A large part of modern geometry concerns the problem of finding a ``good" geometric structure on a topological shape, given a plethora of possibilities. The term ``good" is highly subjective. More broadly however, one may be interested in geometries with a particular property, concerning symmetry or curvature perhaps. Given a geometric constraint, say positive curvature, the problem is to find examples of topological shapes which admit such geometries and to understand what the topological obstructions are in the ones that do not. We know for example that the round geometry is just one of many positive curvature geometries on the sphere. In the standard ``bagel shaped" torus, the inner part has negative curvature. It is a famous theorem of Mathematics that no amount of continuous deformation can give the torus everywhere positive curvature. Thus, because of its topology, the torus can not admit a positive curvature geometry.
One case of this problem is in deciding which smooth manifolds (a particular type of mathematical shape) admit Riemannian metrics (geometric structures) of positive scalar curvature (psc-metrics). This question has attracted a good deal of attention over the years and a number of significant classification results have been achieved. For manifolds which admit psc-metrics there is a related problem, of which far less is known, and which motivates my work. This problem takes the form of the following question.
What is the topology of the space of psc-metrics on a given manifold?
In other words, what is the shape of the space of geometric structures which satisfy the positive scalar curvature condition. This is a highly complicated infinite dimensional space. This problem is analogous to that of trying to understand the shape of all configurations of a robot arm. The arm itself is a $3$-dimensional object, but the space of all configurations may have many more dimensions, depending on factors such as the number of hinges on the arm. One may think of a path through this space of psc-metrics as an ``animation" of the manifold over time, gradually morphing it from one shape to another, but at every stage satisfying the curvature constraint. One may ask if given two such geometries, it is possible to continuously deform one into the other while maintaining positivity of the scalar curvature at every stage. In other words, is the space of psc-metrics path-connected? More generally, what can be said about so-called higher dimensional connectedness? Finally, what about the analogous questions for other curvature notions such as the Ricci curvature?
In recent years significant strides have been made in answering these questions and it is clear that there is a growing interest in this subject. My work forms part of those efforts.
|2020|| The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary.
Mark Walsh (2020) The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary. New York: New York Journal of Mathematics. [Details]
|2011|| Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part 1.
Mark Walsh (2011) Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part 1. Providence, RI, USA: Memoirs of the American Mathematical Society. [Details]
Peer Reviewed Journals
|2021|| 'Homotopy Invariance of the Space of Metrics with Positive Scalar Curvature on Manifolds with Singularities'
B. Botvinnik and M. Walsh (2021) 'Homotopy Invariance of the Space of Metrics with Positive Scalar Curvature on Manifolds with Singularities'. SIGMA, 17 (034):1-27 [DOI] [Details]
|2019|| 'The Observer Moduli Space of Metrics of Positive Ricci Curvature'
B. Botvinnik, M. Walsh and D. Wraith (2019) 'The Observer Moduli Space of Metrics of Positive Ricci Curvature'. Geometry and Topology, 23 :3003-3040 [Details]
|2018|| 'Aspects of Scalar Curvature and Topology, Part 2'
Mark Walsh (2018) 'Aspects of Scalar Curvature and Topology, Part 2'. BULLETIN OF THE IRISH MATHEMATICAL SOCIETY, 81 [Details]
|2017|| 'Aspects of Scalar Curvature and Topology, Part 1'
Mark Walsh (2017) 'Aspects of Scalar Curvature and Topology, Part 1'. BULLETIN OF THE IRISH MATHEMATICAL SOCIETY, 80 [Details]
|2014|| 'Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part 2'
Mark Walsh (2014) 'Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part 2'. Transactions of the American Mathematical Society, [Details]
|2014|| 'H-Spaces, Loop Spaces and Positive Scalar Curvature'
Mark Walsh (2014) 'H-Spaces, Loop Spaces and Positive Scalar Curvature'. Geometry and Topology, 18 (4):2189-2243 [Details]
|2013|| 'Cobordism Invariance of the Homotopy Type of the Space of PSC-Metrics'
Mark Walsh (2013) 'Cobordism Invariance of the Homotopy Type of the Space of PSC-Metrics'. Proceedings of the American Mathematical Society, [Details]
|2010|| 'Homotopy Groups of the Moduli Space of Metrics of Positive Scalar Curvature'
B. Botvinnik, B. Hanke, T. Schick and M. Walsh (2010) 'Homotopy Groups of the Moduli Space of Metrics of Positive Scalar Curvature'. Geometry and Topology, :2047-2076 [Details]
|2017|| The Observer Moduli Space of Metrics of Positive Ricci Curvature
B. Botvinnik, M. Walsh and D. Wraith (2017) The Observer Moduli Space of Metrics of Positive Ricci Curvature Spaces and Moduli Spaces of Riemannian Metrics [Details]
|2020|| H-Space and Loop Space Structures for Intermediate Ricci Curvatures.
