*B.S***
***in
Mathematics*** :**
1987, Al.I.Cuza University of Iasi,
Romania .

** PhD
in Mathematics:** 1994, Michigan
State University, USA.

** Thesis
advisor:** Professor
Thomas H. Parker:

My field of expertise is global analysis with an emphasis on the
geometric applications of elliptic partial differential equations
arising from *gauge theory*, *symplectic* *geometry *and
*index theory for Dirac operators*. In 2010 I asked my self some
questions about “random”Morse functions and since then I turned
into a probabilist enthusiast.

**97. ***Kac-Rice
formula and the number of critical points of random functions.*
These expository notes have a double purpose: (a) describe a complete
proof of the Kac-Rice formula describing the expectation of Z(F)= the
number of critical points of a Gaussian random function F and
(b) present some ideas in the works of
Gass-Stecconi and Ancona-Letendre describing conditions
guaranteeing the finiteness of the p-th moment of Z(F)

**96. ***A
Graduate Course in probability**,
*These are notes that grew out of the one-semester
probability basic graduate course I taught. Now they roughly cover a
year-long basic graduate course. Check the list of contents for more
details.

**95. ***Notes
on Elementary probability*
These are notes for the undergraduate probability class I have
taught at the University of Notre Dame for three years. They have
been continuously update to incorporate remarks and questions from my
students. The list of topics is more or less the list of topics
required for the actuaries Exam-p. I believe that the best way to
understand probability is from examples and by experiencing random
experiments. The book discusses contains many classical examples.
Additionally we have included the short R-programs used for class
simulations. For this reason, the last chapter of the book offers a
very basic introduction to R. We have included many exercises, of
varied difficulty, inspired from undergraduate courses in US and
Europe. The complete solutions are contained in Appendix B of the
book.

**94. ***The
(co)normal cycle and curvature measures *These are
notes for a cycle of lectures I gave in the department. You can see
videos of these lectures here.

**93.** *Wiener
chaos and limit theorems.* I survey the concepts of
Gaussian Hilbert spaces, their chaos decomposition and the
accompanying Malliavin calculus. I then describe how these
ingredients fit in the recent central limit theorems of Nourdin and
Peccati in the Wiener chaos context.

**92. ***Critical
points* *of*
*multi-dimensional random
Fourier series: central limits.*
I complete the work in **87**
and I prove a central limit
theorem concerning the distribution of critical points of a
multidimensional random Fourier series when the lattice of periods is
rescaled by a large factor.

**91. ***A
CLT concerning the critical points of a random function on a
Euclidean space *We
prove a central limit theorem concerning the number of critical
points in large cubes of an isotropic Gaussian random function on a
Euclidean space. **arXiv:
1509.06200**

**90.** *On
the Kac-Rice formula* This is an informal introduction
to the Kac-Rice formula and some of its (mostly one-dimensional)
applications.

**89****.** *A
stochastic Gauss-Bonnet-Chern formula,* We prove that a
Gaussian ensemble of smooth random sections of a real vector bundle
over compact manifold canonically defines a metric on the bundle
together with a connection compatible with it. Additionally, we prove
a refined Gauss-Bonnet-Chern theorem stating that if the bundle and
the manifold are oriented, then the Euler form of the above
connection can be identified, as a current, with the expectation of
the random current defined by the zero-locus of a random section in
the above Gaussian ensemble.

**88****.** *T**he
Gauss-Bonnet-Chern theorem: a probabilistic perspective*,
**this is joint work with Nikhil Savale. **We show that the
Euler form determined by a metric connection on an oriented metric
vector bundle can be identified as a current with the expectation of
the random current defined by the zero locus of a random section of
the bundle. We also explain how to recover the differential geometry
of a metric vector bundle equipped with a metric connection from the
statistics of a certain random section.

**87. ***Critical
points of multidimensional Fourier series: variance estimates*,
**arXiv:
1310.5571**. I show that in the white noise limit the
number critical points of a multidimensional random Fourier series
concentrates with high probability around its expectation.

