Leticia Zárate

I have finished my PH.D. in CINVESTAV-IPN. In my thesis work I studied the BP-homology of Z_2^e × Z_2^e. First I studied the BP-cohomology of finite dimensional lens spaces. In the case of torsion 4, the knowledge of the annihilator ideal of the so-called toral class τ allows to give lower bounds of the topological complexity of lens spaces Lⁿ(4) for particular values of n. For the general case I obtained a family of relations in BP(Z_2^e) &otimes_BP BP(Z_2^e); in particular I obtained a family of elements annihilating the toral class. The main result in this direction stablishes that (2^e, 2^e-1 v₁, 2^e-2 v₁⁴, ... , v₁^{2e + 2e-1-2}) is contained in the annihilator ideal of τ. I conjecture that this family of annihilating elements is in fact a minimal set of generators of the annihilator ideal of τ. Here you can obtain a pdf version of my Thesis.

I am intereseted in homotopy theory and the algebraic properties of formal groups that arise in stable homotopy and its applications to geometric problems. Actually I am working with J. González in some conjectures about the structure, up to extensions, of the ku-homology of the classifying space of the classifying space of the group Z_2^e × Z_2^e. We are also interested in the relation of the ku_*-annihilator ideal of the ku-toral class with its BP-analogue.

I am also interested in Commutative Algebra and some relations with combinatorics. I am working with E. Reyes in the classification of combinatorial structures (graphs, matroids and hypergraphs) that give monomial birational maps. In the case of simple graphs, we have a combinatorial criterion of the graphs that give rise to monomial birational maps. We have an algorithm to construct the rational inverse. In the case of matroids we have a family of birational matroids (conjectured to be all).

I will be applying for postdoctoral positions in Algebraic Topology and/or Commutative ring theory worldwide in fall of 2007.

Here is a list of my publications/preprints.

• E. Reyes, R. Villarreal, L. Zárate, A note on affine toric varieties. Lineal Algebra and its applications 318 (2000) 173-179.
• A. Alcantar, E. Reyes, L. Zárate, On the complete intersection in characteristic p. Aportaciones Matemáticas. Serie Comunicaciones 27 (2000) 133-140.
• Jesús González, Leticia Zárate, BP-theoretic instabilities to the motion planning problem in 4-torsion lens spaces. Osaka Journal of Mathematics 43 Num.3 (2006) 581-596.
• Jesús González, Leticia Zárate, BP(n)-homology of Z_2^e × Z_2^e: the role of v₀ and v₁ divisibility in 2-typical formal group laws. (Preprint.)
• Leticia Zárate, On the BP-homology of Z_2^e × Z_2^e. (Submited.)
• L. Zárate. On the ku-homology of Z_p^e × Z_p^e. (Under preparation.)
• E. Reyes, L. Zárate, Birational graphs and birational matroids. (Under preparation.)

I have a strong background in:

• K-theory
• Generalised homology
• Stable Homotopy
• Formal Groups
• Graded Algebras and Monomial Algebras
• Computational Commutative Algebra
• Cohen-Macaulay Rings
• Algebraic Combinatorics

STUDIES:

• 1993-1996 Ciencias Físico-Matemáticas. BUAP. (Undergraduate student)
• 1996-1999 Departamento de Matemáticas CINVESTAV. (Master student)
• 1999-2002 Departamento de Matemática UFPE - Brasil. (Ph.D. student)
• 2003-2007 Departamento de Matemáticas CINVESTAV. (Ph.D. student)

TALKS:

• Birational maps defined by monomials. XXXVII Congreso Nacional de la Sociedad Matemática Mexicana. Ensenada, B.C. México. (2004).
• Non bipartite graphs and birational maps. ACCOTA 2004. San Miguel de Allende, Guanajuato. México. (2004).
• Spectral Sequences. XXXVIII Congreso Nacional de la Sociedad Matemática Mexicana. México, D.F. México. (2005).
• Birational Matroids. XXXIX Congreso Nacional de la Sociedad Matemática Mexicana Villahermosa, Tabasco. México. (2006).

OTHER ACADEMIC ACTVITIES:

• Assistant of the summer course: Differential Calculus.
  Facultad de Ciencias Físico Matemáticas-BUAP.
  August 1995.
• Assistant of the course Algebraic Structures.
  Centro de Ciencias Exatas e da Natureza. UFPe (Brasil).
  February 2002-June 2002.
• Assistant of the course Commutative Algebra.
  Centro de Ciencias Exatas e da Natureza. UFPe (Brasil).
  February 2002-June 2002.
• Member of the Organizing Committee of Students' Seminary.
  Departamento de Matemáticas. CINVESTAV-IPN.
  August 2004-September 2005.
• Collaborator of Morfismos. (Students and Research Journal.)
  Departamento de Matemáticas. CINVESTAV-IPN.
  August 2005-September 2006.
• Monitor of the XI Workshop in Mathematical Modelling.
  Departamento de Matemáticas. CINVESTAV-IPN.
  August 2006.