M. Walsh and D. J. Wraith (2020) H-Space and Loop Space Structures for Intermediate Ricci Curvatures. Article [Details]
Honours and Awards
|2009||University of Oregon D. K. Harrison Award for Best PhD Thesis||University of Oregon|
|2008||Johnson Fellowship Award||University of Oregon|
|2013||Simons Foundation Collaboration Award||Simons Foundation|
|Committee||Function||From / To|
|Mathematics Dept Course Curriculum Committee||Member||2017 /|
|PhD Committee at IT Tralee||External Examiner||2019 /|
|Faculty of Science and Engineering Research Committee||Mathematics Department Representative||2019 /|
|John Hume Award Evaluation Committee||Mathematics Department Representative||2019 / 2019|
|Transactions Of The American Mathematical Society||Referee|
|Journal Of Computational And Graphical Statistics||Referee|
|Journal Of Mathematical Analysis And Applications||Referee|
|Geometric And Functional Analysis||Referee|
|Differential Geometry And Its Applications||Referee|
|Bulletin Of The London Mathematical Society||Referee|
|2009||University of Oregon||PhD||Mathematics|
Serving as President 2019 - present
Gave Mathematical presentations for local school children during both Maths and Science weeks (2018, 2019). Topics included: "Geometry of Soap Films" and "The Neighbouring Domain Problem".
Helped organise an exhibition on the connections between Mathematics and Music for Kansas public radio station KMUW in Spring 2014.
Between 2012 and 2017, regularly lectured in Mathematics to primary and secondary level students on elementary topics in geometry, topology and number theory.
Worked as a volunteer tutor helping adult learners complete their GED in Mathematics and Science.
Gave lectures to Team Maths students on: ``Geometry and the Shape of the Universe" (2018) and ``Soap Films and Minimal Surfaces" (2019)
|Employer||Position||From / To|
|Oregon State University||Visiting Assistant Professor||01-SEP-10 / 31-JUL-12|
|Wichita State University||Assistant Professor||01-AUG-12 / 30-AUG-16|
|WWU Muenster||Postdoctoral Researcher||01-SEP-09 / 31-AUG-10|
|Maynooth University||Lecturer in Mathematics||01-SEP-17 /|
|Wichita State University||Associate Professor||01-SEP-16 / 31-AUG-17|
MT113S: Linear Algebra
MT202A: Multivariable Integral Calculus
MT342P: Groups 1
MT451P: Differential Geometry
MT212A: Linear Algebra
MT216C: Linear Algebra
MT595R: Reading Course (Geometry and Topology)
- Wichita State University Courses Taught (2012-2017):
Math 242/3: Calculus
Math 513: Abstract Algebra
Math 525: Introduction to Topology
Math 555: Differential Equations
Math 547: Advanced Calculus I
Math 640: Advanced Calculus II
Math 720: Modern Geometry
Math 725: Topology I (Graduate)
Math 679: Differential Topology
Math 534/535: Differential Geometry
Math 341/342: Linear Algebra I, II
Math 255: Vector Calculus II
Math 232: Discrete Mathematics II
Math 355: Discrete Mathematics Through Guided Discovery
Math 333: Knots and Surfaces (Writing Intensive Class)
- University of Muenster (2009-2010): Assisted with, lectured in
Graduate Student Seminar on Atiyah-Singer Index Theorem
Graduate Student Workshop on Infinite Dimensional Lie Groups
- University of Oregon Courses Taught (2002-2009):
Math 256: Ordinary Differential Equations
Math 241/242, 251/252: Differential and Integral Calculus
Math 107/111/112: Calculus, Algebra, Trigonometry
|2021||Matthew Burkemper||Wichita State University||PhD||Gromov-Lawson Cobordism and Positive p-Curvature|
|2022||Cian Hayes||Maynooth University||MSc||Groups of Homotopy Spheres|
|2022||Jacqueline Birkett||Maynooth University||PhD||Positive k-Ricci Curvature, Concordance and Isotopy|
|2015||Sarah Peterson||Wichita State University||MSc||Configuration Spaces|
|2015||Matthew Burkemper||Wichita State University||MSc||Applications of the h-Cobordism Theorem|