Liviu Nicolaescu

1. Area of research.

Most of my research to date has been motivated by three grand themes: index theory, gauge theory and Morse theory. My favorite investigative tools are analytical in nature (pde’s, functional analysis) though my goals are purely geometric

2. Keywords global analysis,

3. Current PhD students.

Brandon Rowekamp

4. Former PhDs

Daniel Cibotaru: Localization formulae in odd K-theory

Current Employer: Universidad Federal Fluminense, Rio de Janeiro, Brasil.

5. Postdocs mentored

Francois Ledrappier

1. Area of research.

I am working in Ergodic Theory, mostly on asymptotic growth rates of the entropy family. Recent papers deal with applications to global geometry of Riemannian manifolds, to actions of discrete subgroups of semi-simple groups, to random walks on discrete groups, and to geometric group theory.

2. Key-words

ergodic theory, entropy

3. Current PhD students.

Pablo Lessa ( Universidad de la Repùblica, Montevideo) is working on entropy relations for foliations of compact spaces with a leafwise Riemannian structure.

4. Former PhD students

N. Anantharaman (Associate Professor in Orsay) and Y. Coudène (Associate Professor in Brest).

5. Postdocs mentored

Seonhee Lim (Seoul National University).

Xiaobo Liu

1. Area of research.

I am interested in differential geometry. I have worked in calibrated geometry and submanifold geometry(isoparametric submanifolds, etc.) before. Most recently, I have been working in Gromov-Witten invariants and Quantum cohomology for symplecitc manifolds, and related problems in moduli space of curves.

2. Keywords

Riemannian geometry, symplectic geometry.

3. Current PhD students.

Tiancong Chen, Youngho Yoon

Karsten Grove

1. Area of research.

Most of my research is concerned with global Riemannian and metric geometry, much of it at the interface between geometry and topology. It includes existence theory of geodesics, comparison theory, relations between curvature, symmetry and topology, properties and constructions of positively and nonnegatively curved manifolds, and most recently relations to Tits geometry and buildings.

2. Keywords

Riemannian geometry, metric geometry

3. Current PhD students.

Joseph Yeager (Maryland),

John Harvey,

Xiaoyang Chen,

Renato Bettiol.

4. Former PhDs

P. Petersen (1987, UCLA),

K.S. Park (1988, Keimyong U., Korea),

C. Plaut (1989, UTenessee),

J. McGowan (1991, Howard U.),

F. Wilhelm (1992, UCRiverside),

C. Searle (1992, UNAM Cuernavaca, Mexico),

M. Cho (1994, Korea Nat. U. Ed., Korea),

L. Guijarro (1995, U Autonoma de Madrid, Spain),

V. Kapovitch (1997, UToronto, Canada),

K. Shankar (1999, UOklahoma),

E. Swarz (1999, Cornell U),

W. Jimenez (2005, NSA),

F. Galaz Garcia (2009, UM\"{u}nster, Germany),

B. Afsari (2009, John Hopkins U)(with Krishnaprasad EE Maryland).

5. Postdocs mentored

C.V. Kim (KIAS, Korea)(2001-03)

E. Proctor (Middlebury College) (October 2009)

Nan Li (visiting assistant professor 2010-12)

Nero Budur

1. Area of research.

My work is in Algebraic Geometry. My focus is on singularities and

the effect they have on the local and global geometry of varieties.

Among my interests are singularities of higher-dimensional varieties,

the Monodromy Conjecture in number theory, and the theory of

hyperplane arrangements.

2. Keywords Multiplier ideals, D-modules, Milnor fibers, motivic zeta functions.

3. Current PhD students. Youngho Yoon

4. Former PhDs

5. Postdocs mentored

Richard Hind

1. Area of research

Symplectic and complex geometry. I am currently studying the Hamiltonian diffeomorphisms, or mechanical motions, of phase spaces. Basic problems include deciding which regions can be mapped into which others under such a motion, and, given this, determining the minimal energy required.

2. Keywords

Symplectic, contact, complex geometry, Hamiltonian, holomorphic curve, complexification.

3. Current PhD students

Jason Nightingale.

4. Former PhD students (with their most recent affiliation)

5. Postdocs mentored.

Jens von Bergmann, 2006-2008.

William Dwyer

1. Area of research. A mixed bag of topics in algebraic topology, including some aspects of stable and unstable homotopy theory, the theory of topological groups, Goodwillie calculus, and operad theory.

