This is the unofficial seminar homepage of the GAPT group at Cardiff University.
The research interests of the GAPT group sweep a broad range of topics, from algebra, geometry, topology, including operator algebras, and non-commutative geometry in pure mathematics, to algebraic and conformal quantum field theory and integrable statistical mechanics in mathematical physics. Usually the seminar takes place every Thursday at 15:10 in room M/0.34. Speakers give 50 minute talks. |

**Thursday 16th May 15:10 - 16:00 M/0.34 - rescheduled**

Alan Thompson (Loughborough University)

**Thursday 21st March 15:10 - 16:00 M/0.34**

Alvaro Torras Casas (Cardiff University)

Input-Distributive Persistent Homology

Abstract:

Persistent Homology has been developed as the main tool of Topological Data Analysis, with numerous applications in science and engineering. However, for very large data sets this tool can be very expensive to compute, both in terms of computational time and hard-disk memory. We will present a new distributive algorithm which takes part directly on the input data. This has some theoretical difficulties since we need to work within the category of persistence modules. In particular, we will see a solution to the extension problem for the Persistent Mayer-Vietoris spectral sequence. At the end we speculate that this approach might give us more information than ordinary Persistent Homology.

Input-Distributive Persistent Homology

Abstract:

Persistent Homology has been developed as the main tool of Topological Data Analysis, with numerous applications in science and engineering. However, for very large data sets this tool can be very expensive to compute, both in terms of computational time and hard-disk memory. We will present a new distributive algorithm which takes part directly on the input data. This has some theoretical difficulties since we need to work within the category of persistence modules. In particular, we will see a solution to the extension problem for the Persistent Mayer-Vietoris spectral sequence. At the end we speculate that this approach might give us more information than ordinary Persistent Homology.

**Thursday 04th April 15:10 - 16:00 M/0.34**

Christian Voigt (Glasgow University)

The Plancherel formula for complex semisimple quantum groups

(joint with R. Yuncken)

Abstract:

The Plancherel formula for complex semisimple Lie groups, due to Gelfand-Naimark and Harish-Chandra, is a basic ingredient in their harmonic analysis. In this talk I’ll present a computation of the Plancherel formula for the quantum deformations of these groups obtained via the Drinfeld double construction. The quantum groups obtained this way have featured prominently in the study of property (T) for tensor categories and subfactors in recent years.

While the “quantum” Plancherel formula itself looks very similar to its classical counterpart – and is essentially a deformation thereof - the proof is completely different; it relies on the BGG-resolution and an application of the Hopf trace formula.

Starting from the classical Plancherel Theorem, I’ll give extensive background/motivation to all of the above, and then outline the key part of our proof.

The Plancherel formula for complex semisimple quantum groups

(joint with R. Yuncken)

Abstract:

The Plancherel formula for complex semisimple Lie groups, due to Gelfand-Naimark and Harish-Chandra, is a basic ingredient in their harmonic analysis. In this talk I’ll present a computation of the Plancherel formula for the quantum deformations of these groups obtained via the Drinfeld double construction. The quantum groups obtained this way have featured prominently in the study of property (T) for tensor categories and subfactors in recent years.

While the “quantum” Plancherel formula itself looks very similar to its classical counterpart – and is essentially a deformation thereof - the proof is completely different; it relies on the BGG-resolution and an application of the Hopf trace formula.

Starting from the classical Plancherel Theorem, I’ll give extensive background/motivation to all of the above, and then outline the key part of our proof.

**Past Seminar Talks**

**Thursday 4th October 15:10 - 16:00 M/0.34**

Stuart White (Glasgow)

Classification of simple nuclear C*-algebras

Abstract:

Recent years have seen repeated striking progress in the structure and classification of simple nuclear C*-algebras. I’ll try and survey what the state of the art is, focusing on recent developments. I’ll try and keep the talk self contained, starting out with what these `simple nuclear C*-algebras’ are and why anyone wants to classify them anyway.

Classification of simple nuclear C*-algebras

Abstract:

Recent years have seen repeated striking progress in the structure and classification of simple nuclear C*-algebras. I’ll try and survey what the state of the art is, focusing on recent developments. I’ll try and keep the talk self contained, starting out with what these `simple nuclear C*-algebras’ are and why anyone wants to classify them anyway.

