m

 

 

Dimiter Vassilev, Professor

Department of Mathematics and Statistics

MSC01 1115

The University of New Mexico

Albuquerque, NM  87131

 

 

 

Office: SMLC, Office 326,

E-Mail: vassilev-at-unm.edu
Phone:
505-277-2136,

Department Fax: (505) 277-5505

 

Physical Address:
University of New Mexico
Department of Mathematics and Statistics (click for map)
311 Terrace Street NE
Albuquerque, NM  87106

 

Research Areas:  Partial Differential Equations, Geometric Analysis, Complex and Real Analysis

Educational Information:

  • Purdue University , West Lafayette , IN, Mathematics, M. S., May 1998, Ph.D., December 2000

  • Université Bordeaux I, Talence-Cedex , France , Mathematics, DEA, May 1994

  • University of Sofia , Sofia , Bulgaria , Mathematics, B.S., 1993

Theses

  • Yamabe type equations on Carnot groups, Ph. D. thesis Purdue University, 2000, advisor Professor Nicola Garofalo.

  • Opérateur à puissances bornées et decroissance de l’énergie locale, DEA thesis Université Bordeaux I, 1994, advisor Professor Vesselin Petkov.

 

     Spring 2024 courses

Spring 2024 Office Hours: I will have office hours through Microsoft Teams on MW 11am-12pm. You can join using the link in Teams or the link that appears in your Outlook or Teams calendar. I will also check regularly the Posts tab in MS Teams for any questions/ comments, etc. You can also stop by my office if you have a question or send me an email to arrange a meeting.


 

Prospective Math Majors, see:

  1. College of Arts and Sciences and Department of Mathematics and Statistics Undergraduate Admission Requirements
  2. Department of Mathematics & Statistics Advisement
  3. Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, Mathematicians and Statisticians, at https://www.bls.gov/ooh/math/mathematicians-and-statisticians.htm
  4. Office of Career Services

Publications

Papers (abstracts are below the list)

 

34. Overdetermined problems in groups of Heisenberg type: conjectures and partial results, with Nicola Garofalo, http://arxiv.org/abs/2309.12567

33. Optimal decay for solutions of nonlocal semilinear equations with critical exponent in homogeneous group, with Nicola Garofalo and Anunziata Loiudice, http://arxiv.org/abs/2210.16893

32. Corrigenda to "$L^p$ estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities, Trans. Amer. Math. Soc. 363 (2011), no. 1, 37--62", arXiv:2210.16888.

31. Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group, with S. Ivanov and I. Minchev,Analysis & PDE, Vol. 16 (2023), No. 3, 839--860.

30. The Obata first eigenvalue theorem on a seven dimensional quaternionic contact manifold, with A. Mohamed, J Geom Anal 33, 13 (2023).

29. On sub-Riemannian and Riemannian spaces associated to a Lorentzian manifold, with R. Sverdlov, in 12th International ISAAC Congress, Birkhäuser “Research Perspectives”, (2022).

28. Non-umbilical quaternionic contact hypersurfaces in hyper-Kähler manifolds, with S. Ivanov and I. Minchev, Int. Math. Res. Not. IMRN, (2019), no. 18, 5649–-5673.

27. The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven, with S. Ivanov and A. Petkov, J. Spectr. Theory, 7 (2017), no. 4, 1119-1170.

26. Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds. Pacific J. Math. 289 (2017), no. 1, 189–201.

25. Quaternionic contact hypersurfaces in hyper-Kahler manifolds, with S. Ivanov and I. Minchev, Ann. Math. Pura Appl., 196 (2017) no. 1, pp 245--267.

24. Quaternionic contact Einstein manifolds, with S. Ivanov and I. Minchev, Math Res. Lett., 23 (2016), no.5, 1405--1432.

23. The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds, with S. Ivanov, Nonlinear Anal. 126 (2015) 262-323.

22. The quaternionic Heisenberg group and Heterotic String Solutions with non-constant dilaton in dimensions 7 and 5, with M.  Fernandez, S. Ivanov, and  L. Ugarte, Comm. Math. Phys., 339, (2015), 199-219.