**86. ***Introduction
to Real Analysis *These are notes for a yearlong
freshman honors course on one-variable real analysis*.*

**85.** *Notes
on linear algebra.* These are notes for the
second half of the yearlong honors course on linear algebra.

**84. ***The
Crofton formula* These are notes for a graduate
class. I give a proof of the Crofton formula for plane curves.

**83**. *Fluctuations
of the number of critical points of random trigonometric polynomials*
I show that the number of critical points of a random trigonometric
polynomial of large degree concentrates with high probability around
its expectation.

**82. ***Random
Morse functions and spectral geometry*
This is a continuation of **79. **To each even, nonnegative
Schwartz function $w(t)$ on the real axis I associate a random
function on a compact Riemann manifold $(M,g)$ of dimension $m$. I
investigate the expected distribution of critical values of such a
random function and its behavior as we rescale $w(t)$ to
$w(\varepsilon t)$ and then let $\varepsilon\to 0$. I prove a central
limit theorem describing what happens to the expected distribution of
critical values when the dimension of the manifold is very large.
Finally, I explain how to use the $\varepsilon\to 0$ behavior of the
random function to recover the Riemannian geometry of $(M,g)$.

**81. ***The
wave group and the spectral geometry of compact manifolds*
I am trying to learn Hormander's estimates of the spectral
function of the Laplacian on a compact Riemannian manifold. This is
more of a struggle than I had anticipated and these time dependent
notes are the result of this struggle. They're supposed to be as
self-contained as possible, but familiarity with basics of
distribution theory is assumed. In the second part, familiarity with
basics of pseudo-differential operators is recommended. The first two
chapters of my lecture notes on these two topics
could serve as a source for this assumed backround.)

**80.** *Combinatorial
Morse flows are hard to find* I investigate the
probability of detecting combinatorial Morse flows on a simplicial
complex via a random search. It is really small, in a quantifiable
way.

**79. ***Complexity
of random smooth functions on compact manifolds*
I relate the distribution of eigenvalues of a random symmetric matrix
in the Gaussian Orthogonal Ensemble to the distribution of critical
values of a random linear combination of eigenfunctions of the
Laplacian on a compact Riemann manifold. I then prove a central limit
theorem describing what happens when the dimension of the manifold is
very large.

**78. ***Jets
of the distance function* If you ever wanted to
know how to compute the jets along the diagonal of the distance
function on a Riemann manifold, then this note might be what you are
looking for. I asked many Riemannian geometers how to compute these
jets but none could give a reference. The usage of the
Hamilton-Jacobi equations makes these computations bearable.

**77. ***The
generalized Mayer-Vietoris principle and spectral sequences**.*
These are notes for a the Fall 2011 Topics class. I use Bott-Tu as
guide but I at various places I use different approaches. You may
also want to consult these notes
on the cohomology of sheaves.

**76. ***Intersection
theory* I cover in
greater details the intersection theoretic results in Bottt-Tu. These
are notes for a Fall 2011 topic class.

**75**. *Pixelations
of planar semialgebraic sets*
**(this is joint work with Brandon
Rowekamp)** We describe an
algorithm that associates to each positive real number $r$ and any
finite collection $C_r$ of planar pixels of size $r$ a planar
piecewise linear set $S_r$ with the following additional property: if
$C_r$ is the collection of pixels of size $r$ that touch a given
compact semialgebraic set $S$, then the output $S_r$ of the algorithm
converges to the set $S$ in a rather strong way. The above strong
converges guarantees that in the limit we can recover the homotopy
type of $S$ and geometric invariants such as area, perimeter and
curvature measures. At its core, this algorithm is a a discrete
version of stratified Morse theory. Here
is a beamer presentation of the juiciest parts.

74. *Flat
currents and their slices.* These are notes for
a talk at the ''Blue Collar Seminar on
Geometric Integration Theory''.

**73. ***The
blow up along the diagonal of the spectral function of the Laplacian*
I describe a surprising
universal behavior of the spectral function of the Laplacian of a
compact Riemann manifold.

**72. ***The
coarea formula*
These are notes for a talk at the ''Blue
Collar Seminar on Geometric Integration Theory''.

**71. ***Critical
sets of random smooth functions on compact manifolds *I
estimate the expected number of critical points of a random linear
combination of eigenfunctions of the Laplacian on certain compact
Riemann manifolds. (Here
is a nice animation with random functions on spheres.)
Here
is a beamer presentation of the most important results.

