Dr Mark Walsh

Mathematics and Statistics

Lecturer

Logic House
132
(01) 708 3991

Biography

I am a Lecturer in Mathematics at Maynooth University. Up until Summer 2017 I was an Associate Professor in Mathematics at Wichita State University, Kansas, USA. I completed my PhD at the University of Oregon at Eugene, under the supervision of Professor Boris Botvinnik, and spent time as a postdoctoral researcher at WWU Muenster, Germany, as well as Oregon State University in Corvallis. My interests are in Geometry, in particular the relationship between Curvature and Topology. My work so far has focused on Positive Scalar Curvature and especially understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. More recently I have shifted my attention to analogous questions for positive Ricci curvature.

Research Interests

The topology of a shape is that which is maintained by continuous deformation, i.e. stretching, shrinking but not tearing. A ``topological" shape may take many geometric forms. For example, while a sphere is usually thought of as round, we may alter its shape in various ways and, provided our alterations are continuous and don't tear or puncture the sphere, we still maintain the topological condition of being a sphere. A torus (surface of a bagel) is topologically distinct from a sphere as there is no continuous way of turning one shape into the other. Such a transformation requires cutting or tearing: i.e. something discontinuous.

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. 

Books

  Year Publication
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

  Year Publication
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]

Conference Publications

  Year Publication
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]

Article

  Year Publication
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

  Year Title Awarding Body
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

Committees

  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

Reviews

  Journal Role
Transactions Of The American Mathematical Society Referee
K-Theory Referee
Journal Of Computational And Graphical Statistics Referee
Journal Of Mathematical Analysis And Applications Referee
Mathematische Annalen Referee
Geometric And Functional Analysis Referee
Differential Geometry And Its Applications Referee
Bulletin Of The London Mathematical Society Referee

Education

  Year Institution Qualification Subject
2009 University of Oregon PhD Mathematics

Outreach Activities

  Description

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)

Employment

  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

Teaching Interests

- Maynooth University Courses Taught (2017-present):
     MT105A: Introduction to Calculus 
     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 344: Vector Calculus 
      Math 415: Introduction to Advanced Mathematics 
      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 825: Topology II (Graduate)    
      Graduate Student Seminars on Riemannian Geometry, Differential
      Topology, Bordism and K-Theory, Spin Geometry
     
- Oregon State University Courses Taught (2010-2012):
      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


Recent Postgraduates

  Graduation Student Name University Degree Thesis
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