Email:  jpinzonc@nd.edu  Office:  116 Hayes Healy 
Phone:  (574) 6317393  Languages:  English, Spanish 
EDUCATION

20082014
Indiana University, Bloomington
PhD in Mathematics
Thesis: Independence of satellite operations of torus knots in the smooth concordance group.
Adviser: Paul Kirk.

20032007
Universidad de Los Andes
BSc in Mathematics
Thesis: Morse Homology.
Adviser: Bernardo Uribe.
Employment

Fall 2020
University of Notre DameAssistant Professor of Mathematics

20192020
Max Planck Institute for MathematicsPostdoctoral Fellow
Mentor: Aru Ray.

20172019
NC State UniversityPostdoctoral Research Scholar
Mentors: Radmila Sazdanovic and Tye Lidman.

20142017
University of Georgia AthensPostdoctoral Teaching and Research Assistant
Mentor: David Gay.
Grants, Honors and Awards
 AIM SQuaRE  Trisections, Knotted Surfaces, and Symplectic 4Manifolds, 2017
 NSF FRG Trisections  New Directions in Lowdimensional Topology, 2016
 OVPR Foreign Travel Assistance Program, University of Georgia 2015
 Provost’s Travel Award for Women in Science, Indiana University 2013
 David A. Rothrock Associate Instructor Award, Indiana University 2012
 Women in Science Program Travel Award, Indiana University 2012
 Women in Science Program Travel Award, Indiana University 2011
 Office of Women’s Affairs Fellowship, Indiana University 2008
 Teaching Assistanship, Universidad de Los Andes 2007
 Academic Excellence, Gimnasio Los Portales 2002
My research investigates the interplay between gaugetheoretic invariants and topological constructions. On the constructive side, my most current efforts have been concentrated around the development of the theory of trisections for 4–manifolds with boundary and of embedded surfaces in B^4. On the gauge theoretical side, I have mostly studied the instanton moduli space. More details about my research can be found on my Research Statement.
PUBLICATIONS
Click here to find me on the arXiv.
Click here to find me on MathSciNet (subscription required.)

Framed instanton homology of surgeries on Lspace knots
Tye Lidman, Juanita PinzonCaicedo, Christopher Scaduto
Submitted
An important class of threemanifolds are Lspaces, which are rational homology spheres with the smallest possible Floer homology. For knots with an instanton Lspace surgery, we compute the framed instanton Floer homology of all integral surgeries. As a consequence, if a knot has a Heegaard Floer and instanton Floer Lspace surgery, then the theories agree for all integral surgeries. In order to prove the main result, we prove that the BaldwinSivek contact invariant in framed instanton Floer homology is homogeneous with respect to the absolute ℤ/2grading, but not the ℤ/4grading.

The Topological Slice Genus of Satellite Knots
Peter Feller, Allison N. Miller, Juanita PinzonCaicedo
Submitted
This paper presents evidence supporting the conjecture that in the topological category the slice genus of a satellite knot P(K) is bounded above by the sum of the slice genera of K and P(U). Our main result establishes this conjecture for a variant of the topological slice genus, the Zslice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern P. This stands in stark contrast with the smooth category, where for example there are many genus 1 knots whose (n, 1) cables have arbitrarily large smooth 4genera. As an application, we show that the (n, 1)cable of any knot of 3genus one (e.g. the figureeight or trefoil knot) has topological slice genus at most 1, regardless of the value of n ∈ N. Further, we show that the lower bounds on the slice genus coming from the TristramLevine and CassonGordon signatures cannot be used to disprove the conjecture.

Satellites of Infinite Rank in the Smooth Concordance Group
Matthew Hedden, Juanita PinzonCaicedo
Submitted
We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use $SO(3)$ gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group $\mathcal{C}$. Our criterion applies widely, notably to scores of unknotted patterns for which the corresponding operators on the topological concordance group are zero. We raise some questions and conjectures regarding satellite operators and their interaction with concordance.

Trisections of 4manifolds with Boundary
Nickolas A. Castro, David T. Gay, Juanita PinzonCaicedo
Proc. Natl. Acad. Sci. USA 115 (2018), no. 43, 10861–10868.
Given a handle decomposition of a 4manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4manifolds with multiple boundary components, and show how trisected 4manifolds with multiple boundary components glue together.

