Dr Mark Walsh

Mathematics and Statistics

Associate Professor

Logic House
127
(01) 708 3791

Biography

I am an Associate Professor 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. 

Book

Year Publication
2020 Mark Walsh (2020) The Space of Metrics of Positive Scalar Curvature on a Manifold with Boundary. New York, USA: New York Journal of Mathematics.
2011 Mark Walsh (2011) Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part I. Providence, RI, USA: Memoirs of the American Mathematical Society.

Peer Reviewed Journal

Year Publication
2022 Walsh M.; Wraith D.J. (2022) 'H-space and loop space structures for intermediate curvatures'. Communications in Contemporary Mathematics, . [DOI] [Full-Text]
2022 Burkemper M.; Searle C.; Walsh M. (2022) 'Positive (p,n)-intermediate scalar curvature and cobordism'. Journal of Geometry and Physics, 181 . [DOI] [Full-Text]
2021 Botvinnik B.; Walsh M.G. (2021) 'Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities'. Symmetry, Integrability And Geometry: Methods And Applications (Sigma), 17 . [DOI] [Full-Text]
2019 B. Botvinnik, M. Walsh and D. Wraith (2019) 'The Observer Moduli Space of Metrics of Positive Ricci Curvature'. Geometry and Topology, 23 :3003-3040.
2018 Mark Walsh (2018) 'Aspects of Scalar Curvature and Topology, Part 2'. BULLETIN OF THE IRISH MATHEMATICAL SOCIETY, 81 .
2017 Mark Walsh (2017) 'Aspects of Scalar Curvature and Topology, Part 1'. BULLETIN OF THE IRISH MATHEMATICAL SOCIETY, 80 .
2014 Mark Walsh (2014) 'Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part 2'. Transactions of the American Mathematical Society, .
2014 Mark Walsh (2014) 'H-Spaces, Loop Spaces and Positive Scalar Curvature'. Geometry and Topology, 18 (4):2189-2243.
2013 Mark Walsh (2013) 'Cobordism Invariance of the Homotopy Type of the Space of PSC-Metrics'. Proceedings of the American Mathematical Society, .
2010 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.

Conference Contribution

Year Publication
2022 Mark Walsh (2022) University of Oregon: Geometric Analysis Seminar Singular Manifolds and Positive Scalar Curvature Eugene, Oregon USA, .
2021 Mark Walsh (2021) ICETPAM 2021, Bahrain Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities via Zoom to audience in Bahrain, .
2020 Mark Walsh (2020) Great Plains Geometry Conference 2020 H-Spaces, Loop Spaces and Positive k-Ricci Curvature University of Oklahoma, .
2020 Mark Walsh (2020) Wichita Mathematical Lecture Series Loop and Curvature Wichita State University, Kansas, USA, .
2019 Mark Walsh (2019) Irish Geometry Conference 2019 Positive Scalar Curvature Metrics on a Manifold with Boundary Maynooth University, .
2019 Mark Walsh (2019) Oregon State University Mathematics Colloquium Algebraic Structures and Curvature Oregon State University, Corvallis OR, USA, .
2017 Mark Walsh (2017) Oberwolfach 2017: Moduli Spaces of Riemannian Metrics The Observer Moduli Space of Metrics of Positive Ricci Curvature Mathematisches Forschungsinstitut Oberwolfach, Germany, .
2016 Mark Walsh (2016) Maynooth University Mathematics Colloquium Loops, Spheres and Scalar Curvature Maynooth University, .
2015 Mark Walsh (2015) Oregon Topology Seminar Loops, Operads and Positive Scalar Curvature Eugene, Oregon USA, .
2014 Mark Walsh (2014) TCU Mathematics Colloquium The Space of PSC-Metrics TCU Dallas, Texas, USA, .
2014 Mark Walsh (2014) Conference on Homological Perturbation Theory Operads and Curvature NUI Galway, .
2014 Mark Walsh (2014) Oregon State Mathematics Colloquium How do we build an exotic manifold? Oregon State University, Corvallis OR, USA, .
2014 Mark Walsh (2014) Cascade Topology Conference Loop spaces, Operads and Positive Scalar Curvature Portland State University, Portland Oregon, USA, .
2013 Mark Walsh (2013) Conference on Curvature and Global Shape Loop Spaces, Operads and Scalar Curvature WWU, Muenster, Germany, .
2013 Mark Walsh (2013) Kansas State University Topology Seminar Moduli Spaces of Positive Scalar Curvature Metrics KSU Manhattan, Kansas, USA, .
2013 Mark Walsh (2013) Irish Geometry Conference 2013 Loops and Curvature NUI Maynooth, .
2012 Mark Walsh (2012) Wichita Mathematical Lecture Series Surgery, Singularities and Scalar Curvature Wichita, Kansas, USA, .
2011 Mark Walsh (2011) Cascade Topology Seminar 2011 Generalised Morse Functions and Positive Scalar Curvature Portland State University, Portland Oregon, USA, .
2011 Mark Walsh (2011) Lewis and Clark College Mathematical Colloquium Topology, Triangles and the Theorem of Toponogov Lewis and Clark College, Portland Oregon, USA, .
2011 Mark Walsh (2011) American Mathematical Society Sectional Meeting 2011 The Moduli Space of Metrics of Positive Scalar Curvature Holy Cross College, Worcester MA, USA, .
2010 Mark Walsh (2010) Workshop on Infinite-Dimensional Lie Groups The Simplicity of the Diffeomorphism Group Zaferna, Austria, .
2010 Mark Walsh (2010) Conference on Scalar Curvature, Geometry and Topology Surgery, Positive Scalar Curvature and Generalised Morse Functions WWU Muenster, Germany, .
2010 Mark Walsh (2010) University of Portland Mathematics Education Colloquium Finding the Right Geometry Portland Oregon, USA, .
2010 Mark Walsh (2010) Willamette University Mathematics Colloquium The Theorem of Gauss-Bonnet Salem, Oregon USA, .
2010 Mark Walsh (2010) Pacific Northwest Geometry Seminar Understanding the Space of Positive Scalar Curvature Metrics University of Oregon, Eugene, USA, .
2009 Mark Walsh (2009) Goettingen Topology Seminar Morse Theory, Wrinkles and Positive Scalar Curvature University of Goettingen, Germany, .
2008 Mark Walsh (2008) American Mathematical Society Joint Meetings 2008 Gromov-Lawson Concordance implies Isotopy San Diego CA, USA, .
2007 Mark Walsh (2007) KTH Stockholm Topology Seminar Unfolding Birth-Death Singularities Royal Technical University (KTH), Stockholm, Sweden, .
2006 Mark Walsh (2006) Irish Geometry Conference 2006 The Space of Generalised Morse Functions NUI Galway, .

