## Portfolio item number 1

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Published in *arXiv:math-ph*, 2002

This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way.

Recommended citation: Daniel D. Ferrante. (2002). "Geometric Calculus and the Fibre Bundle description of Quantum Mechanics." *arXiv:math-ph*. __https://arxiv.org/abs/math-ph/0204024__

Published in *Nuclear Physics. B*, 2002

Using a common technique for approximating distributions [generalized functions], we are able to use standard Monte Carlo methods to compute QFT quantities in Minkowski spacetime, under phase transitions, or when dealing with coalescing stationary points.

Recommended citation: D.D. Ferrante, J. Doll, G.S. Guralnik, D. Sabo. (2003). "Mollified Monte Carlo." *Nuclear Physics. B*. 119(965). __https://arxiv.org/abs/hep-lat/0209053__

Published in *arXiv:hep-lat*, 2002

Two new approaches to numerical QFT are presented.

Recommended citation: G. S. Guralnik, J. Doll, R. Easther, P. Emirdag, D. D. Ferrante, S. Hahn, D. Petrov, D. Sabo. (2002). "Alternative Numerical Techniques." *arXiv:hep-lat*. __https://arxiv.org/abs/hep-lat/0209127__

Published in *arXiv:hep-lat*, 2003

We outline two alternative schemes to perform numerical calculations in quantum field theory. In principle, both of these approaches are better suited to study phase structure than conventional Monte Carlo. The first method, Source Galerkin, is based on a numerical analysis of the Schwinger-Dyson equations using modern computer techniques. The nature of this approach makes dealing with fermions relatively straightforward, particularly since we can work on the continuum. Its ultimate success in non-trivial dimensions will depend on the power of a propagator expansion scheme which also greatly simplifies numerical calculation of traditional perturbation graphs. The second method extends Monte Carlo approaches by introducing a procedure to deal with rapidly oscillating integrals.

Recommended citation: R. Easther, D. D. Ferrante, G. S. Guralnik, D. Petrov. (2003). "A Review Of Two Novel Numerical Methods in QFT." *arXiv:hep-lat*. __https://arxiv.org/abs/hep-lat/0306038__

Published in *arXiv:hep-lat*, 2006

This work develops and applies the concept of mollification in order to smooth out highly oscillatory exponentials. This idea, known for quite a while in the mathematical community (mollifiers are a means to smooth distributions), is new to numerical Quantum Field Theory. It is potentially very useful for calculating phase transitions [highly oscillatory integrands in general], for computations with imaginary chemical potentials and Lattice QFT in Minkowski spacetime.

Recommended citation: D. D. Ferrante, G. S. Guralnik. (2006). "Mollifying Quantum Field Theory or Lattice QFT in Minkowski Spacetime and Symmetry Breaking." *arXiv:hep-lat*. __https://arxiv.org/abs/hep-lat/0602013__

Published in *arXiv:hep-th*, 2008

By means of an appropriate re-scaling of the metric in a Lagrangian, we are able to reduce it to a kinetic term only. This form enables us to examine the extended complexified solution set (complex moduli space) of field theories by finding all possible geodesics of this metric. This new geometrical standpoint sheds light on some foundational issues of QFT and brings to the forefront non-perturbative core aspects of field theory. In this process, we show that different phases of the theory are topologically inequivalent, i.e., their moduli space has distinct topologies. Moreover, the different phases are related by “duality transformations”, which are established by the modular structure of the theory. In conclusion, after the topological structure is elucidated, it is possible to use the Euler Characteristic in order to topologically quantize the theory, in resonance with the content of the Atiyah-Singer Index theorem.

Recommended citation: D. D. Ferrante, G. S. Guralnik. (2008). "Phase Transitions and Moduli Space Topology." *arXiv:hep-th*. __https://arxiv.org/abs/0809.2778__

Published in *Journal 1*, 2010

This paper is about the number 2. The number 3 is left for future work.

Recommended citation: Your Name, You. (2010). "Paper Title Number 2." *Journal 1*. 1(2). __http://academicpages.github.io/files/paper2.pdf__

Published in *Journal 1*, 2010

This paper is about the number 2. The number 3 is left for future work.

Recommended citation: Your Name, You. (2010). "Paper Title Number 2." *Journal 1*. 1(2). __http://academicpages.github.io/files/paper2.pdf__

Published in *Journal 1*, 2010

This paper is about the number 2. The number 3 is left for future work.

Recommended citation: Your Name, You. (2010). "Paper Title Number 2." *Journal 1*. 1(2). __http://academicpages.github.io/files/paper2.pdf__

Published in *Journal 1*, 2015

This paper is about the number 3. The number 4 is left for future work.

Recommended citation: Your Name, You. (2015). "Paper Title Number 3." *Journal 1*. 1(3). __http://academicpages.github.io/files/paper3.pdf__

** Published:**

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** Published:**

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Undergraduate course, *University 1, Department*, 2014

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Workshop, *University 1, Department*, 2015

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