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Causality, local well-posedness, and all that. Notes of a talk I gave in program The Many Faces of Relativistic Fluid Dynamics at the Kavli Institute for Theoretical Physics, UC Santa Barbara, June 2023. This is a non-technical introduction to the topic aimed at an audience of physicists. Here  is a video of the talk.

Recent developments in relativistic fluids. These are notes that grew out of a series of lectures I gave at the semester program General Relativity, Center of Mathematical Sciences and Applications, Harvard University, May 16-17, 2022 (here is a video of the lectures; second video here), and at the Advanced Studies Institutes (ASI) in Uzbekistan, part of the USA-Uzbekistan Collaboration for Research, Education and Student Training, July 25-August 4, 2022.

The Classical Gauge Theoretic Structure of the Standard Model. This is an undergraduate senior thesis which I supervised by Elijah Sheridan. It provides a helpful "dictionary" for mathematicians to understand the basic constructions and concepts of classical gauge theory used by physicists in the standard model.

Introduction to nonlinear wave equations. Short introduction to the topic by Jonathan Luk.

General relativity. Introduction to mathematical general relativity from a modern perspective by Stefanos Aretakis.

Introduction to microlocal analysis. Introduction to the topic by Peter Hintz.

Recent developments in the theory of relativistic fluids. Notes of a series of lectures I gave at the Summer School on Recent Advances in Mathematical Fluid Dynamics, May 20-24, 2019, at USC. Here is a .zip file with the .tex, .bbl, and pictures of the notes. People are welcome to download, modify, and use the .tex file as they see fit (and here are the handwritten notes, just in case). Here is some extra material that complements the notes. Other topics covered in the summer school were Non-uniqueness of weak solutions to the Euler and Navier-Stokes equations, by Tristan Buckmaster, Recent developments on water waves, by Yu Deng, and Nonlinear dynamics of the Schrodinger equations with periodic boundary conditions, by Emanuele Haus.

Recent advances in classical and relativistic fluids. Notes of a series of lectures I gave at the summer school Boston City Limits 2018, June 11-21, at MIT. Here is a .zip file with the .tex, .bbl, and pictures of the notes. People are welcome to download, modify, and use the .tex file as they see fit (and here are the handwritten notes, just in case). 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 PDEs. I review how 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, I then show Egorov's example of a PDE that is not locally solvable at the origin. Some further applications are derived.

Holographic renormalization. Notes of a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook.

Correlation functions in QFT. (Handwritten notes.) 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.

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.

Some ideas in Conformal Field Theory. 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.

Spontaneous symmetry breaking. - Introductory notes on the Higgs mechanism.

Mathematical Foundations of Classical and Quantum Field Theory. Notes of two summer courses I took on the subject. I took these notes before starting my PhD and in hindsight I would have written things differently. But perhaps precisely because of that some uninitiated student might benefit from them. For a sharper introduction to the topic, see Elijah's notes above.

From free-boundary fluids to the constraint equations. Video of a talk I gave in the workshop Mass, the Einstein Constraint Equations, and the Penrose Inequality Conjecture at the Simons Center for Geometry and Physics, jointly with the colloquium of the Stony Brook Department of Mathematics, (2023).

Causality, local well-posedness, and all that. Video of a talk I gave in the program The Many Faces of Relativistic Fluid Dynamics at the Kavli Institute for Theoretical Physics, UC Santa Barbara (2023).

Here is a Vanderbilt news story on the occasion of my Chancellor's Faculty Fellowship award (2023).

General-relativistic viscous fluids. Video of a talk I gave at the General Relativity Seminar, Center of Mathematical Sciences and Applications, Harvard University (2022).

A brief overview of recent developments in relativistic fluids. Video of a six-hours series of lectures at the semester program General Relativity, Center of Mathematical Sciences and Applications, Harvard University (2022). Click here for the second video.

General-relativistic viscous fluids. Video of a talk I gave in the General Relativity Conference, Center of Mathematical Sciences and Applications, Harvard University (2022).

Here is a news story by the Illinois Center for Advanced Studies of the Universe at UIUC, describing my work First-order General Relativistic Viscous Fluid Dynamics (2022), written with Fabio S. Bemfica and Jorge Noronha (2022).

Here is Vanderbilt news story about a NSF Research Trainee award for which I am a Co-Principla Investigator together with Principal Investigator Kelly Holley-Bockelmann. The goal is to establish a graduate certificate program in the emerging field of multimessenger astronomy (2022).

General-relativistic viscous fluids. Video of a talk I gave at the Princeton Gravity Initiative, Princeton University (2021).

General-relativistic viscous fluids. Video of a talk I gave in the Workshop on QGP Phenomenology (2021).

The relativistic Euler equations with a physical vacuum boundary. Video of a talk I gave at the Rocky Mountain Mathematical Physics Seminar, University of Colorado Boulder (2021).

General relativistic viscous fluids. Video of a talk I gave at the Colloquium of the Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University (2021).

The relativistic Euler equations with a physical vacuum boundary. Video of a talk I gave in the Center for Nonlinear Analysis at Carnegie Mellon University (2021).

General-relativistic viscous fluids. Video of a talk I gave in the workshop Relativistic Hydrodynamics: Foundations, Novel Applications and Interdisciplinary Connections that was jointly organized by the Gravity, Quantum Fields and Information group at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) and the Illinois Center for Advanced Studies of the Universe. Click here for the slides of the talk. Here is a discussion session of the workshop; the discussion of my talk starts at 21:57min (2020).

General-relativistic viscous fluids. Video of a talk I gave in the workshop Mathematical and Computational Approaches for the Einstein Field Equations with Matter Fields that took place at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI. Click here for the slides of the talk (2020).

General-relativistic viscous fluids. Video of a talk I gave in the Webinar on Quark Matter and Relativistic Hydrodynamics (Physics Department, Sharif University of Technology, Tehran, Iran). The talk is followed by nearly an hour of discussion with the audience (2020).

Strichartz estimates for the compressible Euler equation with vorticity and low-regularity solutions. Video of a talk I gave in the workshop Dynamics in Geometric Dispersive Equations and the Effects of Trapping, Scattering and Weak Turbulence that took place at the Banff International Research Station in Banff, Canada (2020).

Here is a Vanderbilt news story of a Robert Noyce Scholarship grant (2019-2024) for which I am a Co-Principal Investigator together with Principal Investigators Heather Johnson and Teresa Dunleavy, of the Department of Teaching and Learning, and Co-Principal Investigators David Weintraub of the Physics Department, and Isaac Thompson from Fisk University (2019).

Here is a news story with a short video description (in Portuguese) by the Sociedade Brasileira de Fisica (Brazilian Physical Society) describing my work Causality of the Einstein-Israel-Stewart Theory with Bulk Viscosity (2019), written with Fabio S. Bemfica and Jorge Noronha. Click here to access the video directly (.mov file). The follow-up work, Nonlinear Constraints on Relativistic Fluids Far From Equilibrium (2021), that I wrote with Fabio S. Bemfica, Vu Hoang, Jorge Noronha, and Maria Radosz, was also featured (in Portuguese) by the Sociedade Brasileira de Fisica. (Click here to access the video directly, .mov file.)

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 (2018).

Here is a department news story on the occasion of my 2019 Dean's Faculty Fellowship award (2019).

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 (2016).

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 Piaui (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 (2013).

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 (2013), Teoria de supercordas (2013), A grande ruptura cosmica (big rip) (2017), and a discussion about the work of Stephen Hawking (2018) on the occasion of his passing (mp3 files).

Academic calendar.

Logins.

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.