Id Photo
The canonical hiking photo.


Postal Address

Room b402,
Institut für Algebra, Zahlentheorie
    und Diskrete Mathematik,
Leibniz Universität Hannover,
Welfengarten 1,
30167 Hannover,
Germany.

Email: loughran@math.uni-hannover.de
Tel: 0049 511 762 3519.

About Me

Greetings friends. My name is Daniel Loughran and I am currently a postdoctoral research fellow in mathematics at the Leibniz Universität in Hannover. I am a member of the Institut für Algebra, Zahlentheorie und Diskrete Mathematik. From September 2016, I will be a Turing fellow at the University of Manchester.

A brief CV:

  • 2016- Turing fellow, University of Manchester, UK.
  • 2014-2016 Postdoc, Leibniz Universität Hannover, Germany.
  • 2012-2014 Postdoc, University of Bristol, UK.
  • 2011-2012 Postdoc, Institut de Mathématiques de Jussieu, Paris, France.
  • 2007-2011 PhD, Univerity of Bristol, UK, supervised by Prof Tim Browning.
  • 2003-2007 MMath, Univerity of Sheffield, UK, first class honours.
  • Workshop

    Together with Ulrich Derenthal and Christopher Frei, I organised the workshop

    Analytic number theory and diophantine geometry.

    This took place in Hannover the week of the 7th-11th of September 2015. Please see the workshop webpage for further details.

    Research

    My research lies in the areas of algebraic geometry and number theory, in particular arithmetic geometry, algebraic number theory and analytic number theory. My research interests include rational points, harmonic analysis, sieve theory, Brauer groups, L-functions, class field theory, moduli stacks, Hilbert schemes, abelian varieties, computer algebra, algebraic groups and equivariant geometry.

    My work so far can be roughly sorted into the following themes.

    Manin's conjecture: This conjecture concerns the distribution of rational points of bounded height on algebraic varieties. My papers [2] and [3] prove some cases of this conjecture for certain del Pezzo surfaces. My works [4] and [7] also concern this conjecture, where we construct new counter-examples to Manin's conjecture, and also use the Weil restriction to obtain new cases of this conjecture.

    Equivariant geometry: This topic concerns the study of algebraic varieties equipped with group actions. In my papers [1] and [5], we classify those (possibly singular) del Pezzo surfaces which admit a freely transitive action of some linear algebraic group. Such surfaces are of interest as Manin's conjecture is already known to hold in such cases.

    Rational points on families of varieties: In [6] and [12], I study the problem of counting the number of varieties in a family which contain a rational point. In [6] I formulate some conjectures on this problem and prove some special cases via harmonic analysis on toric varieties, together with answering some cases of a question of Serre on specialisation of Brauer group elements. In [12], we also consider the problem of studying weak approximation for certain families of varieties. In my joint work [14], we find geometric conditions which force very few of the varieties in a family to have a rational point, in a precise quantitative sense.

    Hasse principles: Suppose that some object X over a number field k satisfies some property P over every completion of k. Then does X necessarily satisfy P over k? If the answer is yes, then one says that a Hasse principle holds. We study problems of this type for the existence of lines on surfaces and intersections of two quadrics in [8] and [10], which includes both positive and negative results. In [13] we study problems of this type for the existence of norms for abelian extensions.

    Good reduction of algebraic varieties and the arithmetic of moduli stacks: A famous theorem of Faltings (formally the Shafarevich conjecture), states there are only finitely many abelian varieties of fixed dimension over a given number field k with good reduction outside of a given set of places of k. Such problems are closely related to the study of integral points on moduli stacks. We study problems of this type for flag varieties and complete intersections in projective space in [9] and [11], respectively. We also obtain a version of the Shafarevich conjecture for sextic surfaces in [16].

    Del Pezzo surfaces: A have a soft spot for del Pezzo surfaces. Such surfaces were intensively studied by classical algebraic geometers over the complex numbers, with the most famous examples being cubic surfaces. Over non-algebraically closed fields, however, the arithmetic of del Pezzo surfaces is a very rich topic and an active area of research. Many of my papers concern these surfaces [1],[2],[3],[5],[8], and they also make fleeting appearences in my other works, as useful examples. In my most recent work [17], we study del Pezzo surfaces over finite fields, and give a complete answer to the following question of Serre: over a given finite field k, what are the possibile number of rational points that a smooth cubic surface over k can have?

