Publications

1. Systems of quadratic inequalities (with A. A. Agrachev)
   Proc. Lond. Math. Soc. (2012)

2. Convex Pencils of real quadratic forms
   Discrete Comput. Geom. (2012)

3. The total Betti number of the intersection of three real quadrics
   Adv. Geom. (2014)

4. Random matrices and the average topology of the intersection of two quadrics
   Proc. Amer. Math. Soc. (2015)

5. Statistics on Hilbert’s Sixteenth Problem (with E. Lundberg)
   Int. Math. Res. Not. IMRN (2015)

6. Remarks on Wilmshurst’s Theorem (With S.-Y. Lee and E. Lundberg)
   Indiana Univ. Math. J. (2015)

7. Geodesics and horizontal path spaces in Carnot groups (With A. A. Agrachev and A. Gentile)
   Geom. Topol. (2015)

8. Experiments on the zeroes of harmonic polynomials using certified counting (With J. Hauenstein, E. Lundberg and D. Mehta)
   Exp. Math. (2015)

9. On the number of connected components of random algebraic hypersurfaces (With Y. Fyodorov and E. Lundberg)
   J. Geom. Phys. (2015)

10. Complexity of intersection of real quadrics and topology of symmetric determinantal varieties
    J. Eur. Math. Soc. (JEMS) (2016)

11. Gap probabilities and Betti numbers of a random intersection of quadrics (With E. Lundberg)
    Discrete Comput. Geom. (2016)

12. On the zeroes of random harmonic polynomials: the truncated model (with E. Lundberg)
    J. Math. Anal. Appl. (2016)

13. On the geometry of random lemniscates (with E. Lundberg)
    Proc. Lond. Math. Soc. (2016)

14. How many geodesics join two points on a contact sub-Riemannian manifold? (With L. Rizzi)
    J. Symplectic Geom. (2017)

15. Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry (with F. Boarotto)
    Comm. Anal. Geom. (CAG) (2017)

16. Homotopically invisible singular curves (with A. A. Agrachev and F. Boarotto)
    Calc. Var. Partial Differential Equations (2017)

17. Probabilistic Schubert Calculus (with P. Burgisser)
    J. Reine Angew. Math. (Crelle’s Journal) (2018)

18. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare (with S. Basu and A. Natarajan)
    Q. J. Math. (2019)

19. Homotopy properties of horizontal loop spaces and applications to closed subriemannian geodesics (with A. Mondino)
    Trans. Amer. Math. Soc. (2019)

20. On the geometry of the set of symmetric matrices with repeated eigenvalues (with K. Kozhasov and P. Breiding)
    Arnold Math J. (2019)

21. On the number of flats tangent to convex hypersurfaces in random position (with K. Kozhasov)
    Discrete Comput. Geom. (2019)

22. Random fields and the enumerative geometry of lines on real and complex hypersurfaces (with S. Basu, E. Lundberg and C. Peterson)
    Math. Ann. (2019)

23. Random Spectrahedra (with P. Breiding and K. Kozhasov)
    SIAM J. Optim. (2019)

24. Volume of small balls and Sub-riemannian curvature in 3D contact manifolds (with D. Barilari and I. Beschastnyi)
    J. Symplectic Geom. (2020)

25. Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limit (with A. Auffinger and E. Lundberg)
    Int. Math. Res. Not. IMRN (2020)

26. Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic regime (with R. Mulas)
    Discrete Comput. Geom. (2020)

27. Probabilistic Schubert Calculus: Asymptotics (with L. Mathis)
    Arnold Math J. (2020)

28. Quantitative singularity theory for random polynomials (with P. Breiding and H. Keneshlou)
    Int. Math. Res. Not. IMRN (2020)

29. p-adic integral geometry (with A. Kulkarni)
    SIAM J. Appl. Algebra Geom. (2020)