**Thursday 11th October 15:10 - 16:00 M/0.34**

Fabian Hebestreit (Bonn / INI Cambridge)

Twisted K-theory via retractive symmetric spectra

joint with Steffen Sagave

Abstract:

Twisted K-theory was originally invented to serve as the K-theoretic analogue of singular (co)homology with local coefficients and by design gives explicit Thom- and Poincaré duality isomorphisms. In this formulation it admits a direct description in terms of KK-theory of certain section algebras and thus has tight connections for instance to the geometry of scalar curvature. Modern homotopy theory on the other hand provides a universally twisted companion for every coherently multiplicative cohomology theory by means of parametrised spectra. This construction has very appealing formal properties and, indeed, applied to K-theory allows for much more general twists than those afforded by the operator algebraic one. Necessarily then, such twisted companions are defined in a much more formal manner and thus in general not easily tied to geometry.

The goal of my talk is to briefly explain the category of the title, that naturally houses both constructions and then sketch that, indeed, a suitable restriction of the universal one reproduces the operator theoretic version of twisted K-theory. Time permitting, I shall also sketch how our work strengthens recent results of Dardalat and Pennig, describing the more exotic twists of K-theory via self-absorbing C*-algebras.

Twisted K-theory via retractive symmetric spectra

joint with Steffen Sagave

Abstract:

Twisted K-theory was originally invented to serve as the K-theoretic analogue of singular (co)homology with local coefficients and by design gives explicit Thom- and Poincaré duality isomorphisms. In this formulation it admits a direct description in terms of KK-theory of certain section algebras and thus has tight connections for instance to the geometry of scalar curvature. Modern homotopy theory on the other hand provides a universally twisted companion for every coherently multiplicative cohomology theory by means of parametrised spectra. This construction has very appealing formal properties and, indeed, applied to K-theory allows for much more general twists than those afforded by the operator algebraic one. Necessarily then, such twisted companions are defined in a much more formal manner and thus in general not easily tied to geometry.

The goal of my talk is to briefly explain the category of the title, that naturally houses both constructions and then sketch that, indeed, a suitable restriction of the universal one reproduces the operator theoretic version of twisted K-theory. Time permitting, I shall also sketch how our work strengthens recent results of Dardalat and Pennig, describing the more exotic twists of K-theory via self-absorbing C*-algebras.

**Thursday 18th October 15:10 - 16:00 M/0.34**

Paul Mitchener (Sheffield)

Categories of Unbounded Operators

Abstract:

The Gelfand-Naimark theorem on C*-algebras, which asserts that a C*-algebra, defined axiomatically, is the same thing as a closed sub-algebra of the algebra of bounded linear operators on a Hilbert space, is well-known. Of course, in some cases, for example, mathematical physics, the concern is with unbounded operators such as position and momentum in quantum mechanics.

In this talk, we explore a set of axioms for a mathematical object analogous to a C*-algebra, but for unbounded operators. In particular, our axioms are such that an analogue of the Gelfand-Naimark theorem holds.

Categories of Unbounded Operators

Abstract:

The Gelfand-Naimark theorem on C*-algebras, which asserts that a C*-algebra, defined axiomatically, is the same thing as a closed sub-algebra of the algebra of bounded linear operators on a Hilbert space, is well-known. Of course, in some cases, for example, mathematical physics, the concern is with unbounded operators such as position and momentum in quantum mechanics.

In this talk, we explore a set of axioms for a mathematical object analogous to a C*-algebra, but for unbounded operators. In particular, our axioms are such that an analogue of the Gelfand-Naimark theorem holds.

