Address and contact information:

School of Engineering, Science and Technology

Universidad del Rosario

Carrera 6 No. 12C-16, Bogotá 111711

Office: Edificio Cabal, 508

e-mail: cristianm.martinez@urosario.edu.co

I am a Profesor Principal at the School of Engenieering, Science, and Technology of Universidad del Rosario in Bogotá. Before that, I was a postdoctoral fellow at Universidade Estadual de Campinas (Unicamp) in the Gauge Theory and Algebraic Geometry group of the Institute of Mathematics, Statistics and Scientific Computing under the supervision of Marcos Jardim. Before Brazil, I held several positions: I have been an instructor at Universidad del Norte in Barranquilla, a Postdoctoral Researcher at Universidad de Los Andes in Bogotá, and a Visiting Assistant Professor at the University of California, Santa Barbara under David R. Morrison. I graduated from the University of Utah in May 2015 with a PhD in Mathematics. I wrote my dissertation in Algebraic Geometry under the supervision of Aaron Bertram.

### Research

I am interested in Derived Categories, Bridgeland Stability Conditions, Moduli Spaces of Sheaves, and in general, mathematics inspired by Physics. I have worked on preservation of stability under Fourier-Mukai transforms, construction of stability conditions on higher dimensional varieties and the study of the birational geometry of moduli spaces of sheaves via wall-crossing techniques.

### Papers, preprints

With G. Comaschi, M. Jardim, and D. Mu. Instanton sheaves: the next frontier: Instantons, emerged in particle physics, have been intensely studied since the 1970's and had an enormous impact in mathematics since then. In this paper, we focus on one particular way in which mathematical physics has guided the development of algebraic geometry in the past 40+ years. To be precise, we examine how the notion of mathematical instanton bundles in algebraic geometry has evolved from a class of vector bundles over the complex projective 3-space both to a class of torsion free sheaves on projective varieties of arbitrary dimension, and to a class of objects in the derived category of Fano threefolds. The original results contained in this survey focus precisely on the latter direction; in particular, we prove that the classical rank 2 instanton bundles over the projective 3-space are indeed instanton objects for any suitable chamber in the space of Bridgeland stability conditions. arXiv. São Paulo Journal of Mathematical Sciences. Memorial Volume for Sasha Anan'in, November 2023. https://doi.org/10.1007/s40863-023-00387-3

With J. Lo. Geometric stability conditions under autoequivalences and applications: Elliptic Surfaces: On a Weierstrass elliptic surface, we describe the action of the relative Fourier-Mukai transform on the geometric chamber of Stab(X), and in the K3 case we also study the action on one of its boundary components. Using new estimates for the Gieseker chamber we prove that Gieseker stability for polarizations on certain Friedman chamber is preserved by the derived dual of the relative Fourier-Mukai transform. As an application of our description of the action, we also prove projectivity for some moduli spaces of Bridgeland semistable objects. arXiv. Journal of Geometry and Physics, Volume 194, December 2023, 104994. https://doi.org/10.1016/j.geomphys.2023.104994

With M. Jardim and A. Maciocia. Vertical Asymptotics for Bridgeland Stability Conditions on Threefolds: Let X be a smooth projective threefold of Picard number one for which the generalized Bogomlov-Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to X in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and ch^β(E)=(−R,0,D,0), we prove that there are only a finite number of nested walls in the (α,s)-plane. Moreover, when R=0 the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when β=0 there are no semistable objects in the innermost chamber. In both cases, the only limit semistable objects of the form E or E[1] (where E is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves. International Mathematics Research Notices, rnac236, https://doi.org/10.1093/imrn/rnac236. arXiv.

With W. Liu and J. Lo. Fourier-Mukai Transforms and Stable Sheaves on Weierstrass Elliptic Surfaces: On a Weierstrass elliptic surface X, we define a `limit' of Bridgeland stability conditions, denoted as Z^l-stability, by varying the polarisation along a curve in the ample cone. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a Z^l-stable object, and describe a modification upon which a Z^l-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show that 1-dimensional twisted Gieseker semistable sheaves are taken by a Fourier-Mukai transform to Bridgeland semistable objects. Submitted. arXiv.

With B. Schmidt. Bridgeland stability on blow-ups and counterexamples (with an appendix by Omprokash Das): We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macrì, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstraß elliptic Calabi-Yau threefolds. Furthermore, we show that if the original conjecture, or a minor modification of it, holds on a smooth projective threefold, then the space of stability conditions is non-empty on the blow up at an arbitrary point. More precisely, there are stability conditions on the blow up for which all skyscraper sheaves are semistable. Mathematische Zeitschrift, 2018, DOI 10.1007/s00209-018-2149-3. arXiv.

With A. Bertram. Change of polarization for moduli of sheaves on surfaces as Bridgeland wall-crossing: We prove that the "Thaddeus flips" of L-twisted sheaves constructed by Matsuki and Wentworth can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of 1-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions. International Mathematics Research Notices, rny065, Journal link . arXiv.

Duality, Bridgeland wall-crossing and flips of secant varieties: Let v_d⊂ℙH^0(O(d)) denote the d-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of v_d by embedding the blow-up of ℙH^0(O(d)) along v_d into a suitable moduli space of Bridgeland semistable objects on the complex projective plane. International Journal of Mathematics: Volume 28, Issue 2 (2017), Pages 1750011-40. arXiv.

With A. Bertram and J. Wang. The birational geometry of moduli spaces of sheaves on the projective plane: We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves M(v) on the complex projective plane and mini-wall crossings in the stability manifold Stab(P^2). Geometriae Dedicata: Volume 173, Issue 1 (2014), Pages 37-64. arXiv.

### Students

Daniel Bernal. Universidade Estadual de Campinas. Master Thesis: ''Moduli Spaces of Neural Networks''. Co-advised together with Marcos Jardim. 2023.

Juan Pablo Zuñiga Valencia. Universidad de los Andes. Undergraduate Thesis: "The Derived Category of Coherent Sheaves on P^n". 2019.

Talon Stark. College of Creative Studies at UCSB. Undergraduate Student. "Towards Projectivity of some Bridgeland moduli spaces on Hirzebruch surfaces". Now a graduate student at UCLA. 2018.

### Teaching

This semester (2024-1) I will be teaching Topology, Linear Algebra, and Complex Variables.

Past Teaching

At Urosario: Topology, Differential Equations, Vector Calculus, Complex Variables. At Uniandes: Differential Calculus, Integral Calculus and Differential Equations, Abstract Algebra. At UCSB: Complex Variables, Intro to Real Analysis, Introduction to proof writing, Vector Calculus, Differential Equations.

### Slides from some talks

Universidad de los Andes Colloquium, Fall 2018

Encuentro SCM-SMM, Spring 2018

AMS Sectional Meeting-Portland State University, Spring 2018.

( Washington University in St. Louis-Algebraic Geometry Seminar, Spring 2018)

AMS Sectional Meeting -UGA, Spring 2016

AMS Summer Institute in Algebraic Geometry, Salt Lake City 2015

### Links

School of Engineering, Science and Technology