    Publications

    The following is a list of my publications. My PhD Thesis is essentially a collection of my first three papers, however it also contains a general introduction to Manin's conjecture. It was awarded a postgraduate commendation for excellence in a doctoral thesis by the faculty of science of the University of Bristol.

    1. Del Pezzo surfaces over finite fields and their Frobenius traces (with Barinder Banwait and Francesc Fité),
      submitted, [arXiv:1606.00300], [Code].
    2. Good reduction of Fano threefolds and sextic surfaces (with Ariyan Javanpeykar),
      submitted, [arXiv:1512.07189].
    3. The moduli of smooth hypersurfaces with level structure (with Ariyan Javanpeykar),
      Manuscripta mathematica, to appear, [arXiv:1511.09291].
    4. Fibrations with few rational points (with Arne Smeets),
      submitted, [arXiv:1511.08027].
    5. The Hasse norm principle for abelian extensions (with Christopher Frei and Rachel Newton),
      submitted, [arXiv:1508.02518].
    6. Failures of weak approximation in families (with Martin Bright and Tim Browning),
      Compositio Mathematica, to appear, [arXiv:1506.01817].
    7. Complete intersections: Moduli, Torelli, and good reduction (with Ariyan Javanpeykar),
      Mathematische Annalen, to appear, [arXiv:1505.02249].
    8. The Hasse principle for lines on diagonal surfaces (with Jörg Jahnel),
      Mathematical Proceedings of the Cambridge Philosophical Society, 160(1) (2016), 107–119, [arXiv:1503.05021].
    9. Good reduction of algebraic groups and flag varieties (with Ariyan Javanpeykar),
      Archiv der Mathematik, 104(2) (2015), 133-143, [arXiv:1501.04526].
    10. The Hasse principle for lines on del Pezzo surfaces (with Jörg Jahnel),
      International Mathematics Research Notices, 23 (2015), 12877–12919, [arXiv:1410.5671], [Code].
    11. Varieties with too many rational points (with Tim Browning),
      Mathematische Zeitschrift, to appear. [arXiv:1311.5755].
    12. The number of varieties in a family which contain a rational point,
      Journal of the European Mathematical Society, to appear, [arXiv:1310.6219].
    13. Equivariant compactifications of two-dimensional algebraic groups (with Ulrich Derenthal),
      Proceedings of the Edinburgh Mathematical Society, 58(1) (2015), 149-168, [arXiv:1212.3518].
    14. Rational points of bounded height and the Weil restriction,
      Israel Journal of Mathematics, 210(1) (2015), 47-79, [arXiv:1210.1792].
    15. Manin's conjecture for a singular quartic del Pezzo surface,
      Journal of the London Mathematical Society, 86(2) (2012), 558-584, [arXiv:1109.6253].
    16. Manin's conjecture for a singular sextic del Pezzo surface,
      Journal de Théorie des Nombres de Bordeaux, 22(3) (2010), 675-701, [arXiv:1009.2364].
    17. Singular del Pezzo surfaces that are equivariant compactifications (with Ulrich Derenthal),
      Hausdorff Trimester on Diophantine equations in: Journal of Mathematical Sciences 171 (2010), no. 6, 714-724, [arXiv:0910.2717].
    18. Manin's conjecture for del Pezzo surfaces,
      PhD thesis, 2011, [thesis].

    Research talks

    The following is a list of seminars and meetings where I have given research talks. It also includes upcoming talks.