21. Non-Kahler Heterotic String Solutions with non-zero fluxes and non-constant dilaton, with M.  Fernandez, S. Ivanov, and  L. Ugarte, JHEP, 06 (2014), 073.

20. The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold, with S. Ivanov and A. Petkov, J. Geom. Anal. 24 (2014), no. 2, 756–778.

19. An Obata-type theorem on a three-dimensional CR manifold, with S. Ivanov, Glasg. Math. J. 56 (2014), no. 2, 283–294.

18. Quaternionic Kahler and Spin(7) metrics arising from quaternionic contact Einstein structures, with L. C. de Andres, M.  Fernandez, S. Ivanov, J. A. Santisteban and  L. Ugarte, Ann. Mat. Pura Appl. (4) 193 (2014), no. 1, 261–290.

17. The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension 7, with S. Ivanov and A. Petkov, Nonlinear Analysis: Theory, Methods & Applications 93 (2013) 51–61.

16. An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence free torsion, with S. Ivanov, J. of Geometry, 103,  (2012), no. 3, 475-504.

15. The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group,  with S. Ivanov and I. Minchev, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) Vol. XI (2012), 1-18.

14. Bianchi type A hyper-symplectic and hyper-Kahler metrics in 4D, with L. C. de Andres, M.  Fernandez, S. Ivanov, J. A. Santisteban and L. Ugarte, Classical Quantum Gravity 29 (2012), no. 2, 025003, 18 pp.

13. Lp estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities, Trans. Amer. Math. Soc. 363 (2011), 37--62.

12. Quaternionic contact manifolds with a closed fundamental 4-form, with S. Ivanov, Bull. London Math. Soc. (2010) 42 (6), 1021--1030.

11. Conformal paracontact curvature and the local flatness theorem, with S. Ivanov and S. Zamkovoy, Geom. Dedicata, 144 (2010), no.1, 79--100.

10. Conformal quaternionic contact curvature and the local sphere theorem, with St. Ivanov, J. Math. Pure Appl, 93 (2010), 277--307.

9. Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, with S. Ivanov and I. Minchev, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 1041--1067.

8. Strong unique continuation for generalized Baouendi-Grushin operators, with N. Garofalo , Comm. PDE, 32 (2007), no. 4, 643--663.

7. Regularity near the characteristic boundary for sub-laplacian operators, Pacific J Math., 227 (2006), no. 2, 361--397.

6. A note on the stability of local zeta functions, Proc. Amer. Math. Soc, 134 (2006), 81--91.

5. Overdetermined BVP, quadrature domains, and applications, with D. Khavinson and A. Solynin,  Comput. Methods Funct. Theory, 5 (2005),  19--48.

4. Symmetry properties of positive entire solutions of Yamabe type equations on groups of Heisenberg type , with N. Garofalo, Duke Math. J. 106 (2001), no. 3, 411--448.

3. Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, with N. Garofalo, Math. Ann. 318 (2000), no. 3, 453--516.

2. Strong unique continuation for generalized Baouendi-Grushin operators, with N. Garofalo, Proceedings of the 4th ISAAC Congress, Toronto , 2003, Advances in analysis, 255--263, World Sci. Publ., Hackensack , NJ , 2005

1. The non-linear Dirichlet problem and the CR Yamabe problem, with N. Garofalo, Boundary value problems for elliptic and parabolic operators ( Catania , 1998) Matematiche ( Catania ) 54 (1999), suppl., 75--93.

 

 

                      

Slides of some talks

 

Abstracts of papers

 

31. Obata first eigenvalue theorem on a seven dimensional quaternionic contact manifold, with A. Mohamed, arXiv. We show that a compact quaternionic contact manifold of dimension seven that satisfies a Lichnerowicz-type lower Ricci-type bound and has the P -function of any eigenfunction of the sub-Laplacian non-negative achieves its smallest possible eigenvalue only if the structure is qc-Einstein. In particular, under the stated conditions, the lowest eigenvalue is achieved if and only if the manifold is qc-equivalent to the standard 3-Sasakian sphere.

 

30. Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group, with S. Ivanov and Minchev. A complete solution to the quaternionic contact Yamabe equation on the qc sphere of dimension 4n+3 as well as on the quaternionic Heisenberg group is given. A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown

 

29. On sub-Riemannian and Riemannian spaces associated to a Lorentzian manifold, with R. Sverdlov, 12th International ISAAC Congress, Birkhäuser “Research Perspectives”. We present a certain construction of a sub-Riemannian and Riemannian spaces naturally associated to a Lorentzian manifold. Some additional structures and relations between geometric properties of the corresponding spaces will be explored.

 

28. Non-umbilical quaternionic contact hypersurfaces in hyper-Kähler manifolds, with S. Ivanov and I. Minchev. We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kahler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kahler manifold is endowed with an involutive 7-dimensional distribution whose integral leaves are locally qc-conformal to the standard 3-Sasakian sphere.

 

27. The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven, with S. Ivanov and A. Petkov. We prove quaternionic contact versions of two of Obata's sphere theorems. On a compact quaternionic contact (qc) manifold of dimension bigger than seven and satisfying a Lichnerowicz type lower bound estimate we show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. The same conclusion is shown to hold on a non-compact qc manifold which is complete with respect to the associated Riemannian metric assuming the existence of a function with traceless horizontal Hessian. The third result of the paper is a qc version of Liouville's theorem showing that a qc-conformal diffeomorphism between open connected sets of the 3-Sasakian sphere is a restriction of an element of the qc-conformal automorphism group of the sphere.

 

26. Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds. A proof of the monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds is proven.

 

25. Quaternionic contact hypersurfaces in hyper-Kähler manifolds, with S. Ivanov and I. Minchev. We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in the flat quaternion space is contained in one of the three qc-hyperquadrics in the flat quaternion space. Moreover, we show that an embedded qc-hypersurface in a hyper-Kähler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-Kähler metric is degenerate along the hypersurface.

 

24. Quaternionic contact Einstein manifolds, with S. Ivanov and I. Minchev. The main result is that the qc-scalar curvature of a seven dimensional quaternionic contact Einstein manifold is a constant. In addition, we characterize qc-Einstein structures with certain flat vertical connection and develop their local structure equations. Finally, regular qc-Ricci flat structures are shown to fiber over hyper-Kähler manifolds.

 

23. The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds, with S. Ivanov. We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.

 

22. The quaternionic Heisenberg group and Heterotic String Solutions with non-constant dilaton in dimensions 7 and 5, with M.  Fernandez, S. Ivanov, and  L. Ugarte. New smooth solutions of the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton based on the quaternionic Heisenberg group are constructed. We show that through appropriate contractions the solutions found in the G 2-heterotic case converge to the heterotic solutions on 6-dimensional inner non-K\"ahler spaces previously found by the authors and, moreover, to new heterotic solutions with non-constant dilaton in dimension 5. All solutions satisfy the heterotic equations of motion up to the first order of α .

 

21. Non-Kahler Heterotic String Solutions with non-zero fluxes and non-constant dilaton, with M.  Fernandez, S. Ivanov, and L. Ugarte. Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kähler nilmanifold are presented. In the conformally compact case the dilaton is determined by the real slices of the elliptic Weierstrass function. The dilaton of non-compact complete solutions is given by the fundamental solution of the Laplacian on R4. All solutions satisfy the heterotic equations of motion up to the first order of alpha'.

 

20. The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension 7, with S. Ivanov and A. Petkov, Nonlinear Analysis: Theory, Methods & Applications 93 (2013) 51–61. A version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian on a compact seven dimensional quaternionic contact manifold is proved assuming a lower bound on the $Sp(1)Sp(1)$-components of the qc-Ricci curvature and the positivity of the $P$-function of any eigenfunction. The introduced $P$-function and nonlinear $C$-operator are motivated by the Paneitz operators defined previously in the Riemannian and CR settings and the $P-$function used in the theory of elliptic partial differential equations. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is the round 3-Sasakian sphere.

 

19. An Obata-type theorem on a three-dimensional CR manifold, with S. Ivanov,Glasg. Math. J. 56 (2014), no. 2, 283–294. We prove a CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible valuethen, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three dimensional unit sphere.

 

18. Quaternionic Kahler and Spin(7) metrics arising from quaternionic contact Einstein structures, with L. C. de Andres, M.  Fernandez, S. Ivanov, J. A. Santisteban and  L. Ugarte, Ann. Mat. Pura Appl. (4) 193 (2014), no. 1, 261–290.We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kähler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kähler metrics and Spin(7)-holonomy metrics which seem to be new.  Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds with either a closed fundamental four form or fundamental two forms defining a differential ideal that are not quaternionic Kähler.

 

17. The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold, with S. Ivanov and A. Petkov, J. Geom. Anal. 24 (2014), no. 2, 756–778. The main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a-priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.

 

16. An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence free torsion, with S. Ivanov, J. of Geometry, 103,  (2012), no. 3, 475-504. We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster's metric, pseudohermitian manifold.

 

15. The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group,  with St. Ivanov and I. Minchev,  Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) Vol. XI (2012), 1-18. We determine the best (opimal) constant in the L2 folland-Stein inequality on the quaternionic Heisenberg group and the  non-negative functions for which equality holds.

 

14. Bianchi type A hyper-symplectic and hyper-Kahler metrics in 4D, with L. C. de Andres, M.  Fernandez, S. Ivanov, J. A. Santisteban and L. Ugarte, Classical Quantum Gravity 29 (2012), no. 2, 025003, 18 pp. We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4-D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four.

 

13. Lp estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities, Trans. Amer. Math. Soc. 363 (2011), 37--62. Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp Lq regularity and asymptotic behavior for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of Rn. In addition, we find the best constant and extremals in the case of the considered L2 Hardy-Sobolev inequality. Finally we give some other possible applications and directions to explore concerning non-completeness of metrics with finite volume and bounded scalar curvature and stationary cylindrical states of the Vlasov-Poisson system. The results are related also to Hardy inequalities involving distance to the boundary.

 

12. Quaternionic contact manifolds with a closed fundamental 4-form, with S. Ivanov, Bull. London Math. Soc. (2010) 42 (6), 1021--1030. We show that the fundamental horizontal 4-form on a quaternionic contact manifold of dimension at least eleven is closed if and only if the torsion of the Biquard connection vanishes. This condition characterizes quaternionic contact structures which are locally qc homothetic to a 3-sasakian one.

 

11. Explicit Quaternionic contact structures, Sp(n)-structures and Hyper Kaehler metrics.  with Luis C. de Andres, Marisa  Fernandez, Stefan Ivanov, Jose A. Santisteban and  Luis Ugarte, arXiv:0903.1398, 2009. We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion for which the quaternionic contact conformal curvature tensor does not vanish, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic Heisenberg group. We present a left invariant quaternionic contact structure on a seven dimensional non-nilpotent Lie group, and show that this structure is locally quaternionic contact conformally equivalent to the flat quaternionic contact structure on the quaternionic Heisenberg group. We outline a construction to obtain explicit quaternionic Kähler metrics as well as Spin(7) metrics defining Sp(1)Sp(1)-hypo structures on 7-dimensional manifolds. We present explicit complete quaternionic Kähler metrics and Spin(7)-holonomy metrics on the product of a quaternionic contact structure on a seven dimensional Lie group with the real line which seem to be new.

 

10. Conformal paracontact curvature and the local flatness theorem, with S. Ivanov and S. Zamkovoy, Geom. Dedicata, 144 (2010), no.1, 79--100. A curvature-type tensor invariant called para contact (pc) conformalcurvature is defined on a paracontact manifolds. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group iff the pc conformal curvature vanishes provided the dimension is bigger than three. A different proof of the Chern-Moser-Webster theorem is given, showing that the vanishing of the Chern-Moser invariant is necessary and sufficient condition a CR-structure to be CR equivalent to a quadric in $\mathbb{C}^{n+1}$ provided n>1. The proof establishes also Cartan's result in dimension three. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.

9. Conformal quaternionic contact curvature and the local sphere theorem, with St. Ivanov, J. Math. Pure Appl, 93 (2010), 277--307. A curvature-type tensor invariant called quaternionic contact (qc) conformal curvature is defined on a qc manifolds in terms of the curvature and torsion of the Biquard connection. The discovered tensor is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser invariant in CR geometry. It is shown that a qc manifold is locally qc conformal to the standard flat qc structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure on the sphere if and only if the qc conformal curvature vanishes.

 

8. Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, with S. Ivanov and I. Minchev, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 1041--1067. A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the L2 Folland-Stein embedding theorem is determined.

 

 

7. Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem, with St. Ivanov and I. Minchev, International Centre for Theoretical Physics preprint IC/2006/117, 2006.  The paper is a study of the conformal geometry of quaternionic contact manifolds with the associated Biquard connection. We give a partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. In particular, the scalar curvature of the Biquard connection with vanishing torsion is a global constant. We consider interesting classes of functions on hypercomplex manifold and their restrictions to hypersurfaces. We show a "3-Hamiltonian form" of infinitesimal automorphisms of quaternionic contact structures and transformations preserving the trace-free part of the horizontal Ricci tensor of the Biquard connection. All conformal deformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing trace-free part of the horizontal Ricci tensor of the Biquard connection are explicitly described.

  

 

6Strong unique continuation for generalized Baouendi-Grushin operators, with N. Garofalo , Comm. PDE, 32 (2007), no. 4, 643--663. Our main result gives a quantitative control of the order of zero of a weak solution to perturbations of the Baouendi-Grushin operator.  The perturbations are tailored on the geometry of the Baouendi-Grushin operator and should be interpreted as a sort of Lipschitz continuity with respect to a suitable pseudo-distance associated to the system of vector fields. Our result generalizes the famous result due to Aronszaijn, Krzywicki and Szarski valid for elliptic operators in divergence form with Lipschitz continuous coefficients.

 

 

5. Regularity near the characteristic boundary for sub-laplacian operators, Pacific J Math.,  227 (2006), no. 2, 361--397. We prove that the best constant in the Folland-Stein embedding theorem on Carnot groups is achieved. This implies the existence of a positive solution of the Yamabe type equation on Carnot groups. The second goal of the paper is to show  regularity of the Green's function and solutions of the Yamabe equation involving the sub-Laplacian near the characteristic boundary of a domain in the considered groups.

 

 

4. A note on the stability of local zeta functions, Proc. Amer. Math. Soc, 134 (2006), 81-91.

We show the existence of an interval of stability under small perturbations of local zeta functions corresponding to non-trivial local solutions of an elliptic equation with Lipschitz coefficients.

 

 

3. Overdetermined BVP, quadrature domains, and applications, with D. Khavinson and A. Solynin,  Comput. Methods Funct. Theory, 5 (2005), 19-48. We consider an overdetermined problem in planar multiply connected domains. This problem is solvable if and only if the domain is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution of the problem in some special cases and discuss some applications concerning the shape of electrified droplets and small air bubbles in a fluid flow.  Other cases, that are left open, include a variation of the  Serrin problem but with moving boundaries. This case is also of interest to computing the sharp constant in an inequality, finer than the isoperimetric inequality, concerning the analytic content of a set.

 

2. Symmetry properties of positive entire solutions of Yamabe type equations on groups of Heisenberg type , with N. Garofalo, Duke Math. J. 106 (2001), no. 3, 411--448.

The main result of this paper is the determination of the entire solutions with partial symmetry of the Yamabe equation on Iwasawa type groups. As a corollary we find the best constant in the L2 Folland-Stein Sobolev type inequality restricted to functions with partial symmetry.

 

 

1. Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groupsp , with N. Garofalo, Math. Ann. 318 (2000), no. 3, 453--516. A part of this paper is devoted to the boundary regularity of weak solutions to the Yamabe equation on Carnot groups near characteristic points of the boundary. Another part concerns non-existence of solutions to the Yamabe equation results on certain domains in Heisenberg or Iwasawa type groups.

 

 

 


Conferences and Meetings

 

Special Session on Theoretical and Applied perspectives in Machine Learning, AMS Fall Western Sectional Meeting 2021

 

 

Conference on Geometric Structures in Mathematical Physics, Golden Sands, Bulgaria, September 19 - 26, 2011, Program, abstracts and slides of talks.

 

(click on image for a full-size poster) 

Special session on geometric structures and PDEs, AMS Spring Western Session meeting, 2010
 
 

Past Courses

 

Fall 2023 courses for Vassilev, D. 

  • 53) Vector Analysis - 67897 - MATH 311 - 003. Hybrid Instructional Method: Face-to-Face Plus DSH 226 2:00 pm - 3:15 pm T;  Remote Set Day/Time 2:00 pm - 3:15 pm R Remote Instruction Microsoft Teams. Course Description. Vector algebra, lines, planes; vector valued functions, curves, tangent lines, arc length, line integrals; directional derivative and gradient; divergence, curl, Gauss’ and Stokes’ theorems, geometric interpretations. Prerequisite: Calculus III. See this web page for further information
  • 52) Introduction to Topology - MATH 431 002 & Foundations of Topology - MATH 535 002,   Face-to-Face SMLS 352 11:00 am - 12:15 pm T, Remote Set Day/Time 11:00 am - 12:15 pm R Location MS Teams. Description: Metric spaces, topological spaces, continuity, algebraic topology. Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces. Prerequisites: MATH321. Pre- or co requisites: MATH322. Texts: 1. Munkres J. Topology (Pearson) 2nd ed 2. Introduction to Topology, Gamelin & R.E.Greene, Dover Publ., 2nd ed. See this web page for further information

Spring 2023Schedule for Vassilev, D.

  • 51) Advanced Calculus II - MATH 402 002 CRN6016. Hybrid Instructional Method: Face-to-Face Plus: 2:00 pm - 3:15 pm T SMLC B81, Remote Set Day/Time 2:00 pm - 3:15 pm R Remote Instruction Microsoft Teams. Course Description. Generalization of 401/501 to several variables and metric spaces: sequences, limits, compactness and continuity on metric spaces; interchange of limit operations; series, power series; partial derivatives; fixed point, implicit and inverse function theorems; multiple integrals. Prerequisite: MATH401. See this page for further information.

Fall 2022 Schedule for Vassilev, D. 

Spring 2022 Schedule for Vassilev, D.

  • 48) MATH439 ST: Intro to Diff Manifold & MATH536 Introduction to Differentiable Manifolds , Time TTh 8:00-9:15.  Description: The main goal of the course is to familiarize you with topological and geometric notions and structures through a gentle introduction to some topics motivated by physics and geometry. Specific topics will include basic topology, smooth manifolds, vector field and their flows, differential forms, Stokes theorem, DeRham theory, Lie groups and algebras, Riemannian and semi-Riemannian metrics, vector and fiber bundles, connections. Some applications will be given to Maxwell's Equations and the Yang-Mills Equation. Prerequisites: Linear Algebra and Calculus III, preferably also MATH431, 322 will be helpful. Course web page

     

  • 47) MATH 533 Algebraic Topology II & MATH538 Riemannian Geometry II, Time TTh 1400-1515. Description: Morse theory in finite and infinite dimensions. Applications in geometry and existence of solutions to variational problems and partial differential equations. Course Web Page

Fall 2021 Schedule for Vassilev, D. 

  • 46) MATH311 Vector Analysis - 67897 - MATH 311 - 003. Hybrid Instructional Method: Face-to-Face Plus 8:00 am - 9:15 am T, Remote Set Day/Time 8:00 am - 9:15 am R Remote Instruction Microsoft Teams. Course Description. Vector algebra, lines, planes; vector valued functions, curves, tangent lines, arc length, line integrals; directional derivative and gradient; divergence, curl, Gauss’ and Stokes’ theorems, geometric interpretations. Prerequisite: Calculus III. See the Course web page for further information.
  • 45) MATH 537 Riemannian Geometry I, Time: M/W/F: 12:00 pm – 12:50 pm, Location: MS Teams. Riemannian Geometry I, Time MW 1230-1345, Location SMLC-124. Description: Theory of connections, curvature, Riemannian metrics, Hopf-Rinow theorem, geodesics. Riemannian submanifolds. Prerequisite: 536.
  • 44) Math 431 Introduction to Topology & Math 535  Foundations of Topology, MWF 1400-1450, Location MS Teams. Description:  Metric spaces, topological spaces, continuity, algebraic topology.  Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces.  Prerequisite: MATH401.  Text: Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd edition.

Spring 2020 Schedule for Vassilev, D.

 

Fall 2019 Schedule for Vassilev, D.

Spring 2019 Schedule for Vassilev, D.

Fall 2018 Schedule for Vassilev, D.

  • 37) MATH313-001 44939 Complex Variables for Engineering. (3) Time TTh 12:30-13:45, Location DSH-233.  Course web page.
  • 36) MATH 539-001 62064 Selected topics Geometry & Topology, Time TTh 14:00-15:15, Location SMLC352. Course web page

  • 35) MATH 551-001, Problems, F 16:00-17:00, Location SMLC 356.

  • Algebra & Geometry Seminar, W 15:00-16:00, Location SMLC 352.

Spring 2018 Schedule for Vassilev, D.

  • 34) MATH439-001 40631 ST: Intro to Diff Manifold, Time TTh 8:00-9:15 SMLC-120.  Description:Maxwell's Equations, Manifolds, Vector Fields, Differential Forms and DeRham Theory, Lie groups and Lie Algebras. Basics on Bundles and Connections, Gauge transformations, Holonomy, Curvature and the Yang-Mills Equation. Prerequisites: 264, 321. 322 will be helpful. Course web page.

     

  • 33) MATH 538-001 44849 Riemannian Geometry II, Time TTh 1230-1345, Location SARAR-107. Description: Continuation of MATH 537 with emphasis on adding more structures. Comparison theorems in Riemannian geometry, Riemannian submersions, Bochner theorems with relation to topology of manifolds, Riemannian Foliations, Complex and Kaehler geometry, Sasakian and contact geometry.

    Prerequisite: 537. Course web page.

Fall 2017  Schedule for Vassilev, D.

  • 32) MATH306-001 59458 College Geometry, Time MW 1600-1715 SMLC-124.  Description: An axiomatic approach to fundamentals of geometry, both Euclidean and non-Euclidean. Emphasis on historical development of geometry. (T). Prerequisites: 162 or 215. Course web page with Syllabus & Homework

     

  • 31) MATH 537-003 60880 Riemannian Geometry I, Time MW 1230-1345, Location SMLC-124. Description: Theory of connections, curvature, Riemannian metrics, Hopf-Rinow theorem, geodesics. Riemannian submanifolds. Prerequisite: 536. Course web page with Syllabus & Homework.

Spring 2017  Schedule for Vassilev, D.

Fall 2016  Schedule for Vassilev, D.

Spring 2016  Schedule for Vassilev, D.

  • 26) MATH 439-001 55949 ST: Intro to Diff Manifold & MATH 536 - 001 51950 Introduction to Differentiable Manifolds. Class time  11:00 am-12:15 pm. Location SMLC 352. Course web page with Syllabus & Homework.

Fall 2015  Schedule for Vassilev, D.

  • 25) 44643-MATH 431 001 Intro To Topology & 44644-MATH 535 001 Foundations of Topology MWF 1400-1450  Location Dane Smith Hall DSH-233 Description:  Metric spaces, topological spaces, continuity, algebraic topology.  Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces.  Prerequisite: MATH401.  Text: Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd edition. Syllabus and Homework.

  • 24) 54094-MATH 579 002 ST: Part Diff Equas 3 MWF  1100-1150 Location Dane Smith Hall DSH126.

    Description: The course will focus on fully non-linear equations, mainly related to the Monge-Ampere equation. Syllabus.

Spring 2015 Schedule for Vassilev, D.

Spring 2014 Schedule for Vassilev, D.

  • 22) MATH 538 Riemannian Geometry II. Lectures TR 1100-1215 SMLC 124. Course Readings: https://learn.unm.edu/ Catalog Description: Continuation of MATH 537 Riemannian Geometry I with emphasis on adding more structures. Riemannian submersions, Bochner theorems with relation to topology of manifolds, Riemannian Foliations, Complex and Kaehler geometry, Sasakian and contact geometry. Prerequisite: 537. Syllabus.

Fall 2013

  • 21) Math264 Sections 003 and 004, Calculus III, Lectures MWF 1100-1150 DSH-326. Recitation Sections  003 T 1100-1215 DSH-234 & 004 T 1100-1215 DSH-234. Syllabus and Homework. Description: Vector operations, vector representation of planes and curves, functions of several variables, partial derivatives, gradient, tangent planes, optimization, multiple integrals in Cartesian cylindrical and spherical coordinates, vector fields, line integrals and Green’s theorem. Prerequisite: C (not C-) or better in 163.

