Charles Epstein - US grants

Studies will be continued on the mechanisms whereby chromosome imbalance results in abnormalities of development and function. The research goals for the coming year are: a) The isolation and analysis of trisomic and normal cell populations from normal reversibly yields trisomy 17 mouse chimeras for H-2 antigen synthesis and integration into membranes. b) The formation and analysis of normal reversibly yields monosomy 19 (1,12) chimeras to determine whether monosomy results in cell lethality. c) The analysis of protein synthesis in monosomic mouse embryos by 2-dimensional gel electrophoresis and ultrastructural examination of monosomy 19 embryos. d) The quantitation of interferon induced products in normal and trisomy 21 cells.

The principal investigator will study global invariants on strictly pseudoconvex manifolds that arise as renormalized Chern classes for complete Kaehler metrics. He will continue his study of pseudodifferential operator calculi adapted to the study of degenerate operators similiar to the Bergman-Laplace operator. A goal will be to obtain index formulae relating the renormalized Chern numbers to "analytic" indices for degenerate elliptic complexes. And he will continue an investigation of the problem of embedding three dimensional CR-manifolds. For many years mathematicians have studied the behavior of certain differential operators on hypersurfaces which extend off to infinity. This study has its origins in the calculus of Newton. In this project the principal investigator will analyze operators which have large variations over very large domains "near" infinity. In the past, only progress on operators which have small local variations was made.

oxidative stress; enzyme activity; superoxide dismutase; aging; age difference; metalloenzyme; free radicals; genetically modified animals; laboratory mouse;

The overall objective of this research program is to discover the mechanisms by which the presence of an extra copy of human chromosome 21 (HSA 210 produces the phenotype of DS. In addition to mental retardation and mild dysmorphic features, important compounds of this phenotype are the neurodegenerative and functional changes of Alzheimer's disease (AD), a T- lymphocyte immunodeficiency, increased frequency of childhood leukemia, and congenital heart disease. The approach Dr. Epstein's group uses is based on the premise that it will be possible to related specific components of the trisomic phenotype to the increased expression of genes or sets of genes present on HSA 21. Dr Epstein has previously pioneered important studies of the trisomy 16 (Ts16) mouse as an animal model of DS, and a partial Ts16 model has been developed recently by others. In this proposal, a series of approaches to the phenotypic mapping of both partial and complete Ts16 are proposed. These approaches are subtractive in nature initially in that they are based on the analysis of the changes of the trisomic region by telomere insertion or irradiation, the resulting progeny will be assessed with regard to which phenotypic features of complete or partial Ts16 disappear as extra copies of particular regions of the chromosome are not longer present. Regions so identified will then be analyzed further to identify potential candidate genes, and the true role of these regions and genes in producing phenotypic changes in trisomy established by transgenic and homologous recombination techniques. To accomplish these goals, Dr. Epstein and colleagues will generate mouse Ts16 embryonic stem cells with progressive truncations of chromosome 16 (MMU) 165) and analyze the phenotypic (immunologic, hematopoietic, central nervous system) of the partial Ts animals derived from these cells. He and colleagues will further generate and characterize Ts16 embryos and fetuses (derived from stem cell lines with complete and truncated chromosome 16s) with regard to overall morphogenesis, to the viability and gene expression of central nervous system neurons, to development of immune and hematopoietic systems, and to cardiac morphogenesis.