2. Keywords. Algebraic topology, homotopy theory, stable homotopy theory

4. Former Ph.D. students : John Harper, Julie Bergner, Bernard Badziuch, Marian Anton, Wojchech Chacolski, Michele Intermont, Krzistof Trautman, Alexei Strounine, Jan Spalinski, Thomas Fischer, Stanislav Betley, John Sawka.

Bruce Williams

1. Area of research

Automorphisms of topological and smooth manifolds, manifold structure on fibrations, Reidemeister torsion for smooth bundles, applications of higher algebraic K-theory to geometric topology, Riemann-Roch type theorems in geometric topology

2. Keywords

Fixed point theory for fiber bundles, equivariant fixed point theory, connections between cyclotomic trace and periodic point theory

3. Current PhD students

Allegra Reiber

4. Former PhD’s

George Kennedy, National Security Agency

Tomasz Zukowski, U. of Warsaw(Poland)

Nancy Cardim, U. Federal Fluminense(Brazil)

Padmini Kamath, motherhood

Damjan Kobal, U. Ljubljana(Solvenia)

Vesta Coufal,Gonzaga University, Washington State

Stacy Hoehn, Vanderbilt(NSF postdoc)

Gun Sunyeekhan, Chulalongkorn University(Thailand)

5. Postdocs mentored

Wojciech Dorabiala, Penn. State, Altoona

Kate Ponto, U. of Kentucky

Stephan Stolz

1. Research interests:

Topological and geometric aspects of quantum field theories. One goal of my research is to contribute to a mathematical definition of quantum field theory that is versatile enough to include many different flavors of field theories (e.g., topological field theories, conformal field theories, Euclidean field theories and supersymmetric versions of these field theories).

A conjecture that my collaborator Peter Teichner (UC Berkeley and Max-Planck-Institut Bonn, Germany) pursue is to show that families of supersymmetric Euclidean field theories parametrized by a manifold $X$ represent elements of $TMF^*(X)$, the generalized cohomology theory constructed by Hopkins-Miller known as the "topolological modular form theory". Another conjecture that motivated much of my interest in field theories is the following: Let $X$ be a closed Riemannian manifold of dimension $4k$ with string structure (i.e., its tangent bundle restricted to the 4-skeleton is trivial). Then if the Ricci curvature is positive, the Witten genus of $X$ vanishes. The Witten genus is a topologically defined invariant of $M$. Heuristically, it is the "partition function" of the "non-linear $\sigma$-model of $X$, a field theory associated to $X$ that has yet to be defined rigorously.

2. Keywords: topology, differential geometry mathematical physics

3. Current graduate students:

Stuart Ambler, Ryan Grady, Santosh Kandel, Augusto Stoffel and Peter Ulrickson

4. Former graduate students:

Herman Serrano (Ph.D. 1994, Professor at the Universidad Tecnologica de Pereira, Columbia

Cherng-Yi Yu (Ph.D. 1995 Tamkang University, Taiwan)

David Wraith (Ph.D. 1995, Senior Lecturer at Maynooth University, Ireland)

Michael Joachim (Ph.D. 1997, Professor at the University of M\"unster, Germany)

Laszlo Feher (Ph.D. 1997, Professor at E\"otv\"os University, Hungary)

Mike Dekker (Ph.D. 2003, Associate Professor at Ferris State University)

Vlad Chernysh (Ph.D. 2003, unknown)

Elke Markert (Ph.D. 2005, Simons Center for Systems Biology, IAS, Princeton)

Corbett Redden (Ph.D. 2006, Hausdorff Institute, Bonn, Germany)

Florin Dumitrescu (Ph.D. 2006, Post-Doc at the University of Hamburg, Germany)

5. Postdocs mentored:

Arlo Caine (postdoc at Notre Dame 2008-2011, now assistant professor at Cal Poly Ponoma)

Justin Thomas (postdoc since Fall 2010)

Gabor Szekelyhidi

1. Area of research ( include a paragraph describing your research)

I'm interested in complex differential geometry, in particular the

existence of extremal metrics on Kahler manifolds. Extremal metrics

were introduced by Calabi in the 80s, and they are minimizers of

various curvature functionals on Kahler manifolds. In the past 10-15

years much progress has been made in relating the existence of such

metrics to algebro-geometric properties of the underlying complex

manifold. I work on both the algebraic aspects related to geometric

invariant theory, and the analytic questions that arise when trying to

solve the relevant partial differential equation.

2. Keywords (a few keywords that best describe your research)

Extremal metrics, stability of varieties, geometric flows.

