*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*.

**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

**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.

*__________________________________________________________________________________________________________________________________________________________
*

*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