Ecology and Consciousness
I am interested in the interplay between consciousness and ecology. I see that a change in consciousness is needed to face the ecological problems of our time. We are in a confused culture that teach us that we are different from our surroundings. This has produced a hyper-competitive culture that affects many forms of life. In the words of David Bohm, "to be confused about what is different and what is not, is to be confused about everything". Thus, the consequences of our fragmented way of thinking spread to all what we do. The symptoms are vast and overwhelming. This inconsistency suggests that the truth is other, that what we do is being done to us; that our existence is relational.
The shift in consciousness needed to move from a fragmented to a wholeness cosmovision requires careful attention. Any suggested solution to the ecological problems standing from a fragmented perspective will only be palliative. One of my main interests is to study the process of this movement in consciousness and explore holistic solutions to our ecological crisis.
Inverse Problems
Inverse problems are mathematical problems in abductive reasoning, where from the observation of an event we look for the best and most likely explanation. A classical example in abductive reasoning is the job of a forensic detective, that by observing the crime scence he formulates a hypothesis of the sequence of events leading to the crime. The detective reconstructs an approximation of the crime scene through methods of interrogation, careful observation and analysis.
In inverse problems our subject of study are different types of waves. We observe, question and analyze them to obtain information about their past (e.g., where they where?, at what velocity did they travel?, what did they encounter?, etc.) In several applications, we want to analyze physical and chemical properties of a model, but we only have access to indirect information of this properties. Inverse problems deals with recovering such information from a set of accessible data. This area is of growing interest for disciplines like medical imaging, geophysics, quantum mechanics, and many more.
Many of these mathematical models are described by means of a differential equation. Thus, a typical inverse problem is to recover the coefficients of a differential equation (i.e., physical properties like medium propagation, internal density distribution, etc), from measurements of the solution at the boundary of the domain, or at infinity. In general, these problems are highly non-linear and ill-posed, demanding tools from different areas of mathematics (e.g., differential geometry, microlocal analysis, functional analysis).
During my PhD, I worked on three different stability problems. The first one, related to the wave equation; the second to the conductivity equation with application to medical imaging and the last one was a joint work with the department of computer science in a software solution to improve image perception. While on my posdoctoral reseach I worked mainly in the problem of Photoacoustic Tomography (PAT). Currently I am working on the joint reconstruction problem in PAT.
Contact Information
- Email address: carlos.montalto@ucr.ac.cr
- Office: 252 ECCI
- Telephone: (506) 2511-8038
- Address:
Mailbox 48
Escuela de Matemáticas
Universidad de Costa Rica
San Pedro de Montes de Oca
San José, Costa Rica