Mei-Chi Shaw

1. Area of research ( include a paragraph describing your research)

My research interests are in several complex variables, partial differential equations and complex geometry. I am currently working on the Cauchy-Riemann operator and CR geometry on complex projective spaces and negatively curved manifolds. The goal is to understand how the presence of positive or negative curvature will influence solutions to the Cauchy-Riemann equations and function theory on complex manifolds.

2. Keywords several complex variables, complex geometry

3. Current PhD students

Lien-Yung Kao

4. Former PhD students

Deyun Wu, Ph.D. 1994. Professor, Shanghai School of Business, China.

Sophia Vassiliadou, Ph.D. 1997, tenured associate professor, Georgetown University.

Phillip Harrington, Ph.D. 2004, tenured associate professor, University of Arkansas, Fayettville.

5. Postdocs mentored.

Dan Coman(Syracuse University)

Ovidiu Calin (Eastern Michigan University)

Wayne Eby (Temple University)

Robert Juhlin (University of Vienna, Austria)

Debraj Chakrabarti (Bangalore, Tata Institute)

Gerard Misiolek

1. Area of research

My main interests are in analysis of PDE, geometry of groups of diffeomorphisms and connections to infinite-dimensional Hamiltonian systems.

2. Keywords

Global Analysis

3. Current PhD students

James Benn

4. Former PhD students (with their most recent affiliation)

Feride Tiglay, Fields Institute, Toronto, Canada

Matt Gurski

1. Area of research

Broadly speaking, my research is in geometric analysis: conformal geometry, partial differential equations,

the calculus of variations, and global differential geometry.

Some of my recent work has focused on higher order variational problems which arise in mathematical physics.

For example, the regularized determinant is a quantity defined in terms of the spectrum of a geometric

partial differential operator. We are interested in the global variational properties of this action, and the existence

and geometric properties of critical points.

2. Keywords

Differential Geometry, partial differential equations, calculus of variations.

3. Current PhD students

Yeuh-Ju Lin

Gang Li

4. Former PhD students (with their most recent affiliation)

Raymond Jensen, Trine University

Sujin Khomrutai, Chulalongkorn University (Thailand)

Yen-Chang Huang, National Taiwan University

Nancy Stanton

Area of research

My research is in complex differential geometry, especially the

interplay between differential geometry, partial differential

equations and several complex variables. In particular, I have done

research on geometry and the spectrum of the ∂̅-Laplacian on

complex manifolds, on geometry and the spectrum of the

∂̅-Neumann problem on strictly pseudoconvex domains, and on

geometry and the spectrum of the ∂̅<sub>b</sub>-Laplacian on

compact strictly pseudoconvex CR manifolds. I have also done research

on the local geometry of real hypersurfaces in complex space.

Key words

complex differential geometry, spectrum of the Laplacian, heat kernel

invariants

Former PhD students

James Faran, University of Buffalo (with S. S. Chern)

Roger Olson, St. Joseph's College

Kaining Wang

Larry Taylor

1. Area of research

I work in topology. This has involved at various times

work in K theory and L theory as these areas contain the

obstructions to problems in manifold topology. I have

done work in high dimensional manifolds, low dimensional manifolds

and stratified spaces. I have also done work in calculating

the homology/cohomology of interesting classes of spaces

such as function spaces, spaces of sections and configuration

spaces.

2. Keywords

Algebraic topology, geometric topology, manifolds, K-theory, L-theory,

stratified spaces

3. Current PhD students

None

4. Former PhD students

Constance Elko

Emanouil Magiroupoulos

Zarko Bizaca

Dimitrios Kodokostas

Sam Evens

1. Area of research

I work on representation theory and geometry of Lie groups and algebraic groups, using tools from differential geometry,

algebraic geometry, and topology. In recent years, I have worked on problems from Poisson geometry of homogeneous

spaces, geometry and combinatorics of orbits of symmetric subgroups on the flag variety, the complex Gelfand-Zeitlin

system on the Lie algebra of matrices, and degenerations of the cup product on the cohomology of the flag variety

of a compact Lie group.

2. Keywords (

Poisson geometry, representation theory, flag variety

3. Current PhD students

Martha Precup (4th year)

Nicole Kroege (3rd year)

Brian Shourd (3rd year)

4. Former PhD students (with their most recent affiliation)

Benjamin Jones (University of Wisconsin-Stout)

Oleksandra Lyapina (Hong Kong University)

5. Postdocs mentored.

Mark Colarusso

Arlo Caine