We develop a model of phonological contrast in natural

language. Specifically, the model describes the maintenance of

contrast between different words in a language, and the elimination of

such contrast when sounds in the words merge.  An example of such a

contrast is that provided by the two vowel sounds "i" and "e", which

distinguish pairs of words such as "pin" and "pen" in most dialects of

English.  We model language users' knowledge of the pronunciation of a

word as consisting of collections of labeled exemplars stored in

memory.  Each exemplar is a detailed memory of a particular utterance

of the word in question.  In our model an exemplar is represented one

or two phonetic variables along with a weight indicating how strong

the memory of the utterance is.   Starting from an exemplar-level

model we derive integro-differential equations for the evolution of

exemplar density fields in phonetic space. Using these latter

equations we investigate under what conditions two sounds merge

causing words to no longer contrast. Our main conclusion is that for

the preservation of phonological contrast, it is necessary that

anomalous utterances of a given word are discarded, and not merely

stored in memory as an exemplar of another word.

We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.

 Study of equivalences and symmetries of real submanifolds in
complex space goes back to the classical work of Poincar\'e  and Cartan
and was deeply developed in later work of Tanaka and Chern and Moser. This
work initiated far going research in the area (since 1970's till present),
which is dedicated to questions of regularity of mappings between real
submanifolds in complex space, unique jet determination of mappings,
solution of the equivalence problem, and study of automorphism groups of
real submanifolds.

Current state of the art and methods involved provide satisfactory (and
sometimes complete) solution for the above mentioned problems in
nondegenerate settings. However, very little is known for more degenerate
situations, i.e., when real submanifolds under consideration admit certain
degeneracies of the CR-structure.

The recent CR (Cauchey-Riemann Manifolds) - DS (Dynamical Systems)
technique, developed in our joint work with Shafikov and Lamel, suggests
to replace a real submanifold with a CR-singularity by an appropriate
dynamical systems. This technique has recently enabled us to solve a
number of long-standing problems in CR-geometry, in particular, related to
a Conjecture by Treves and that by Ebenfelt and Huang.

The technique also has applications to Dynamics.

In this talk, we give an overview of the technique and the results
obtained recently by using it.

Ricci solitons, as self-similar solutions to the Ricci flows, are important for understanding both the dynamic and singularity formation of the Ricci flows. The talk will primarily focus on the four dimensional shrinking Ricci solitons. We will explain some of the recent progress, made jointly with Ovidiu Munteanu, toward the structure at infinity of such solitons.