**70.
***Statistics
of linear families of smooth functions on knots.*
Given a finite dimensional vector space of smooth functions on a
knot, I compute the expected number of critical points of a random
function in this space. As a consequence, I show that the expected
number of critical points of the restriction to the unit circle in
the plane of a random homogeneous polynomial of given degree k is
equal to
.

57. *On
the total curvature of semialgebraic graphs**_
*arXiv: 0806.3683 We
define the total curvature of a semialgebraic graph $\Gamma\subset
\bR^3$ as an integral $K(\Gamma)=\int_\Gamma d\mu$, where $\mu$ is a
certain Borel measure completely determined by the local extrinsic
geometry of $\Gamma$. We prove that it satisfies the Chern-Lashof
inequality $K(\Gamma)\geq b(\Gamma)$, where
$b(\Gamma)=b_0(\Gamma)+b_1(\Gamma)$, and we completely characterize
those graphs for which we have equality. We also prove the following
unknottedness result: if $\Gamma\subset \bR^3$ is homeomorphic to the
suspension of an $n$-point set, and satisfies the inequality
$K(\Gamma) <2+b(\Gamma)$, then $\Gamma$ is unknotted. Moreover, we
describe a simple planar graph $G$such that for any $\varepsilon>0$
there exists a knotted semialgebraic embedding $\Gamma$ of $G$ in
$\bR^3$ satisfying $K(\Gamma)<\varepsilon +b(\Gamma)$.

**56. ***On
a Bruhat-like poset**_*
**arXiv: 0711.0735**
We investigate the poset of strata of the stratification
introduced in the paper **55. **

**55.** *Schubert
calculus on the Grassmannian of hermitian lagrangian subspaces**_*
**arXiv: 0708.2669, Adv. Math. 2010,
doi:10.1016/j.aim.2010.02.003** We describe a Schubert like
stratification on the Grassmannian of hermitian lagrangian spaces in
$\mathbb{C}^n\oplus \mathbb{C}^n$ which is natural compactification
of the space of hermitian $n\times n$ matrices. The closures of the
strata define integral cycles, and we investigate their intersection
theoretic properties. The methods employed are Morse theoretic.

**54.** *Tame
flows***
****math.GT/0702424, (Memoirs A.M.S.) ****The
tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame)
sets are the pfaffian sets introduced by Khovanski, and a flow $\Phi:
\mathbb{R}\times X\rightarrow X$ on pfaffian set $X$ is tame if the
graph of $\Phi$ is a pfaffian subset of $\mathbb{R}\times X\times X$.
Any compact tame set admits plenty tame flows. We prove that the flow
determined by the gradient of a generic real analytic function with
respect to a generic real analytic metric is tame. The typical tame
gradient flow satisfies the Morse-Smale condition, and we prove that
in the tame context, under certain spectral constraints, the
Morse-Smale condition implies the fact that the stratification by
unstable manifolds is Verdier and Whitney regular. We explain how to
compute the Conley indices of isolated stationary points of tame
flows in terms of their unstable varieties, and then give a complete
classification of gradient like tame flows with finitely many
stationary points. We use this technology to produce a Morse theory
on posets generalizing R. Forman's discrete Morse theory. Finally, we
use the Harvey-Lawson finite volume flow technique to produce a
homotopy between the DeRham complex of a smooth manifold and the
simplicial chain complex associated to a triangulation. Here
is a beamer presentation of the most important results.**

**53. ***Curvature
measures* I like
very much how these notes turned out. This is an old subject,_ and in
these notes I dress it in some new clothes.

**52. ***Metoda*
*functiilor
generatoare* (“*Gazeta Matematica’’,
2007’*). Try this if you can read Romanian.

**51. ***Morse
functions statistics***,
****math.GT/0604437,****
****_**Functional
Analysis and Other Applications**, 1(2006), 97-103. ****I**
answer a question of V.I.
Arnold concerning the growth of the number of Morse function on
the 2-sphere. Read more about this
problem. Also, you can
watch Arnold talking about this.

