What kind of things are you working (thinking) on?

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• 
  Giovanni Acquaviva · Università degli Studi di Trento

  Hola Raul, in a few words, what I'm trying to do in my research is to analyze quantum fields coupled to classical curved metrics. A field in its vacuum state, coupled to a changing geometry, has a probability to jump to excited states and the cool (but sometimes disturbing ;-)) thing is that many quantities in this "curved space QFT" have a lot in common with thermodynamics. For example, both the static black hole metric and the de Sitter expanding metric excite the field in a thermal way: you can actually associate a temperature to those spacetimes (the first case is the well known Hawking radiation, the second one is related to the inflationary scenario in the early universe). So, my curiosity goes mainly to the connections between general relativity and thermodynamics. What are you interested in/ working on, Raul?

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  Hola Raul,
  
  in a few words, what I'm trying to do in my research is to analyze quantum fields coupled to classical curved metrics. A field in its vacuum state, coupled to a changing geometry, has a probability to jump to excited states and the cool (but sometimes disturbing ;-)) thing is that many quantities in this "curved space QFT" have a lot in common with thermodynamics.
  For example, both the static black hole metric and the de Sitter expanding metric excite the field in a thermal way: you can actually associate a temperature to those spacetimes (the first case is the well known Hawking radiation, the second one is related to the inflationary scenario in the early universe).
  So, my curiosity goes mainly to the connections between general relativity and thermodynamics.
  
  What are you interested in/ working on, Raul?

  Feb 15, 2012
• 
  Raul Alvarez · National Autonomous University of Mexico

  That sounds pretty cool Giovanni, actually when i took my QFT course some words were said about the relation between termodinamics and QFT in curved backgrounds, but sadly where only a few ideas and i don't have a good taste of that area. So, i hope to share some thought's along those lines. Right now i'm writing my bcs tesis in the so called "Seiberg-witten invariants". They are "numbers" constructed via the space of solutions of the "monopole" equations, derived by N. Seiberg and E. Witten in 94. They are interesting for at least two reasons. First, they are, until now, the only invariants that detect diferences between the "differentiable structure" of four dimentional manifold, that's a whole industry!. But i'm more astonished by the way the invariants where derived , they (as you probably suspect for the name) are a certain limit of a supersymmetric quantum field theory, namely, N = 2 SYM. And that, i think, is a goldmine of research both in physics and math, that's my reason number two. Best wishes Raúl

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  That sounds pretty cool Giovanni, actually when i took my QFT course some words were said about the relation between termodinamics and QFT in curved backgrounds, but sadly where only a few ideas and i don't have a good taste of that area. So, i hope to share some thought's along those lines.
  
  Right now i'm writing my bcs tesis in the so called "Seiberg-witten invariants". They are "numbers" constructed via the space of solutions of the "monopole" equations, derived by N. Seiberg and E. Witten in 94. They are interesting for at least two reasons. First, they are, until now, the only invariants that detect diferences between the "differentiable structure" of four dimentional manifold, that's a whole industry!. But i'm more astonished by the way the invariants where derived , they (as you probably suspect for the name) are a certain limit of a supersymmetric quantum field theory, namely, N = 2 SYM. And that, i think, is a goldmine of research both in physics and math, that's my reason number two.
  
  Best wishes
  Raúl

  Feb 17, 2012
• 
  Giovanni Acquaviva · Università degli Studi di Trento

  Dear Raúl, Seiberg-Witten invariants: that seems a really interesting topic, even though my knowledge of how it actually works is nearly zero... when you say that they detect differences between the differentiable structure of 4-manifolds, it means that through them you are able to (in some sense) classify 4-manifolds? Or simply you can compare different 4-manifolds on the "differentiable structure" level? Sorry for the obviously elementary questions, but (not being an expert in supersymmetry nor in most of the huge world of mathematics) I'm trying to grasp some information without some of the basics that you have. I would be pleased to share some knowledge about our research subjects, knowing that, in the end, it's usual to find some connections and similarities between the topics (that's one of the most amazing things in nature). Greetings! Giovanni

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  Dear Raúl,
  
  Seiberg-Witten invariants: that seems a really interesting topic, even though my knowledge of how it actually works is nearly zero... when you say that they detect differences between the differentiable structure of 4-manifolds, it means that through them you are able to (in some sense) classify 4-manifolds? Or simply you can compare different 4-manifolds on the "differentiable structure" level? Sorry for the obviously elementary questions, but (not being an expert in supersymmetry nor in most of the huge world of mathematics) I'm trying to grasp some information without some of the basics that you have.
  I would be pleased to share some knowledge about our research subjects, knowing that, in the end, it's usual to find some connections and similarities between the topics (that's one of the most amazing things in nature).
  
  Greetings!
  
  Giovanni

  Feb 23, 2012
• 
  Raul Alvarez · National Autonomous University of Mexico

  Actually the situation in four dimensions is poorly understood, and just get's worse when you restrict the attention on the differentiable case. Let me just give you an idea of how bad is the situation in the latter case with the following example. The canonical prototype of smooth manifold is certainly the euclidean space, it is a vector space an because of that is topologically trivial. Well, the most astonishing result in 4 dimentional smooth topology is that there are an infinite number of "non-equivalent" diferentiable structures over euclidean 4-space. The classification problem goes beyond the scope of SW, the best thing that the theory can do for us is the assigment of a number for each 4-manifold in such a way that does not change under difeomorphisms, but there are technical subtleties (not too restrictive) that forbids the theory to be defined in all 4-manifolds. The thing is that even when the theory is limited to some subclass of 4-manifolds, the scenario can be very surprising there. Also, SW theory is fully loaded of mathematical mistery, the invariants where derived from higly non-rigurus methods of supersymetric QFT . It's hoped that the understanding of this methods someday clarify formal aspects of gauge theory. There is also a conjecture, due to Witten, that claims that SW theory and Donnaldson's instanton theory are exactle the same theory... Greetings Raúl

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  Actually the situation in four dimensions is poorly understood, and just get's worse when you restrict the attention on the differentiable case. Let me just give you an idea of how bad is the situation in the latter case with the following example. The canonical prototype of smooth manifold is certainly the euclidean space, it is a vector space an because of that is topologically trivial. Well, the most astonishing result in 4 dimentional smooth topology is that there are an infinite number of "non-equivalent" diferentiable structures over euclidean 4-space.
  
  The classification problem goes beyond the scope of SW, the best thing that the theory can do for us is the assigment of a number for each 4-manifold in such a way that does not change under difeomorphisms, but there are technical subtleties (not too restrictive) that forbids the theory to be defined in all 4-manifolds. The thing is that even when the theory is limited to some subclass of 4-manifolds, the scenario can be very surprising there.
  Also, SW theory is fully loaded of mathematical mistery, the invariants where derived from higly non-rigurus methods of supersymetric QFT . It's hoped that the understanding of this methods someday clarify formal aspects of gauge theory. There is also a conjecture, due to Witten, that claims that SW theory and Donnaldson's instanton theory are exactle the same theory...
  
  Greetings
  Raúl

  Feb 24, 2012