers, 207–218, Fields Inst. Commun. 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015
• Symmetries of rational functions arising in Ecalle's study of multiple zeta values (with A. Salerno and A. Tucker), in Women in Numbers Europe, Springer 2015, pp 153--166.
• Bihomogeneous forms in many variables, J. Th. Nombres Bordeaux 26 (2014), 483-506
• Norms as products of linear polynomials (with A. N. Skorobogatov), J. London Math. Soc. 89 (2014), 559-580

Reports

Field Arithmetic, Oberwolfach report 25/2018, 'The local-global principle for rational points and a conjecture of Harpaz and Wittenberg'

At the moment I'm scheduled to lecture the following courses

Summer term 2024: Introduction to Number theory

Winter term 2024/25: Differenzial- und Integralrechnung I

If you would like to write a Bachelor's or Master's thesis with me, then please write me an e-mail and we will start discussing. In this case it would be valuable for me to know which courses you have already attended, which courses you plan to take in the coming semester say, what your mathematical interests so far are and anything else that I should know and would be relevant in finding a topic. Normally, after first discussions about what your goals are and what your background is, we will then have a first meeting to discuss potential topics. If you already have something concrete in mind, that you would like to look at, then please let me know (though this is certainly not necessary, I'm very happy to suggest topics).

Courses which would be ideal to take before starting a Bachelor thesis under my supervision: algebra and number theory, as well as a seminar in this direction

Courses which would be ideal to take before starting a Master thesis under my supervision: 2 parts of a cycle and a seminar in the area of analytic number theory, number theory, diophantine geometry

Here are a couple of examples of thesis that I supervised in the past

Bachelor thesis: