Marcelo Mendes Disconzi

Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu

office: Stevenson Center 1515

phone: (615) 322 7147 fax: (615) 343 0215

mail to: 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240

Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu

office: Stevenson Center 1515

phone: (615) 322 7147 fax: (615) 343 0215

mail to: 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240

General

Here you find information about my academic work and education, some notes I have written, and some links.

My interests are partial differential equations, geometric analysis, and mathematical physics. Currently the main topics of my research are the study of compressible and incompressible Euler equations (including generalizations such as free boundary problems), and the evolution problem for Einstein's equations coupled to matter (especially, Einstein's equations coupled to the equations of fluid dynamics). Other topics I have worked on are: geometric constraints induced by Einstein's equations on three-dimensional slices (for instance, problems related to the Penrose inequality and the positive mass theorem); variational and analytic aspects of effective potentials arising in compactifications of string theory (for example, the study of equations of motion derived from effective actions); and conformal deformations of Riemannian metrics (mainly, the Yamabe problem).

Click here for my CV.

I organize the Partial Differential Equations Seminar. Click here for the Partial Differential Equations research group at Vanderbilt. Click here for past events organized at Vanderbilt.

Here you find information about my academic work and education, some notes I have written, and some links.

My interests are partial differential equations, geometric analysis, and mathematical physics. Currently the main topics of my research are the study of compressible and incompressible Euler equations (including generalizations such as free boundary problems), and the evolution problem for Einstein's equations coupled to matter (especially, Einstein's equations coupled to the equations of fluid dynamics). Other topics I have worked on are: geometric constraints induced by Einstein's equations on three-dimensional slices (for instance, problems related to the Penrose inequality and the positive mass theorem); variational and analytic aspects of effective potentials arising in compactifications of string theory (for example, the study of equations of motion derived from effective actions); and conformal deformations of Riemannian metrics (mainly, the Yamabe problem).

Click here for my CV.

I organize the Partial Differential Equations Seminar. Click here for the Partial Differential Equations research group at Vanderbilt. Click here for past events organized at Vanderbilt.

Selected
papers

Below is a sample of my papers, together with a short description of each of them. For a complete list of my publications, see my CV.

Motion of slightly compressible fluids in a bounded domain, II. (with David G. Ebin.) Communications in Contemporary Mathematics, Vol. 19, No. 4, pp. 1650054, 57 pages (2017).

Compactness and Non-Compactness for Yamabe Problem on Manifolds with Boundary. (with Marcus A. Khuri.) J. Reine Angew. Math. (Crelle's Journal), Vol. 2017, Issue 724, pp. 145-201 (2017).

On the boundedness of effective potentials arising from string compactifications. (with Michael R. Douglas and Vamsi P. Pingali.) Communications in Mathematical Physics, Vol. 325, Issue 3, pp. 847-878 (2014).

Below is a sample of my papers, together with a short description of each of them. For a complete list of my publications, see my CV.

The
free boundary Euler equations with large surface tension. (with David
G. Ebin.) Journal of Differential Equations, Vol. 261, Issue 2,
pp. 821-889 (2016).

We study the free
boundary Euler equations with surface tension in three spatial
dimensions, showing that the equations are well-posed if the
coefficient of surface tension is positive. Then we prove that under
natural assumptions, the solutions of the free boundary motion
converge to solutions of the Euler equations in a domain with fixed
boundary when the coefficient of surface tension tends to infinity.

Using a simple and
well-motivated modification of the stress-energy tensor for a viscous
fluid proposed by Lichnerowicz, we prove that Einstein's equations
coupled to a relativistic version of the Navier-Stokes equations are
well-posed in a suitable Gevrey class if the fluid is incompressible
and irrotational. These last two conditions are given an appropriate
relativistic interpretation. The solutions enjoy the domain of
dependence or finite propagation speed property. We also derive a full
set of equations, describing a relativistic fluid that is not
necessarily incompressible or irrotational, which is well-suited for
comparisons with the system of an inviscid fluid.

Motion of slightly compressible fluids in a bounded domain, II. (with David G. Ebin.) Communications in Contemporary Mathematics, Vol. 19, No. 4, pp. 1650054, 57 pages (2017).

We study the problem of
inviscid slightly compressible fluids in a bounded domain. We find a
unique solution to the initial-boundary value problem and show that it
is near the analogous solution for an incompressible fluid provided
the initial conditions for the two problems are close. In particular,
the divergence of the initial velocity of the compressible flow at
time zero is assumed to be small. Furthermore we find that solutions
to the compressible motion problem in Lagrangian coordinates depend
differentiably on their initial data, an unexpected result for this
type of non-linear equations.