Conference Publication

Year Publication
2017 B. Botvinnik, M. Walsh and D. Wraith (2017) Spaces and Moduli Spaces of Riemannian Metrics The Observer Moduli Space of Metrics of Positive Ricci Curvature
Certain data included herein are derived from the © Web of Science (2024) of Clarivate. All rights reserved.

Honors and Awards

Date Title Awarding Body
01/05/2009 University of Oregon D. K. Harrison Award for Best PhD Thesis University of Oregon
01/01/2008 Johnson Fellowship Award University of Oregon
01/01/2013 Simons Foundation Collaboration Award Simons Foundation

Committees

Committee Function From / To
Mathematics Dept Course Curriculum Committee Member 25/09/2017 -
PhD Committee at IT Tralee External Examiner 01/03/2019 -
Faculty of Science and Engineering Research Committee Mathematics Department Representative 23/09/2019 -
John Hume Award Evaluation Committee Mathematics Department Representative 01/05/2019 - 31/05/2019

Employment

Employer Position From / To
Oregon State University Visiting Assistant Professor 01/09/2010 - 31/07/2012
Wichita State University Assistant Professor 01/08/2012 - 30/08/2016
WWU Muenster Postdoctoral Researcher 01/09/2009 - 31/08/2010
Maynooth University Lecturer in Mathematics 01/09/2017 -
Wichita State University Associate Professor 01/09/2016 - 31/08/2017

Education

Start date Institution Qualification Subject
University of Oregon PhD Mathematics

Outreach Activities

Organisation Type Description
North Kildare Maths Problem Solving Club Civic Society Served as President 2019 - 2022
Maths and Science Week Civic Society 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".
National Public Radio (NPR) Civic Society Helped organise an exhibition on the connections between Mathematics and Music for Kansas public radio station KMUW in Spring 2014.
Wichita State Math Circle Civic Society Between 2012 and 2017, regularly lectured in Mathematics to primary and secondary level students on elementary topics in geometry, topology and number theory.
Kansas Mental Health Association Civic Society Worked as a volunteer tutor helping adult learners complete their GED in Mathematics and Science.
Team Maths Civic Society Gave lectures to Team Maths students on: ``Geometry and the Shape of the Universe" (2018) and ``Soap Films and Minimal Surfaces" (2019)

Teaching Interests

- Maynooth University Courses Taught (2017-present):
     MT105A: Introduction to Calculus 
     MT113S: Linear Algebra 1
     MT201A: Multivariable Differential Calculus
     MT202A: Multivariable Integral Calculus
     MT212A: Linear Algebra 2 for Mathematical Studies
     MT216C: Linear Algebra 2
     MT333P: Complex Analysis 1
     MT342P: Groups 1
     MT412C: Graph Theory
     MT451P: Differential Geometry
     MT212A: Linear Algebra
     MT216C: Linear Algebra
     MT595R: Reading Courses (Riemannian Geometry, Differential Topology, Algebraic 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 Students

Graduation date Name Degree
2021 Matthew Burkemper PhD
2022 Cian Hayes MSc
2022 Jacqueline Birkett PhD
2015 Sarah Peterson MSc
2015 Matthew Burkemper MSc