30. Low degree approximation of random polynomials (with D. Niang Diatta)
    Found. Comput. Math. (2021)

31. Maximal and typical topology of real polynomial singularities (with M. Stecconi)
    Ann. Inst. Fourier. (2021)

32. What is the degree of a smooth hypersurface? (with M. Stecconi)
    J. Singul. (2021)

33. On tameness of zonoids (with L. Mathis)
    J. Funct. Anal. (2022)

34. The zonoid algebra, generalized mixed volumes and random determinants (with P. Burgisser, P. Breiding and L. Mathis)
    Adv. Math. (2022)

35. Betti numbers of random arrangements (with S. Basu and A. Natarajan)
    J. Lond. Math. Soc. (2022)

36. Hausdorff approximations and volumes of tubes of singular algebraic sets (with S. Basu)
    Math. Ann. (2022)

37. Probabilisitc enumerative geometry over p-adic numbers: linear spaces on complete intersections (with R. Ait El Manssour)
    Ann. Henri Lebesgue (2022)

38. Symplectic instability of Bezout’s Theorem (with M. Ancona)
    Israel J.  Math. (2023)

39. Multicomplexes on Carnot groups and their associated spectral sequence (with F. Tripaldi)
    J. Geom. Anal. (2023)

40. Moduli spaces of geometric graphs (with M. Belotti and A. Newman)
    Algebr. Geom. Topol. (to appear)

41. A proof of the Hamiltonian Thom Isotopy Lemma (with P. Antonini and F. Cavalletti)
    Note di Matematica (to appear)

42. The probabilistic method in real singularity theory (with M. Stecconi)
    Arnold Math J. (to appear)

43. On smooth functions with two critical values (with C. Meroni and D. Zuddas)
    Rev. Mat. Complut., (to appear)

44. What is the probability that a random symmetric tensor is close to rank-one? (with A. Cazzaniga and A. Rosana)
    SIAM J. Appl. Algebra Geom.,  (to appear)

45. Differential topology of Gaussian Random fields (with M. Stecconi)
    preprint on arxiv, submitted

46. Nonarchimedean integral geometry (with P. Burgisser and A. Kulkarni)
    preprint on arxiv, submitted

47. Geometry and topology of Spin random fields (with D. Marinucci, M. Rossi and M. Stecconi)
    preprint on arxiv, submitted

48. A non-Archimedean version of Kostlan's theorem (with Y. El Maazouz)
    preprint on arxiv, submitted

49. Optimal transport between algebraic hypersurfaces (with P. Antonini and F. Cavalletti)
    preprint on arxiv, submitted

50. How regularization affects the geometry of loss functions (with N. Bottman and Y. Cooper)
    preprint on arxiv, submitted

51. Morse Theory of Euclidean distance functions and applications to real algebraic geometry (with A. Guidolin, I. Ren and M. Scolamiero)
    preprint on arxiv

Reports, surveys and other papers

1. A real viewpoint on the intersection of complex quadrics and their topology
   preprint on arXiv

2. Intrinisic volumes of sets of singular matrices and applications to real algebraic geometry
   preprint on arXiv

3. Plenty of Morse functions by perturbing with sums of squares
   Proc. Gokova Geometry and Topology (2014)

4. (with D. Barilari) Geometry of Maslov cycles
   Geometric Control and Sub–Riemannian Geometry (2014)

5. Convex bodies in real enumerative geometry
   Oberwolfach Report (2017)

6. The codimension
   Snaphosts of modern mathematics from Oberwolfach (2017)

7. What is...Random Algebraic Geometry?

Papers from my students

• Nodal sets of random high dimensional spherical harmonics by F. Fusero
  Master thesis of my former student Flavio Fusero. It is proved the concentration around the mean of the number of connected components of the zero set of a random spherical harmonic of large degree on the n sphere; this generalizes a previous result by Nazarov and Sodin
  
  

• Real lines on random cubic surfaces by R. Ait El Manssour, M. Belotti, C. Meroni
  Generalizing the results from Random fields and the enumerative geometry of lines on real and complex hypersurfaces, in this paper the authors give a closed formula for the expected number of real lines on random cubic surfaces defined by a probability distribution invariant under the action of orthogonal change of variables.