**Thursday 01 November 15:10 - 16:00 M/0.34**

Andreas Aaserud (Cardiff)

K-theory of some AF-algebras from braided categories

Abstract:

In the 1980s, Renault, Wassermann, Handelman and Rossmann explicitly described the K-theory of the fixed point algebras of certain actions of compact groups on AF-algebras as polynomial rings. Similarly, Evans and Gould in 1994 explicitly described the K-theory of certain AF-algebras related to SU(2) as quotients of polynomial rings. In this talk, I will explain how, in all of these cases, the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ (where $A$ denotes the AF-algebra whose K-theory is being computed) arising from a unitary braiding on an underlying C*-tensor category and essentially defined by Erlijman and Wenzl in 2007. I will also give new explicit descriptions of the K-theory of certain AF-algebras related to SU(3), Sp(4) and G$_2$ as quotients of polynomial rings. Finally, I will attempt to explain how this work is motivated by the Freed-Hopkins-Teleman formula for the fusion rings of WZW-models in conformal field theory. This is all based on joint work with David E. Evans.

K-theory of some AF-algebras from braided categories

Abstract:

In the 1980s, Renault, Wassermann, Handelman and Rossmann explicitly described the K-theory of the fixed point algebras of certain actions of compact groups on AF-algebras as polynomial rings. Similarly, Evans and Gould in 1994 explicitly described the K-theory of certain AF-algebras related to SU(2) as quotients of polynomial rings. In this talk, I will explain how, in all of these cases, the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ (where $A$ denotes the AF-algebra whose K-theory is being computed) arising from a unitary braiding on an underlying C*-tensor category and essentially defined by Erlijman and Wenzl in 2007. I will also give new explicit descriptions of the K-theory of certain AF-algebras related to SU(3), Sp(4) and G$_2$ as quotients of polynomial rings. Finally, I will attempt to explain how this work is motivated by the Freed-Hopkins-Teleman formula for the fusion rings of WZW-models in conformal field theory. This is all based on joint work with David E. Evans.

**Wednesday 14th November 13:10 - 14:00 M/2.06**

Vladimir Dotsenko (Trinity College Dublin)

Noncommutative analogues of cohomological field theories

Abstract:

Algebraic structures that are usually referred to as cohomological field theories arise from geometry of Deligne-Mumford compactifications of moduli spaces of curves with marked points. I shall talk about some new rather remarkable algebraic varieties that have a lot in common with [genus 0] Deligne-Mumford spaces, and several new algebraic structures that naturally arise from studying those varieties.

Noncommutative analogues of cohomological field theories

Abstract:

Algebraic structures that are usually referred to as cohomological field theories arise from geometry of Deligne-Mumford compactifications of moduli spaces of curves with marked points. I shall talk about some new rather remarkable algebraic varieties that have a lot in common with [genus 0] Deligne-Mumford spaces, and several new algebraic structures that naturally arise from studying those varieties.

**Thursday 22nd November 15:10 - 16:00 M/0.34**

Gandalf Lechner (Cardiff)

The Yang-Baxter equation and extremal characters of the infinite braid group

Abstract:

The Yang-Baxter equation (YBE) is a cubic matrix equation which plays a prominent in several fields such as quantum groups, braid groups, knot theory, quantum field theory, and statistical mechanics. Its invertible normal solutions ("R-matrices") define representations and extremal characters of the infinite braid group. These characters define a natural equivalence relation on the family of all R-matrices, and I will describe a research programme aiming at classifying all solutions of the YBE up to this equivalence.

I will then describe the current state of this programme. In the special case of normal involutive R-matrices, the classification is complete (joint work with Simon and Ulrich). The more general case of R-matrices with two arbitrary eigenvalues is currently work in progress, and I will present some partial results, including a classification of all R-matrices defining representations of the Temperley-Lieb algebra and a deformation theorem for involutive R-matrices.

The Yang-Baxter equation and extremal characters of the infinite braid group

Abstract:

The Yang-Baxter equation (YBE) is a cubic matrix equation which plays a prominent in several fields such as quantum groups, braid groups, knot theory, quantum field theory, and statistical mechanics. Its invertible normal solutions ("R-matrices") define representations and extremal characters of the infinite braid group. These characters define a natural equivalence relation on the family of all R-matrices, and I will describe a research programme aiming at classifying all solutions of the YBE up to this equivalence.

I will then describe the current state of this programme. In the special case of normal involutive R-matrices, the classification is complete (joint work with Simon and Ulrich). The more general case of R-matrices with two arbitrary eigenvalues is currently work in progress, and I will present some partial results, including a classification of all R-matrices defining representations of the Temperley-Lieb algebra and a deformation theorem for involutive R-matrices.