ABSTRACT Proposal: DMS-962304 PI: Epstein CR-geometry is the natural odd dimensional analogue of complex geometry. From work of Kohn, Boutet de Monvel, and Harvey and Lawson it is known that any compact strictly pseudoconvex CR-manifold of dimension 5 or greater can be realized as the boundary of a compact normal Stein space. A CR-manifold with such a realization is called embeddable. It has been known since the 1960s that the situation in 3-dimensions is quite different: the generic perturbation of an embeddable CR-structure is not embeddable. Our goal is to describe the set of embeddable CR-structures on a compact three manifold as a subset of the set of all CR-structures. For the case of deformations of the unit 3-sphere the small embeddable perturbations are essentially an infinite dimensional and infinite codimensional analytic submanifold of the set of all CR-structures. The research outlined in this proposal is directed towards extending such results to more general classes of 3-manifolds. In earlier work the investigator introduced a stratification of the set of embeddable CR-structures. Locally the stratification is defined by formally analytic relations. A major thrust of the proposed work is to analyze these equations using methods arising from the Nash-Moser implicit function theorem. It is hoped that it can be shown that the strata have a transverse analytic structure. General position arguments and a generalization of a result of Kiremidjian being sought by the investigator and G. Henkin would then show that the stratification has only finitely many distinct strata. This would imply that the set of embeddable structures is a closed subset in a reasonable topology. Using extremal determinants analogous to those used by Osgood, Phillips and Sarnak it is hoped to define a natural exhaustion function for the space of embeddable structures. The importance of mathematics in physics, economics, chemistry, engineering, etc. stems from the fact that EQUATIONS describe the rela tionships among the variables that arise in these fields. Making predictions is thereby reduced to solving the equations. One therefore needs criteria to determine whether the equations have solutions, and algorithms for solving them. In practice this can only be done approximately so it is important to have an estimate for the size of he errors one is making. In this proposal we consider a FAMILY of related equations that arise naturally in analytic geometry. These equations often do not have solutions. Even though the equations in the family are closely related, the property of solvability can change very wildly as one moves through the family. Our principal aim is to understand this instability and give criteria that describe the well behaved members of the family. There are similarities between the issues that arise in this analysis and problems encountered in image reconstruction techniques used, e.g., in CAT scans. It is hoped that a better understanding of the case at hand will provide insight into other cases where similar instabilities arise.

The overall objective of this research program is to discover the mechanisms by which the presence of an extra copy of human chromosome 21 (HSA 210 produces the phenotype of DS. In addition to mental retardation and mild dysmorphic features, important compounds of this phenotype are the neurodegenerative and functional changes of Alzheimer's disease (AD), a T- lymphocyte immunodeficiency, increased frequency of childhood leukemia, and congenital heart disease. The approach Dr. Epstein's group uses is based on the premise that it will be possible to related specific components of the trisomic phenotype to the increased expression of genes or sets of genes present on HSA 21. Dr Epstein has previously pioneered important studies of the trisomy 16 (Ts16) mouse as an animal model of DS, and a partial Ts16 model has been developed recently by others. In this proposal, a series of approaches to the phenotypic mapping of both partial and complete Ts16 are proposed. These approaches are subtractive in nature initially in that they are based on the analysis of the changes of the trisomic region by telomere insertion or irradiation, the resulting progeny will be assessed with regard to which phenotypic features of complete or partial Ts16 disappear as extra copies of particular regions of the chromosome are not longer present. Regions so identified will then be analyzed further to identify potential candidate genes, and the true role of these regions and genes in producing phenotypic changes in trisomy established by transgenic and homologous recombination techniques. To accomplish these goals, Dr. Epstein and colleagues will generate mouse Ts16 embryonic stem cells with progressive truncations of chromosome 16 (MMU) 165) and analyze the phenotypic (immunologic, hematopoietic, central nervous system) of the partial Ts animals derived from these cells. He and colleagues will further generate and characterize Ts16 embryos and fetuses (derived from stem cell lines with complete and truncated chromosome 16s) with regard to overall morphogenesis, to the viability and gene expression of central nervous system neurons, to development of immune and hematopoietic systems, and to cardiac morphogenesis.