Compactness and Non-Compactness for Yamabe Problem on Manifolds with Boundary. (with Marcus A. Khuri.) J. Reine Angew. Math. (Crelle's Journal), Vol. 2017, Issue 724, pp. 145-201 (2017).

We study the problem of
conformal deformation of Riemannian structure to constant scalar
curvature with zero mean curvature on the boundary. We prove
compactness for the full set of solutions when the boundary is umbilic
and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also
established under these hypotheses, and we provide counter-examples to
compactness when $n \geq 25$. Lastly, our methods point towards a
vanishing theorem for the umbilicity tensor, which is anticipated to
be fundamental for a study of the nonumbilic case.

On the boundedness of effective potentials arising from string compactifications. (with Michael R. Douglas and Vamsi P. Pingali.) Communications in Mathematical Physics, Vol. 325, Issue 3, pp. 847-878 (2014).

We study effective
potentials coming from compactifications of string theory. We show
that, under mild assumptions, such potentials are bounded from below
in four dimensions, giving an affirmative answer to a conjecture
proposed by the second author in arXiv:0911.3378v4
[hep-th]. We also derive some sufficient
conditions for the existence of critical points. All proofs and
mathematical hypotheses are discussed in the context of their
relevance to the physics of the problem.

Selected
pre-prints

Below is a sample of my pre-prints, together with a short description of each of them. For a complete list, see my CV.

*The
relativistic Euler equations: Remarkable null structures and
regularity properties.* (with Jared
Speck.) arXiv: 1809.06204 [math.AP], 83 pages (2018).

Below is a sample of my pre-prints, together with a short description of each of them. For a complete list, see my CV.

We derive a new
formulation of the relativistic Euler equations that exhibits
remarkable properties. This new formulation consists of a coupled
system of geometric wave, transport, and elliptic equations, sourced
by nonlinearities that are null forms relative to the acoustical
metric. Our new formulation is well-suited for various applications,
in particular for the study of stable shock formation, as it is
surveyed in the paper. Moreover, using the new formulation presented
here, we establish a local well-posedness result showing that the
vorticity and the entropy of the fluid are one degree more
differentiable compared to the regularity guaranteed by standard
estimates (assuming that the initial data enjoy the extra
differentiability). This gain in regularity is essential for the study
of shock formation without symmetry assumptions. Our results hold for
an arbitrary equation of state, not necessarily of barotropic type.

Notes

Here are some notes that I (and other people) have written.

Correlation functions in QFT - (handwritten). The basic ideas and concepts of quantum field theory are discussed with the intent of making physics books and papers on the subject more accessible to a mathematical audience. The focus is on correlation functions for the scalar field: what they are, how to compute them, their Feynman diagrams and renormalization properties. For a more details, see the table of contents.

Elementary realization of of BRST symmetry and gauge fixing - notes of a series of lectures given by Martin Rocek. All ideas of BRST symmetry and BV formalism are developed at a very basic level using finite dimensional integrals instead of path integrals. Excellent for those interested in the general idea of the formalism (pdf file).

Some algebraic structures in physics - notes from a series of informal meetings that I and some other students organized with the goal of sharing our different background in physics and mathematics (pdf file).

Some ideas in Conformal Field Theory - (handwritten) notes from a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook (.zip file with a bunch of .jpg files, or click here to access each file separetely).

Mathematical Foundations of Classical and Quantum Field Theory - notes of two summer courses I took on the subject (pdf file).

Here are some notes that I (and other people) have written.

Recent
advances in classical and relativistic fluids - (handwritten).
Notes of a series of lectures I gave at the summer school Boston
City Limits 2018, June 11-21, at MIT.
Other topics covered in the summer school were mathematical general
relativity, by Stefanos
Aretakis (notes here),
the formation of singularities in general relativity, by Jared
Speck (notes TBP), and solitions, bubbling, and blow-up for
semilinear PDEs, by Andrew
Lawrie (notes here).

Some
advanced
techniques on PDE's - we review how the negative norm Sobolev
spaces can be used to derive a necessary and sufficient condition for
existence of weak solutions of any linear PDE. Using this, to show
Egorov's example of a PDE that is not locally solvable at the origin.
Some further applications are derived (pdf file).

Holographic
renormalization - notes of a talk I gave in the RTG
Seminar in Geometry and Physics at Stony Brook (pdf file).