**Thursday 29th November 15:10 - 16:00 M/0.34**

Tomasz Brzezinski (Swansea)

Twisted reality

Abstract:

Recently two approaches to twisting of the real structure of spectral triples were introduced. In one approach, the definition of a twisted real structure of an ordinary spectral triple was presented in [T Brzeziński, N Ciccoli, L Dąbrowski, A Sitarz, Twisted reality condition for Dirac operators, Math. Phys. Anal. Geom. 19 (2016), no. 3, Art. 16]. In the second approach [G Landi, P Martinetti, On twisting real spectral triples by algebra automorphisms, Lett. Math. Phys. 106 (2016), no. 11, 1499–1530] the notion of real structure for a twisted spectral triple was proposed. In this talk we present and compare these two approaches.

Twisted reality

Abstract:

Recently two approaches to twisting of the real structure of spectral triples were introduced. In one approach, the definition of a twisted real structure of an ordinary spectral triple was presented in [T Brzeziński, N Ciccoli, L Dąbrowski, A Sitarz, Twisted reality condition for Dirac operators, Math. Phys. Anal. Geom. 19 (2016), no. 3, Art. 16]. In the second approach [G Landi, P Martinetti, On twisting real spectral triples by algebra automorphisms, Lett. Math. Phys. 106 (2016), no. 11, 1499–1530] the notion of real structure for a twisted spectral triple was proposed. In this talk we present and compare these two approaches.

**Thursday 06th December 15:10 - 16:00 M/0.34**

Ashley Montanaro (Bristol)

Quantum algorithms for search problems

Abstract:

Quantum computers are designed to use quantum mechanics to outperform any standard, "classical" computer based only on the laws of classical physics. Following many years of experimental and theoretical developments, it is anticipated that quantum computers will soon be built that cannot be simulated by today's most powerful supercomputers. In this talk, I will begin by introducing the quantum computational model, and describing the famous quantum algorithm due to Grover that solves unstructured search problems in approximately the square root of the time required classically. I will then go on to describe more recent work on a quantum algorithm to speed up classical search algorithms based on the technique known as backtracking ("trial and error"), and very recent work on calculating the level of quantum speedup anticipated when applying this algorithm to practically relevant problems. The talk will aim to give a flavour of the mathematics involved in quantum algorithm design, rather than going into the full details.

The talk will be based on the papers

Quantum walk speedup of backtracking algorithms, Theory of Computing (to appear); arXiv:1509.02374

Applying quantum algorithms to constraint satisfaction problems (with Earl Campbell and Ankur Khurana); arXiv:1810.05582

Quantum algorithms for search problems

Abstract:

Quantum computers are designed to use quantum mechanics to outperform any standard, "classical" computer based only on the laws of classical physics. Following many years of experimental and theoretical developments, it is anticipated that quantum computers will soon be built that cannot be simulated by today's most powerful supercomputers. In this talk, I will begin by introducing the quantum computational model, and describing the famous quantum algorithm due to Grover that solves unstructured search problems in approximately the square root of the time required classically. I will then go on to describe more recent work on a quantum algorithm to speed up classical search algorithms based on the technique known as backtracking ("trial and error"), and very recent work on calculating the level of quantum speedup anticipated when applying this algorithm to practically relevant problems. The talk will aim to give a flavour of the mathematics involved in quantum algorithm design, rather than going into the full details.

The talk will be based on the papers

Quantum walk speedup of backtracking algorithms, Theory of Computing (to appear); arXiv:1509.02374

Applying quantum algorithms to constraint satisfaction problems (with Earl Campbell and Ankur Khurana); arXiv:1810.05582

**Thursday 13th December 15:10 - 16:00 M/0.34**

Matthew Buican (Queen Mary, London)

From 4D Supersymmetry to 2D RCFT via Logarithmic Theories

Abstract:

I will discuss recent progress connecting the physics of certain large classes of 4D superconformal field theories with logarithmic conformal field theories. I will then use this connection to discuss a bridge between the physics of these 4D theories and certain more familiar 2D rational conformal field theories.