  • 20) Math431 Introduction to Topology & Math 535  Foundations of Topology, MWF 900-950, Location SMLC-356 Syllabus and Homework. Description:  Metric spaces, topological spaces, continuity, algebraic topology.  Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces.  Prerequisite: MATH401.  Text: Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd edition.

Spring 2013 Schedule for Vassilev, D.

  • 19) Math 536 001 Intro Diff Manifolds MWF 1100-1150, CENT-1028 . Description: Concept of a manifold, differential structures, vector bundles, tangent and cotangent bundles, embedding, immersions and submersions, transversality, Stokes' theorem. Prerequisite: Math 511.  Suggested Textbooks: Lee J. M., Introduction to smooth manifolds (Springer, Graduate Texts in Mathematics) & Morita S., Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201). Syllabus and homework.

Fall 2012 Schedule for Vassilev, D.

  • 18) Math 431 Introduction to Topology & Math 535  Foundations of Topology, MWF 1300-1350, Location DSH-327. Syllabus and Homework & WebCT Description:  Metric spaces, topological spaces, continuity, algebraic topology.  Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces.  Prerequisite: MATH401.  Text: Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd edition.

Spring 2012 Schedule for Vassilev, D.

Description: The Mittag-Leffler theorem, series and product expansions, introduction to asymptotics and the properties of the gamma and zeta functions. The Riemann mapping theorem, harmonic functions and Dirichlet's problem. Introduction to elliptic functions. Selected topics.  Prerequisite: MATH561. MATH562 is a continuation of  Math 561, so it might be useful to check  the  HW for MATH561 Fall 2010. Textbook: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) (Hardcover) Robert E. Greene and Steven G. Krantz.

Spring 2011 Schedule for Vassilev, D.

  • 16) 25493 Math 562 001 Func of Cmplx Var II TR 1230-1345 MITCH 212.  Syllabus and HW Description: The Mittag-Leffler theorem, series and product expansions, introduction to asymptotics and the properties of the gamma and zeta functions. The Riemann mapping theorem, harmonic functions and Dirichlet's problem. Introduction to elliptic functions. Selected topics.  Prerequisite: MATH561. MATH562 is a continuation of  Math 561, so it might be useful to check  the  HW for MATH561 Fall 2010. Textbook: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) (Hardcover) Robert E. Greene and Steven G. Krantz

Fall 2010 Schedule for Vassilev, D.

  • 15) 37221 Math 561 001 Func of Cmplx Var I MWF 1400-1450 HUM 428 Syllabus and HW Description: Analyticity, Cauchy theorem and formulas, Taylor and Laurent series, singularities and residues, conformal mapping, selected topics.  Prerequisite: Math 311 or 402. Textbook: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) (Hardcover) Robert E. Greene and Steven G. Krantz
  • 14) 39785 Math 180 016 Elements of Calc I MWF 0900-0950 ECON 1002,   Syllabus and HW,   WebCT link for current grade. Description: Limits of functions and continuity, intuitive concepts and basic properties; derivative as rate of change, basic differentiation techniques; application of differential calculus to graphing and minima-maxima problems; exponential and logarithmic functions with applications. Meets New Mexico Lower Division General Education Common Core Curriculum Area II: Mathematics (NMCCN 1613).  Prerequisite: ACT=>26 or SAT=>600 or MATH 121 or MATH 150 or Compass College Algebra >66. 
 

Textbook: Applied Calculus for the Managerial, Life, and Social Sciences, 8th Edition

Soo T. Tan
ISBN-10: 0495559695

ISBN-13: 9780495559696

© 2011

Department support page: http://math.unm.edu/courses/math180/index.php

 

Spring 2010

  • 13) 30243 Math 536 001 Intro Diff Manifolds MWF 1000-1050 HUM 422 Syllabus and homework Description: Concept of a manifold, differential structures, vector bundles, tangent and cotangent bundles, embedding, immersions and submersions, transversality, Stokes' theorem. Prerequisite: Math 511.

 

Fall 2009

Spring 2009

Fall 2008

Spring 2008