Abstract

Proposal: DMS-9970487
Principal Investigator: Charles Epstein

Let (X,H) denote a compact, odd dimensional manifold with a
contact structure, H. The basic examples are (1) The co-sphere
bundle of a compact manifold, (2) The strictly pseudoconvex
boundary of a complex manifold. Any contact manifold has a
distinguished algebra of pseudo-differential operators called the
Heisenberg algebra. Epstein and Melrose, building on work of
Boutet de Monvel and Guillemin, identified a family of order zero
projection operators in the Heisenberg algebra, generalizing the
Szego projector defined on the boundary of a complex manifold.
They established that if S, S' are two such projectors then the
projection defined by S from the image of S' to the image of S is
a Fredholm map. Let i(S,S') denote the index of this
restriction. We call this the relative index. By considering
Heisenberg operators which act on vector bundles, many classical
index problems can be re-phrased as relative index problems for
pairs of generalized Szego projectors. The main thrust of the
proposed research is to obtain formulae for the relative index in
terms of more classical geometric invariants, such as indices of
Dirac operators and holomorphic Euler characteristics. A general
relative index formula of this type will allow Epstein and
Melrose to completely settle the general case of the
Atiyah-Weinstein problem, asking for a geometric formula for the
index of an FIO defined by a contact diffeomorphism between two
co-sphere bundles. Beyond this the investigator plans to search
for analytic invariants of contact structures. Using an approach
suggested by Lempert and an extension of the Heisenberg algebra
he hopes to obtain a second formula for i(S,S'), which should be
useful in studying the small embeddable deformations of the
CR-structure on a strictly pseudoconvex, compact CR-manifold.

Partial differential equations are the principal tool for
mathematical descriptions of problems in physics, chemistry,
economics, etc., and it is a fundamental problem to decide how
many solutions, if any, such an equation has. In general this
problem is too delicate to solve exactly, but there is a number
called the Fredholm index which can estimate the number of
solutions. The main result in this subject is the "Atiyah-Singer
Index Theorem." The research proposed by Epstein is concerned
with extending these ideas to more general classes of equations
which arise in the context of "contact manifolds," a natural
arena for problems in mechanics and control theory. Analytical
tools are being developed which should have many applications
beyond the formula for the Fredholm index.

Dr. Epstein is presently a professor in the Mathematics department at the University of Pennsylvania. He proposes to spend a year visiting the magnetic resonance research group in the Radiology Department of the Hospital of the University of Pennsylvania, where Dr. Felix Wehrli has agreed to be his host. While there he will study topics related to magnetic resonance imaging and gain hands on experience with imaging apparatus. His study will be directed toward determining the practical feasibility of magnetic resonance imaging using an inhomogeneous background field and non-linear gradient fields. The initial part of this problem involves the analysis of certain special solutions of Maxwell's equations. The next step is to study the practical invertibility of a class of integral transforms which are perturbations of the standard Fourier transform. Hopefully this will lead up to designs for magnets whose fields, though inhomogeneous, can still be used for magnetic resonance imaging. The ultimate goal is to build a working prototype. If inhomogeneous field imaging should prove feasible it could reduce the cost of the imaging apparatus and increase the flexibility and applicability of this technology. This IGMS project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).

transcranial magnetic stimulation; nervous system disorder therapy; clinical depression; Parkinson's disease; gait; neuropsychology; muscle rigidity; tremor; abnormal involuntary movement; patient oriented research; clinical research; human subject;

Complex analysis forms one of the great organizing principles of both pure and applied mathematics. In this conference, to be held from June 18 to 22 of 2007 at the facilities of the Institute Henri Poincare in Paris, we will explore analysis of functions of one and several complex variables and the ways in which complex analysis is used to address problems both inside and outside of pure mathematics, for example, in geometry, topology, inverse scattering theory, and mathematical physics. The meeting marks the sixty-fifth birthday of Gennadi Khenkin, and for young American mathematicians this conference represents a golden opportunity to be introduced to the "Russian" approach to complex analysis and its applications. Despite the more than fifteen years since the collapse of Soviet Union, this approach to the subject is still not well known amongst American students. The list of plenary speakers includes some of the finest products of
the Russian school of complex analysis.

The purpose of this grant proposal is to provide support for US-based participants in this important international meeting. We are requesting funds to defray the travel expenses for several of the plenary speakers, as well as for several promising young mathematicians. The plenary talks will be aimed at nonexperts and cover topics ranging from representation theory to complex dynamics, transcendental methods in algebraic geometry to inverse scattering theory. Reflecting a remarkable feature of Professor Khenkin's career, this meeting will emphasize the interplay between techniques from complex analysis and its applications to inverse problems, mathematical physics, dynamics, representation theory, and topology. Details are available at the conference website (http://www.institut.math.jussieu.fr/projets/ac/evenements/2006/henkin/).

Metaphors are not just ossified expressions in grammar books. Rather, they are living, evolving expressions that pepper our language far more heavily than one might think. Many expressions that we take for granted are actually metaphorical, that is, they represent one concept by referring to another. For instance, the sentence 'He was feeling down' actually involves metaphorical usage of the spatial word 'down.' Given their ubiquity, processes in the brain that mediate comprehension and production of metaphors are critically important to human language. Dr. Krishnankutty Sathian of Emory University and his colleagues are carrying out a project to determine how metaphors are processed by the brain. It turns out that metaphors very often represent a somewhat abstract concept by implicitly invoking similarity to a more concrete concept that is grounded in our sensory experiences. For example, in the sentence, 'She had a rough day,' the word 'rough' connotes something unpleasant, similar to the way a rough surface feels unpleasant. Much research has already established that various sensory domains, such as color, shape, texture and spatial location, are each processed in relatively distinct sectors of the brain (specifically, within parts known collectively as "sensory cortex"). Dr. Sathian's project is testing whether or not understanding metaphors involves activation of those particular sensory cortical regions (e.g. the region specialized for processing texture in the case of metaphorical usage of 'rough'). In order to do this, brain scans using functional magnetic resonance imaging (fMRI) are being carried out while healthy volunteers listen to sentences containing metaphors. The brain responses to metaphors are being contrasted with responses during presentation of sentences of similar difficulty but lacking metaphors. How different brain regions interact is also being examined using measures of neural connectivity. Even if the predicted sensory cortical activity is found, it does not mean that the activity is actually necessary for metaphor comprehension. To test the necessity of cortical activity, another technique known as 'transcranial magnetic' stimulation (TMS) is being used. Tiny magnetic pulses are applied to specific brain regions to transiently disrupt their function. If TMS over particular sensory cortical regions affects understanding of the corresponding metaphors, then those sensory regions are truly necessary for understanding metaphors.

This work will enable new insights into the basis of metaphorical thought. Especially because the use of metaphors in language is pervasive, this research is contributing to fundamental understanding of human cognition. This research is helping to answer the question whether comprehending metaphors requires activity, not only in classic language regions of the brain, but also more widely in other sensory regions. The findings are providing evidence concerning theories (so-called 'grounded cognition' theories), which argue that knowledge is grounded in basic sensory and motor experiences (as first theorized by Greek philosophers like Aristotle and Epicurus). The project is also showing how different brain regions, specialized for different purposes, interact during high-level thinking. This research is relevant to a variety of human activities, including literary writing, pedagogical teaching, political speeches, and product marketing, in all of which the best expositors typically make innovative and highly effective use of metaphors. It is also relevant to many neurological disorders such as autism, stroke and Alzheimer's disease, which impair communication between brain regions, resulting in problems with figurative thinking.

This award provides funding for participation in the conference "Operator Algebras in the Twenty-first Century" held March 30-31, 2019 at the Department of Mathematics, University of Pennsylvania. The conference focuses on recent developments in analysis, especially in the field of operator algebras and its applications in physics. The conference aims to facilitate both the growth of the operator algebras field per se, as well as to inform non-specialists of the power of these methods for addressing questions outside of this field. Several distinguished mathematicians have agreed to attend and speak at this conference. This award gives early career researchers, members of underrepresented groups, and researchers without other sources of funding an opportunity to attend and participate in this conference.

Operator algebras provide a powerful perspective on problems throughout mathematics. This meeting will review some of the many accomplishments in this field, especially as related to the work of Richard V. Kadison, and their relationships to problems of current interest in mathematics and mathematical physics. Topics include: relations with topological quantum field theory, advances in condensed matter physics, the Kadison-Singer conjecture, and elaboration of the non-commutative geometry paradigm, which places operator algebras at the center of problems in fields from geometry, to mathematical physics, to analytic number theory and beyond.

More information is available at the conference web site
https://www.math.upenn.edu/~deturck/kadison-conference/index.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.