From 4D Supersymmetry to 2D RCFT via Logarithmic Theories

Abstract:

I will discuss recent progress connecting the physics of certain large classes of 4D superconformal field theories with logarithmic conformal field theories. I will then use this connection to discuss a bridge between the physics of these 4D theories and certain more familiar 2D rational conformal field theories.

**Thursday 31st January 15:10 - 16:00 M/0.34**

Vincenzo Morinelli (Tor Vergata, Rome)

Scale and Möbius covariance in two-dimensional Haag-Kastler net

Abstract:

The relation between conformal and dilation covariance is a controversial problem in QFT. Although many models which are dilation covariant are indeed conformal covariant a complete understanding of this implication in the algebraic approach to QFT is missing. In this talk we present the following result: Given a two-dimensional Haag-Kastler net which is Poincare-dilation covariant with additional properties, we prove that it can be extended to a conformal (Möbius) covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. Time permitting we further discuss counterexamples.

Based on a joint work with Yoh Tanimoto arXiv:1807.04707.

Scale and Möbius covariance in two-dimensional Haag-Kastler net

Abstract:

The relation between conformal and dilation covariance is a controversial problem in QFT. Although many models which are dilation covariant are indeed conformal covariant a complete understanding of this implication in the algebraic approach to QFT is missing. In this talk we present the following result: Given a two-dimensional Haag-Kastler net which is Poincare-dilation covariant with additional properties, we prove that it can be extended to a conformal (Möbius) covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. Time permitting we further discuss counterexamples.

Based on a joint work with Yoh Tanimoto arXiv:1807.04707.

**Thursday 07th February 15:10 - 16:00 M/0.34**

Ulrich Pennig (Cardiff University)

Exponential functors, R-matrices and higher twists

Abstract:

R-matrices are solutions to the Yang-Baxter equation, which was introduced as a consistency equation in statistical mechanics, but has since then appeared in many other research areas, for example integrable quantum field theory, knot theory, the study of Hopf algebras and quantum information theory. Twisted K-theory on the other hand is a variant of topological K-theory that allows local coefficient systems called twists. Twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the twisted equivariant K-theory of the group to the Verlinde ring of the associated loop group. In this talk I will discuss how involutive R-matrices give rise to a natural generalisation of the twist appearing in this theorem via exponential functors.

Exponential functors, R-matrices and higher twists

Abstract:

R-matrices are solutions to the Yang-Baxter equation, which was introduced as a consistency equation in statistical mechanics, but has since then appeared in many other research areas, for example integrable quantum field theory, knot theory, the study of Hopf algebras and quantum information theory. Twisted K-theory on the other hand is a variant of topological K-theory that allows local coefficient systems called twists. Twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the twisted equivariant K-theory of the group to the Verlinde ring of the associated loop group. In this talk I will discuss how involutive R-matrices give rise to a natural generalisation of the twist appearing in this theorem via exponential functors.

**Thursday 14th February 15:10 - 16:00 M/0.34**

Enrico Fatighenti (Loughborough University)

Fano varieties of K3 type and IHS manifolds

Abstract:

Subvarieties of Grassmannians (and especially Fano varieties) obtained from sections of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperkähler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi.

Fano varieties of K3 type and IHS manifolds

Abstract:

Subvarieties of Grassmannians (and especially Fano varieties) obtained from sections of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperkähler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi.

**Thursday 21st February 15:10 - 16:00 M/0.34**

Tyler Kelly (University of Birmingham)

Open Mirror Symmetry for Landau-Ginzburg Models

Abstract:

Mirror Symmetry provides a link between different suites of data in geometry. On one hand, one has a lot of enumerative data that is associated to curve counts, telling you about important intersection theory in an interesting moduli problem. On the other, one has a variation of Hodge structure, that is, complex algebro-geometric structure given by computing special integrals. While typically one has focussed on the case where we study the enumerative data for a symplectic manifold, we here will instead study the enumerative geometry of a Landau-Ginzburg model. A Landau-Ginzburg model is essentially a triplet of data: an affine variety $X$ [think $\mathbb{C}^n$] with a group $G$ acting on it and a $G$-invariant algebraic function $W$ from $X$ to the complex numbers. We will describe what open enumerative geometry looks like for this gadget for the simplest examples ($W=x^r$) and explain what mirror symmetry means in this context. This is joint work in preparation with Mark Gross and Ran Tessler.