Independence of Iterated Whitehead Doubles
Juanita PinzonCaicedo
Proc. Amer. Math. Soc. 147 (2019), no. 3, 1313–1324.
A theorem of Furuta and FintushelStern provides a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3spheres. In this article we use these results and some 4dimensional constructions to produce infinite families of positive torus knots whose iterated Whitehead doubles are independent in the smooth concordance group.

Diagrams for Relative Trisections
Nickolas A. Castro, David T. Gay, Juanita PinzonCaicedo
Pacific J. Math. 294 (2018), no. 2, 275–305.
We establish a correspondence between trisections of smooth, compact, oriented 4manifolds with connected boundary and diagrams describing these trisected 4manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise standardness condition. This should be thought of as the 4dimensional analog of a sutured Heegaard diagram for a sutured 3manifold. We also give many foundational examples.

Independence of Satellites of Torus Knots in the Smooth Concordance Group
Juanita PinzonCaicedo
Geom. Topol. 21 (2017), no. 6, 3191–3211.
The main goal of this article is to obtain a condition under which an infinite collection ℱ of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining both the instanton moduli space over a 4manifold with tubular ends and the corresponding ChernSimons invariant of the adequate 3dimensional portion of the 4manifold. More specifically, the result is derived from Furuta's criterion for the independence of Seifert fibred homology spheres in the homology cobordism group of oriented homology 3spheres. Indeed, we first associate to ℱ the corresponding collection of 2fold covers of the 3sphere branched over the elements of ℱ and then introduce definite cobordisms from the aforementioned covers of the satellites to a number of Seifert fibered homology spheres. This allows us to apply Furuta's criterion and thus obtain a condition that guarantees the independence of the family ℱ in the smooth concordance group.

Traceless SU(2) representations of 2stranded tangles
Yoshihiro Fukumoto, Paul Kirk, Juanita PinzonCaicedo
Math. Proc. Cambridge Philos. Soc. 162 (2017), no. 1, 101–129.
Given a 2stranded tangle T contained in a ℤhomology ball Y, we investigate the character variety R(Y,T) of conjugacy classes of traceless SU(2) representations of π1(Y \ T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y, T) for many tangles naturally associated to knots in S 3. Moreover, we determine the image of the restriction map from R(T, Y) to the traceless SU(2) character variety of the 4punctured 2sphere (the pillowcase). We give examples to show this image can be nonlinear in general, and show it is linear for tangles associated to pretzel knots.
SELECTED TALKS

Gauge Theory and Knot Concordance
Latinx in Math, Duke Topology Seminar, MSU Topology Seminar
Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K0 and K1 are said to be smoothly concordant if there is a smooth embedding of the 2—dimensional annulus S1 × [0, 1] into the 4—dimensional cylinder S3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set of smooth concordance classes of knots, C, is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in lowdimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4manifolds can be used to better understand these objects. In particular, the study of antiself dual connections on 4manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.
Poster 
La dimensión 4 y la concordancia de nudos
XXI Congreso Colombiano de Matematicas
La meta principal de la topología geométrica es la clasificación de las variedades según su estructura (topológica, suave, simpléctica, etc.). En dimensión 4 la historia es especial: es la única dimensión en la que una variedad puede admitir un número infinito de estructuras suaves no equivalentes y la única dimensión en la que existen variedades homeomorfas pero no difeomorfas a R^4. A su vez, la concordancia de nudos es el estudio de los nudos como fronteras de superficies encajadas en espacios de dimensión 4. La diferencia entre estructuras topológicas y estructuras suaves en dimensión 4 está estrechamente relacionada con la concordancia de nudos. En la charla explicaré una faceta de ésta relación.

Examples of relative trisections
AMS Sectional NCSU, Georgia Tech, AMS Sectional St. Thomas University, HIM
Trisections of 4manifolds relative to their boundary were introduced by Gay and Kirby in 2012. They are decompositions of 4manifolds that induce open book decomposition in the bounding 3manifolds. This talk will focus on diagrams of relative trisections and will be divided in two. In the first half I will focus on trisections as fillings of open book decompositions and I will present different fillings of different open book decompositions of the Poincare homology sphere. In the second half I will show examples of trisections of pieces of some of the surgery techniques that result in exotic 4manifolds.
Slides1 Slides2 
Iterated Whitehead Doubles are Independent
MPIM
In the 1980’s Furuta and FintushelStern applied the theory of instantons and ChernSimons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3spheres. These results, together with some 4dimensional constructions can be used to show that iterated Whitehead doubles of positive torus knots are independent in the smooth concordance group.