    1. 26-30/09/2016 - Rational points and algebraic geometry, CIRM, Luminy.
    2. 05-09/09/2016 - Elementare und Analytische Zahlentheorie (ELAZ 2016), Strobl am Wolfgangsee, Austria.
    3. 20-22/07/2016 - Zeta functions and motivic integration, Heinrich Heine Universität Düsseldorf.
    4. 08/06/2016 - Oberseminar Arithmetische und Algebraische Geometrie, Ludwig-Maximilians-Universität München.
    5. 20/05/2016 - Number theory seminar, Copenhagen.
    6. 09/05/2016 - Oberseminar analytic number theory, Georg-August-Universität Göttingen.
    7. 31/03/2016 - Séminaire de géométrie algébrique, Université de Rennes 1, Rennes.
    8. 03/03/2016 - Number theory seminar, University of Manchester.
    9. 01/03/2016 - Number theory seminar, University of York.
    10. 24/02/2016 - Linfoot seminar, Bristol.
    11. 17/02/2016 - London number theory seminar, UCL.
    12. 05/02/2016 - Intercity number theory, Leiden.
    13. 01/02/2016 - Algebra, geometry and number theory seminar, Leiden.
    14. 21/01/2016 - Oberseminar für Algebraische Geometrie und Arithmetik, Universität Duisburg-Essen.
    15. 03/12/2015 - Oberseminar Algebra und Geometrie, Oldenburg.
    16. 01/12/2015 - Arithmetic geometry seminar, HU Berlin.
    17. 28/06-05/07/2015 - Rational points 2015, Franken-Akademie Schloss Schney, Bayreuth.
    18. 22-26/06/2015 - Quantitative arithmetic geometry, Mittag-Leffler Institute, Stockholm.
    19. 15/05/2015 - Number theory seminar, Copenhagen.
    20. 24/04/2015 - Göttingen-Hannover number theory workshop, Georg-August-Universität Göttingen.
    21. 02/04/2015 - Séminaire d'algèbre et de théorie des nombres, EPFL, Lausanne.
    22. 11/03/2015 - Number theory lunch seminar, Max Planck Institute for Mathematics, Bonn.
    23. 17/11/2014 - Rencontres de théorie analytique et élémentaire des nombres, Institut Henri Poincaré, Paris.
    24. 08/10/2014 - Linfoot seminar, Bristol.
    25. 30/09/2014 - SFB-Kolloquium, Mainz.
    26. 11-12/07/2014 - Exciting new faces in analytic number theory, Hausdorff center for mathematics, Bonn.
    27. 13/06/2014 - Séminaire variétés rationnelles, Institut Henri Poincaré, Paris.
    28. 23/05/2014 - The locally free geometry seminar, Amsterdam.
    29. 22/05/2014 - Algebra, geometry and number theory seminar, Leiden.
    30. 12-16/05/2014 - Rational and integral points on higher-dimensional varieties, AIM, US.
    31. 17-21/03/2014 - Geometry and arithmetic of surfaces, Munich.
    32. 04/03/2014 - Number theory seminar, University of York.
    33. 06-10/01/2014 - Collaborations around rational points, Belairs research institute, Barbados.
    34. 03/12/2013 - Number theory seminar, University of Cambridge.
    35. 28/03/2013 - Research seminar, HIM, Bonn.
    36. 28/02/2013 - Pure mathematics seminar, Exeter University.
    37. 11/02/2013 - Number theory seminar, Warwick University.
    38. 06/12/2012 - Séminaire d'arithmétique, Université Lille 1, Lille.
    39. 03/12/2012 - Pure mathematics seminar, Royal Holloway, London.
    40. 27/11/2012 - Number theory seminar, Sheffield University.
    41. 22/10/2012 - Pure maths research seminar, UEA, Norwich.
    42. 18/10/2012 - Number theory seminar, Oxford University.
    43. 04-08/06/2012 - Analytic methods for diophantine equations, Göttingen.
    44. 21/05/2012 - Arithmétique en plat pays, Ghent.
    45. 27/04/2012 - Séminaire de théorie des nombres de Caen, Caen.
    46. 09/03/2012 - Séminaire variétés rationnelles, Ecole normale supérieure, Paris.
    47. 22/02/2012 - Séminaire de théorie des nombres, Institut Fourier, Grenoble.
    48. 02/02/2012 - Séminaire de géométrie algébrique, Université de Rennes 1, Rennes.
    49. 05/12/2011 - Rencontres de théorie analytique et élémentaire des nombres, Institut Henri Poincaré, Paris.
    50. 14/11/2011 - Séminaire de théorie des nombres, Institut de Mathématiques de Jussieu, Paris.