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From the begininng of my PhD in 1999 to 2012 I have worked on "Spectral geometry of Dirac operators on fiber bundles of compact flat manifolds" (spectrum, isospectrality, spin structures, Dirac and Laplace operators) with a touch of number theory (eta invariants, class numbers, L-functions).
From the begininng of my PhD in 1999 to 2012 I have worked on "Spectral geometry of Dirac operators on fiber bundles of compact flat manifolds" (spectrum, isospectrality, spin structures, Dirac and Laplace operators) with a touch of number theory (eta invariants, class numbers, L-functions).
Since 2013 I am interested in Coding Theory on a wide sense, but mainly in cyclic codes and their weight distributions. This involves finite fields and rings, algebraic curves, exponential sums, etc.
Since 2013 I am interested in Coding Theory on a wide sense, but mainly in cyclic codes and their weight distributions. This involves finite fields and rings, algebraic curves, exponential sums, etc.
Also, I have always love Combinatorics, Graph theory and Discrete mathematics in general.
Also, I have always love Combinatorics, Graph theory and Discrete mathematics in general.
Research Projects
Research Projects
Spectral geometry
Spectral geometry
Problems: Study the spectral properties of Dirac and Laplace operators acting on fiber bundles of compact flat manifolds. Compute the spectrum explicitly. Isospectrality problem: construct pairs of isospectral manifolds having different geometry or topology.
People: Roberto Mialtello, Juan Pablo Rossetti, Peter Gilkey.
Finite fields: graphs, codes and curves
Finite fields: graphs, codes and curves
Problems: Interrelations betweem cyclic codes, algebraic curves and Cayley graphs. Spectral properties of generalized Paley graphs (espectrum, Ramanuja graphs, equienergy) Waring numbers over finite rins. Diagonal equations. Number of rational points of Artin-Schreier curves. Cyclic codes associated to quadratic forms or q-linearized polynomials. Special functions (Bent, APN, planar). Functional graphs.
People: Denis Videla, Paula Chiapparoli; Herivelto Borges, Lucas Reis; Fabio Brochero, Daniela Oliveira; Claudio Qureshi.
AG-codes and towers of function fields
AG-codes and towers of function fields
Problems: New construcion of AG-codes. Cyclic AG-codes. Lifting of AG-codes. Isodual AG-codes. Towers of function fields.
People: María Chara, Ricardo Toledano, Gustavo Cabaña, Luciane Quoos.
Invariant metrics on finite groups
Invariant metrics on finite groups
Problems: Invariant metrics on finite groups. Lee metrics. Poset metrics. Hierarchical poset metrics. Chain metrics.
People: Maximiliano Vides. Claudio Qureshi.
Krawtchouk polynomials
Krawtchouk polynomials
Problems: New identities for Krawtchouk polynomials. Relation with binomial numbers. Congruence values.
Grants
Grants
I had and have grants from SECYT (UNC), CAI+D (UNL), FONCYT (ANPCyT) y CONICET.
I participate in research projects with Claudio Qureshi (UdelaR, Uruguay) and with Herivelto Borges (USP - Sao Carlo, Brazil). I used to participate in projects with Javier Cilleruelo and Orlando Villamayor from Madrid (UAM, Spain).
However, I know that in life, one does not may take things for granted, specially research projects.
Students and thesis
Students and thesis
Maximiliano VIDES. Graduate Final Work, 2013. Códigos de Kerdock y Preparata: linealidad y dualidad sobre el anillo Z4.
Eduardo BARSEGHIAN. Graduate Final Work, 2015. Funciones zeta y series armónicas alternantes
Maximiliano VIDES. PhD Thesis, 2018. Métricas sobre anillos y grupos con aplicaciones a la teoría de códigos.
Denis VIDELA. PhD Thesis, 2018. El espectro de códigos cíclicos y grafos asociados.
Paula CHIAPPAROLI. Graduate Final Work, 2020. Distribución de pesos de códigos cíclicos a partir de sumas exponenciales y curvas algebraicas
Paula CHIAPPAROLI. PhD Thesis (ongoing).
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