**50.
***Counting
Morse functions on the 2-sphere***,
****math.GT/0512496. ****Compositio
Math., 144(2008), 1081-1106. This **paper**
****is inspired by
earlier work of **V.I.
Arnold**. We compute the
number of excellent Morse functions on the two sphere. Here
are beamer slides on this topic.**

**49. ***The
anatomy of a singularity** ,
*These are notes for a Felix Klein Seminar.

**48. ***Notes
on Morse theory,*These are notes for
the Fall 2005,“*Topics in topology” *course.

**47. ***On
a multidimensional checkers game** ,
*This is a generalization of a problem from the
Budapest semester in math that I learned from A. Stipsicz. I suspect
somebody else has thought of this generalization before, but in any
case, here it is.

**46. ***The
Euler characteristic***, ****Some
nice consequences of really elementary tricks.**

**45. ***Characteristic
currents of singular connections*

**43.** *Notes
on the Atiyah-Singer index theorem**,*
These are notes for the Math 658 , Spring 2004 course, “*Topics
in topology”.*

**42.** *Microlocal
studies of shapes* I
begin by describing two classical instances where the conormal cycle
makes its appearance and then I describe its construction in the
special case of simplicial complexes.

**41. ***Derangements
and asymptotics of Laplace transforms of polynomials
**__***math.CO/0401281
& **New
York Journal of Mathematics**, v.10(2004), 117-131. **This
is an amusing elementar problem with surprising ramifications.

**40.** “*Motivic”
Integral Geometry**_*
Here I am trying to understand a beautiful little paper of
Pierre Schapira. For an approach to this problem using tame geometry
check * here*.

**39.** *The
Poincare-Verdier duality*

**38.** *Chern*
*classes
of singular algebraic varieties** ,*
Try this link from time to time.

**37.** *On
the curvature of singular complex hypersurfaces,*
Talk at the *Felix Klein Geometry
Seminar*, Fall 2003, Notre Dame.

**36.** *The
many faces of the Gauss-Bonnet theorem**,*
This little expository paper is an extended version of my talk at the
*Graduate Student Seminar*, Fall
2003.

**35.** ** Three-dimensional
Seiberg-Witten theory,** Notes
for a mini-course in Grenoble.

**34. ***Notes
on the topology of complex singularities **,***
**(PDF, Revised 07/11/2001) This is an ongoing process.
I have included lots of NEW examples.

**32.** *Homeomorphisms
vs. Diffeomorphisms***
**This little expository paper is an extended version
of my talk at the 2003 `*`First year
graduate student seminar''.*

**31.** *Residues
and Hodge Theory***
**This expository paper is an extended version of
my talks at Karen Chandler's *``Absolutely
fabulous algebraic geometry seminar''.*

**30.** *Solutions
to some problems in Hatcher's Algebraic Topology book.*

**29.**
*Seiberg-Witten
invariants and surface singularities III. Splicings and cyclic
covers**,
***math.AG/02071018 (joint with
Andras Nemethi)
**We prove the conjecture formulated in **25 **for
suspension sigularities, i.e. isolated surface singularities of the
form z^n+ f(x,y)=0, where, f(x,y)=0 is an irreducible plane curve.

**28.** *Seiberg**-Witten
invariants and surface singularities.***
**This is a summary of **25 **and **26.**

**27. ***Of
shapes, differentials and integrals.***
**This little expository paper is an extended version
of my talk at the 2002 *``First year
graduate student seminar''*.

**26.**
*Seiberg-Witten
invariants and surface singularities II. Singularities with good
$C^*$-action**,
***math.AG/0201120 (joint with Andras
Nemethi) **We prove that the geometric genus of an
isolated quasihomogeneous surface singularity whose link is a
rational homology sphere is completely detremined by the
Seiberg-Witten invariant and the Gompf
invariant of the canonical spin^c structure on the link of this
singularity. When the sigularity is smoothable this implies that the
Seiberg-Witten invariant of the canonical spin^c structure is -1/8
the signature of the Milnor fiber. Note although these singularities
are analytically rigid some are neither Gorenstein, nor rational.
Which is then the best rigidity condition?