Correlation functions in QFT - (handwritten). The basic ideas and concepts of quantum field theory are discussed with the intent of making physics books and papers on the subject more accessible to a mathematical audience. The focus is on correlation functions for the scalar field: what they are, how to compute them, their Feynman diagrams and renormalization properties. For a more details, see the table of contents.

Elementary realization of of BRST symmetry and gauge fixing - notes of a series of lectures given by Martin Rocek. All ideas of BRST symmetry and BV formalism are developed at a very basic level using finite dimensional integrals instead of path integrals. Excellent for those interested in the general idea of the formalism (pdf file).

Some algebraic structures in physics - notes from a series of informal meetings that I and some other students organized with the goal of sharing our different background in physics and mathematics (pdf file).

Some ideas in Conformal Field Theory - (handwritten) notes from a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook (.zip file with a bunch of .jpg files, or click here to access each file separetely).

Topics
in
Differential Topology - notes by Somnath
Basu of a course taught by Blaine
Lawson (pdf file).

Mathematical Foundations of Classical and Quantum Field Theory - notes of two summer courses I took on the subject (pdf file).

Links
and
useful material

Simons Center for Geometry and Physics - the intellectual focus of the Center is at the interface of Mathematics, in particular Geometry, and Theoretical Physics.

The
Comprehensive
LaTex symbol list - excellent material by Scott Pakin (pdf file).

Simons Center for Geometry and Physics - the intellectual focus of the Center is at the interface of Mathematics, in particular Geometry, and Theoretical Physics.

Media
and outreach

I am occasionally a guest in the radio program Fronteiras da Ciencia, a radio program (in Portuguese) dedicated to science discussions for the general public. I participated in the episodes Gravitacao quantica, Teoria de supercordas, A grande ruptura cosmica (big rip), and a discussion about the work of Stephen Hawking on the occasion of his passing (mp3 files).

Here is
a department
news story on the occasion of my 2018
Sloan Fellowship award. The announcement of the 2018 Sloan Fellows
also appeared in the
New York Times and in the American
Mathematical Society website. Here the Vanderbilt
news story.

Here is a department news story on the occasion of my 2018 Dean's Faculty Fellowship award.

The three-dimensional free boundary Euler equations with surface tension. Video of a talk I gave in the workshop Recent Advances in Hydrodynamics that took place at the Banff International Research Station in Banff, Canada.

The "sticky" universe. A news story on a paper that I wrote with Robert Scherrer and Thomas Kephart. In 2015, the year the paper was published, it received widespread media coverage, including from The Guardian, Redorbit, New Statesman, The Huffington Post, among many others. This unexpected media attention led me to write some reflections on science and the media. In 2016, the paper was again in the news with stories in the Wired, BBC Brazil (in Portuguese), Revista Piauí (in Portuguese), and TV Cultura (in Portuguese). This story was in the cover of the 2017 department newsletter, Spectrum.

The Einstein system for inviscid and viscid relativistic fluids (.flv file). Video of a talk I presented at the Colloquium of the Department of Applied Mathematics at USP (Brazil). The talk was in Ensligh, although the introduction and Q&A were in Portuguese.

Here is a department news story on the occasion of my 2018 Dean's Faculty Fellowship award.

The three-dimensional free boundary Euler equations with surface tension. Video of a talk I gave in the workshop Recent Advances in Hydrodynamics that took place at the Banff International Research Station in Banff, Canada.

The "sticky" universe. A news story on a paper that I wrote with Robert Scherrer and Thomas Kephart. In 2015, the year the paper was published, it received widespread media coverage, including from The Guardian, Redorbit, New Statesman, The Huffington Post, among many others. This unexpected media attention led me to write some reflections on science and the media. In 2016, the paper was again in the news with stories in the Wired, BBC Brazil (in Portuguese), Revista Piauí (in Portuguese), and TV Cultura (in Portuguese). This story was in the cover of the 2017 department newsletter, Spectrum.

The Einstein system for inviscid and viscid relativistic fluids (.flv file). Video of a talk I presented at the Colloquium of the Department of Applied Mathematics at USP (Brazil). The talk was in Ensligh, although the introduction and Q&A were in Portuguese.

I am occasionally a guest in the radio program Fronteiras da Ciencia, a radio program (in Portuguese) dedicated to science discussions for the general public. I participated in the episodes Gravitacao quantica, Teoria de supercordas, A grande ruptura cosmica (big rip), and a discussion about the work of Stephen Hawking on the occasion of his passing (mp3 files).