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Department of Mathematical Sciences

# Research Seminar Series

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.

## Pure Maths Colloquium: Stable homotopy theory and spectra

Presented by David Barnes, Sheffield University

7 May 2012 16:00 in CM221

If homotopy theory is the study of spaces up to homotopy, then stable homotopy theory is the study of homotopy theory up to `suspension'. Suspension is a method of making spaces larger, for example the suspension of the n-sphere is the n+1-sphere. There are many fascinating patterns and structures within homotopy theory that are only revealed when looking through the lens of stable homotopy theory, such as the Freundenthal suspension theorem. Another example of stable data is that coming from cohomology theories.

Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. To study cohomology theories, one looks at their representing objects, called spectra. The category of spectra is called the stable homotopy category and is central to the study of algebraic topology.