Open Mirror Symmetry for Landau-Ginzburg Models

Abstract:

Mirror Symmetry provides a link between different suites of data in geometry. On one hand, one has a lot of enumerative data that is associated to curve counts, telling you about important intersection theory in an interesting moduli problem. On the other, one has a variation of Hodge structure, that is, complex algebro-geometric structure given by computing special integrals. While typically one has focussed on the case where we study the enumerative data for a symplectic manifold, we here will instead study the enumerative geometry of a Landau-Ginzburg model. A Landau-Ginzburg model is essentially a triplet of data: an affine variety $X$ [think $\mathbb{C}^n$] with a group $G$ acting on it and a $G$-invariant algebraic function $W$ from $X$ to the complex numbers. We will describe what open enumerative geometry looks like for this gadget for the simplest examples ($W=x^r$) and explain what mirror symmetry means in this context. This is joint work in preparation with Mark Gross and Ran Tessler.

**Thursday 28th February 15:10 - 16:00 M/0.34**

Clelia Pech (University of Kent)

Mirror symmetry for cominuscule homogeneous varieties

Abstract:

In this talk reporting on joint work with K. Rietsch and L. Williams, I will explain a new version of the construction by Rietsch of a mirror for some varieties with a homogeneous Lie group action. The varieties we study include quadrics and Lagrangian Grassmannians (i.e., Grassmannians of Lagrangian vector subspaces of a symplectic vector space). The mirror takes the shape of a rational function, the superpotential, defined on a Langlands dual homogeneous variety. I will show that the mirror manifold has a particular combinatorial structure called a cluster structure, and that the superpotential is expressed in coordinates dual to the cohomology classes of the original variety.

I will also explain how these properties lead to new relations in the quantum cohomology, and a conjectural formula expressing solutions of the quantum differential equation in terms of the superpotential. If time allows, I will also explain how these results should extend to a larger family of homogeneous spaces called cominuscule homogeneous spaces.

Mirror symmetry for cominuscule homogeneous varieties

Abstract:

In this talk reporting on joint work with K. Rietsch and L. Williams, I will explain a new version of the construction by Rietsch of a mirror for some varieties with a homogeneous Lie group action. The varieties we study include quadrics and Lagrangian Grassmannians (i.e., Grassmannians of Lagrangian vector subspaces of a symplectic vector space). The mirror takes the shape of a rational function, the superpotential, defined on a Langlands dual homogeneous variety. I will show that the mirror manifold has a particular combinatorial structure called a cluster structure, and that the superpotential is expressed in coordinates dual to the cohomology classes of the original variety.

I will also explain how these properties lead to new relations in the quantum cohomology, and a conjectural formula expressing solutions of the quantum differential equation in terms of the superpotential. If time allows, I will also explain how these results should extend to a larger family of homogeneous spaces called cominuscule homogeneous spaces.

**Thursday 07th March 15:10 - 16:00 M/0.34**

Farzad Fathizadeh (Swansea University)

Heat kernel expansion of the Dirac-Laplacian of multifractal Robertson-Walker cosmologies

Abstract:

I will talk about a recent work in which we find an explicit formula for each Seeley-deWitt coefficient in the full heat kernel expansion of the Dirac-Laplacian of a Robertson-Walker metric with a general cosmic expansion factor. We use the Feynman-Kac formula and combinatorics of Brownian bridge integrals heavily. The extension of the result to the inhomogeneous case, where the spatial part of the model has a fractal structure, will also be presented. This is joint work with Yeorgia Kafkoulis and Matilde Marcolli.

Heat kernel expansion of the Dirac-Laplacian of multifractal Robertson-Walker cosmologies

Abstract:

I will talk about a recent work in which we find an explicit formula for each Seeley-deWitt coefficient in the full heat kernel expansion of the Dirac-Laplacian of a Robertson-Walker metric with a general cosmic expansion factor. We use the Feynman-Kac formula and combinatorics of Brownian bridge integrals heavily. The extension of the result to the inhomogeneous case, where the spatial part of the model has a fractal structure, will also be presented. This is joint work with Yeorgia Kafkoulis and Matilde Marcolli.