**25.** *Seiberg**-Witten
invariants and surface singularities I**,
***math.AG/0111298, Geometry and
Topology, 6(2002), 269-328, (joint with Andras
Nemethi) **We formulate a conjecture relating analytic
invariants of isolated surface singularities to topological
invariants of the links of those singularities, which are assumed to
be rational homology 3-spheres. More precisely we show that the link
of such a singularity is equipped with a canonical spin^c structure.
We conjectured that if the singularity is Gorenstein or rational then
its geometric genus is determined in a simple explicit fashion by the
Seiberg-Witten invariant and the Gompf
invariant of this canonical spin^c structure. To support this
conjecture we establish its validity on many classes of
singularities: Brieskorn, cyclic quotient singularities, A-D-E
singularities, polygonal singularities etc. This considerably
generalizes earlier results of Fintushel-Stern, Neuwmann-Wahl.

**24.**** ***Seiberg**-Witten
invariants of 3-manifolds. Part 1***
**This unfinished
manuscript describes in great detail the construction of SW theory of
3-manifolds, closed or with boundary. It has considerable overlap
with published work of Yuhan Lim and Marcolli-Wang (which is why I
never finished it and I will never publish it). However, I adopt a
sufficiently new point of view to make it useful for references. In
particular, I am careful in figuring out the various signs. There are
a couple of other folklore results for which I could not find
references.

**23.** *Seiberg-Witten
invariants of rational homology spheres***
****math.GT/0103020, Comm. Contem. Math. ****I
prove that for rational homology 3-spheres SW**
invariant <==> Casson-Walker invariant + Reidemeister torsion.

**22.** *Geometric
connections and geometric Dirac operators on contact manifolds,***
****math.DG/0101155, Diff. Geom and its Appl. ****I**
construct several natural
connections and Dirac type operators on a general metric contact
manifold which are more sensitive to the geometric background. In the
special case of CR manifolds these connections are also compatible
with the CR structure and include among them the Webster connection.
We also describe several Weitzenbock type formulae. Our method is
based on work of P. Gauduchon applied to two almost complex manifolds
naturally associated to a given metric contact manifold.

**21.** *On
the Reidemeister torsion of rational homology spheres**,***
***(PDF) ***math.GT/0006181, Int. J. of Math
and Math Sci. 25(2001), 11-17. **I
prove that the mod **Z
**reduction of the
torsion of a rational homology 3-sphere is completely determined by
three data: a certain canonical spin^c structure, the linking form
and a **Q/Z**-valued
constant *c*.
This constant is a new topological invariant of the rational homology
sphere. Experimentations with lens spaces suggest this constant may
be as powerful an invariant as the torsion itself.

**20.** ** Notes
on Seiberg-Witten theory**,

**19.** ** The
Reidemeister Torsion of 3-Manifolds**,
Walter de Guyter, January 2003. I
survey, mainly through very concrete examples, various
interpretations of this topological invariant. I think you will find
many nice old things and new ones.

**18.** *On
the space of Fredholm selfadjoint operators**,***
***(PDF),*_
**An. Sti. Univ.
Iasi**, **53** (2007), p.209-227. *This is a paper is
dedicated to the memory of my good friend ***Gheorghe
Ionesei***, the ultimate math poet. You can find ***here***his cute proof of the Abel-Jacobi theorem. *In
this paper I compare various natural topologies on the space of
(*possibly unbounded*) Fredholm selfadjoint operators and
describe which one is relevant for K-theory. In the process, I fix a
gap in **7 **indicated to me by Bernhelm Booss-Bavnbeck
and K. Furutani.

**17.** ** Eta
invariants, spectral flows and finite energy Seiberg-Witten
monopoles**, ( PDF file, 21
pages),

**16.** *Seiberg-Witten
invariants of lens spaces**.*
**math.DG/9901071**, **Canad. J. Math., 53(2001),
780-808. **It contains an algorithm based on the ideas in **13**,
for computing Seiberg-Witten invariants of lens spaces. We apply this
algorithm to two problems: (i) to compute the Froyshov invariants of
many families of lens spaces; (ii) to show that the knowledge of the
Seiberg-Witen invariants of a lens space is topologically equivalent
to the knowledge of its Casson-Walker invariant + the Milnor-Turaev
torsion. Problem (i) has interesting applications concerning the
negative definite manifolds bounding a given lens space. (The
published version contains less information that the version on the
network.)

**15.** ** On
the Cappell-Lee-Miller gluing theorem**,

**14.** ** Lattice
points inside rational simplices and the Casson invariant of
Brieskorn spheres**,

**13.** ** Finite
energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert
fibrations**,

**12.**
*On
a theorem of Henri Cartan concerning the equivariant cohomology***
**, **math.DG/0005068**, **An.
Sti. Univ. Iasi**, **45**(1999), 1-17. I give a
new proof of a theorem of H. Cartan concerning the equivariant
cohomology of the infinitesimally free smooth actions of compact Lie
groups. The approach is based on a BRST isomorphism of J. Kalkman and
the unusual strategy of connecting two genuine geometric objects
(connections in a principal bundle) via a path of virtual (formal)
ones. The resulting isomorphism is described at the ** chain
level.** My interest in equivariant cohomology is due to
its presence in gluing formulae for gauge theoretic invariants.

**11.**** ***Eta
invariants of Dirac operators on circle bundles over Riemann surfaces
and virtual dimensions of finite energy Seiberg-Witten moduli spaces*,
**math.DG/9805046**, **Israel J. Math**., **114**(1999),
61-123. This is a continuation of the paper **10.** I am primarily
interested in the virtual dimensions of finite energy Seiberg-Witten
moduli spaces on 4-manifolds bounding a disjoint union of circle
bundles over Riemann surfaces. In the special case when the manifold
is a cylinder based on such a circle bundle this dimension was
determined by Mrowka-Ozsvath-Yu by algebraic-geometric techniques. I
use a differential-geometric approach based on the
Atiyah-Patodi-Singer theorem. This requires firstly the determination
of the eta invariants of some Dirac operators. I present two entirely
distinct methods to achieve this. The first method relies on the
Bismut-Cheeger-Dai adiabatic results while the second method is more
elementary and allows me to determine the *entire eta function*.
An important part of the virtual dimension formula is a certain
spectral flow. This is determined at the end of a very delicate
perturbation analysis.

**10.** ** Adiabatic
limits of the Seiberg-Witten equations on Seifert manifolds**,

**9.** ** Lectures
On The Geometry Of Manifolds**,

**8.** *On
the cobordism invariance of the index of Dirac operators*,
**Proceedings A.M.S.**, **125**(1997), 2797-2801. I show that
any cobordism between two Dirac operators $D_0$, $D_1$ defines a
natural tunneling isomorphism $\ker D_0 \oplus \ker D_1^* \rightarrow
\ker D_1 \oplus \ker D_0^*$. In particular, this provides a very
intuitive explanation for the equality $ ind D_0= ind D_1$. This
tunneling map has potential applications in the recent $spin^c$
quantization theory of V. Guillemin and his school.

**7.** *Generalized
symplectic geometries and the index of families of elliptic problems**,***Memoirs A.M.S.**, vol. **128**, Number 609, 1997. This is a
far reaching generalization of the first half of my thesis. I prove a
``surgery'' formula for the index of an *arbitrary* family of
Dirac operators. This arbitrariness encompasses two aspects. The
family is parameterized by an *arbitrary compact CW-complex *and
moreover we also discuss *higher K-theoretic* indices. I
proposed a method which deals *simultaneously* with *both
*difficulties. I first cast the problem in a symplectic framework
and then I deal with it using a generalization of the symplectic
reduction technique. Again a central player is a higher dimensional
version of the Maslov index which is given a dual interpretation,
*K*-theoretic and symplectic. A nice byproduct of this work is
an ``adiabatic proof'' of a very general result concerning the
cobordism invariance of the index of families.

**6.** ** Morse
theory on grassmannians**,
An.
Sti. Univ. Iasi,

**5.** *The
spectral flow, the Maslov index and decompositions of manifolds**,***
****Duke Univ.J **,
**80**(1995), p.485-534. This is essentially my PhD dissertation.
I prove a gluing formula for the spectral flow of a path of
selfadjoint Dirac operators (on a manifold cut in two parts by a
hypersurface) in terms of an infinite dimensional Maslov index. To
get a better understanding of this infinite-dimensional contribution
I ``stretch'' indefinitely the neck of the separating hypersurface
and then *explicitly* describe the behavior of this Maslov index
in the adiabatic limit.

**4.** *Rigidity
of generalized laplacians and some geometric applications*,
**Aequationes** **Mathematicae**, **48**(1994), p.143-162.
This is my first paper as a graduate student. One consequence of this
work is that a Riemann metric is determined (up to a homothety) by
its associated sheaf of (locally defined) harmonic functions.

**3.** *A
weighted semilinear elliptic equation involving critical Sobolev
exponents*,
**Differential and Integral Equations**, **4**(1991),
p.653-671. I study the critical situations for the equations
considered in the previous paper. This work extends previous results
of H. Brezis and L. Nirenberg.

**2.** *Existence
and regularity for a singular semilinear Sturm-Liouville problem*,
**Differential and Integral Equations**, **3**(1990),
p.305-322. I study the radially symmetric solutions of a semilinear
elliptic problem on a ball equipped with a radially symmetric but
possibly singular metric. The singular metric defines a new critical
Sobolev exponent (with respect to suitable weighted Sobolev spaces).
This paper is concerned with with existence and regularity in the
(singular) sub-critical case.

**1.** *Optimal
control for a nonlinear diffusion equation*,
**Richerche** **di matematice**, vol. **37** (1988), p.3-27.
This is essentially my senior thesis. I solve a problem posed by J.
I. Diaz and A. Friedman concerning a reaction-diffusion process at
equilibrium controlled by a catalyst. The goal is to determine the
optimal distribution of catalyst which produces the maximum amount of
output. One also has to take into account some cost considerations.
After deriving first-order optimality conditions in the form of
nonlinear variational inequalities I then solve them *explicitly*
using the technique of symmetric rearrangements.

*__________________________________________________________________________________________________________________________________________________________
*

*10. **Concentration
of measures* This is the senior thesis of * Misha
Sweeney. *He surveys some classical and modern
concentration of measure results and presents a few applications

*9. **Information
theory* This is the senior thesis of * Patrick
LeBlanc.* He studies the entropy of probability
measures on finite sets (alphabets) and its relationtip to coding
various ergodic communications.

*8. **Crofton
formulae, *This is the senior thesis of * Paul
Sweeney*. He proves The Croftonformula for curves and
hypersurfaces in an Euclidean space.

*7. **Markov
chains:A Random Walk Through Particles, Cryptography, Websites and
Card Shuffling**, *This is the senior
thesis of * Mike McCafferty*.
He goes through the basic facts about Markov chains, proves
ergodicity and various of the limit theorems and then looks at
various examples of Markov chains produced via the
Metropolis-Hastings algorithm. He has also implement in R a
decryption procedure discussed by Perci Diaconis in a Bulletin AMS
survey.

6. *A
study of planar pixelation*, This is the
PhD dissertation of * Brandon
Rowekamp*. He describes how to recognize the
essential geometric features (area, perimeter, curvature, Betti
numbers) of a planar set from its pixellations.

5. *Regularization
of divergent series and Tauberian theorems**,
*This is the senior thesis of * Jack
Enyart*. Nice introduction to the subject.

4. *Localization
formulae in odd K-theory**,*This
is the PhD dissertation of * Daniel
Cibotaru*. I like very much how it turned out,
and I am sure it will find applications.

**3. ***Two
proofs of the DeRham theorem**,*This
is the *senior thesis* of my
student * Andrew Fanoe*. He
goes through two different proofs of the classical DeRham theorem,
the geometric one by H. Whitney, and the more algebraic one due to A.
Weil. I think he improved on Weil’s presentation by using the
concept of (semi)simplicial sets.

**2._ ***Asymptotics*
*of
oscillatory integrals*
This is the *senior
thesis* of my student * R. Zach
Lamberty*. He describes on a concrete two dimensional
example Varcheko’s technique of investigating the asymptotics of
oscillatory integrals via toric resolutions of the phase. Check it
out ! I like how it came out.

**1.** *Geometric
Valuations* Very nice notes by the REU students
* Cordelia
Csar, Ryan Johnson*_
and