An Overview of Relative Trisections
University of Iowa, AMS Sectional UGA, BIRS, SUNY Buffalo
Based on the notion of trisections of closed 4manifolds of Gay and Kirby, Nick Castro and myself developed a rigorous definition of relative trisections as a generalization of trisections to 4manifolds with boundary. In the talk I will present the basics of relative trisections starting with their relationship to open book decompositions of the bounding manifolds. I will then introduce a stabilization operation that gives rise to a statement about the uniqueness of relative trisections, thus complementing Gay and Kirby's proof of the existence of relative trisections. Finally, I will introduce the notion of diagrams of relative trisections and describe a method to recover the open book decomposition of the bounding manifold from the trisection diagrams.
Slides Video 
Construcciones de 3variedades y la esfera de Poincaré
Matemáticas por estudiantes, Universidad de Los Andes
En 1910 Henri Poincare publicó un articulo en el que afirmaba que los grupos de homología eran suficientes para determinar si una 3variedad es la 3esfera S3. Cuatro años más adelante Poincaré mismo publicó la descripción de una 3variedad con los mismos grupos de homología que S3 pero distinta a esta última. Hoy en día este espacio se conoce como la esfera de Poincaré y parte de su encanto radica en que se puede describir de distintas maneras, todas construcciones de enorme importancia en el estudio de 3variedades. La idea de la charla es describir algunas de estas construcciones a través de la esfera de Poincaré. Algunas de las construcciones que intentaré describir son cirugía de Dehn, espacios recubridores ramificados y espacios cocientes.

Independence of Whitehead Doubles of Torus Knots in the Smooth Concordance Group
Georgia Tech, Brandeis University, Knots in the Triangle, University of Georgia, IAS, University of Minnesota
In the 1980’s Furuta and FintushelStern applied the theory of instantons and ChernSimons invariants to develop a criterion for a collection of special homology spheres to be independent in the homology cobordism group of oriented homology 3spheres. Hedden and Kirk then use the aforementioned criterion to establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group. In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that simplify and extend their argument.
Slides 
3Esferas de Homología
Oaxaca
Una 3esfera de homología es una variedad diferencial cerrada y orientada que tiene mismos grupos de homología de la 3esfera. En el minicurso presentaré diferentes construcciones de este tipo de objetos, introduciré los invariantes de Rokhlin y Casson para finalmente hablar de la estructura algebraica del grupo de cobordismo de 3esferas de homología.

Traceless SU(2) representations of 2stranded tangles
Michigan State, Louisiana State University
Given a codimension 2 submanifold A⊂X define R(X,A) as the space of traceless SU(2) representations of π_1(X\A) modulo conjugation. For Y a 3manifold and K⊂Y a knot, KronheimerMrowka defined the Instanton Knot Homology of (Y,K) as the homology of a chain complex whose groups are generated by the elements of R(Y,K). In the talk we describe a method to determine R(S^3,K) whenever K is a torus or pretzel knot.
Slides
I find the most effective way to impart knowledge is through dialogue: I view teaching as a circle of communication between the students and myself, and in my classroom I work for this mutual interaction. In practice, this means that I direct the discussion toward exploration of concepts and propose pertinent questions that prompt the students to be active participants. By building upon class participation and mutual interaction I work to turn students into active agents, going from passively receiving knowledge to generating it.
20+
Classes
10+ years
Teaching Experience
900+
Students
Teaching Experience

Mathematics Courses
Courses typically taken by math majors or graduate students in mathematics. Vector Bundles and Characteristic Classes
PhD level course.
 Algebraic Topology
PhD level course. One semester as a teaching assistant.
 Pointset Topology
One semester as the main instructor. Mixed undergraduate/graduate course.
 Calculus on Manifolds
One semester experience as a teaching assistant.
 Vector Bundles and Characteristic Classes

Mathematics Requirement
Courses taken by nonmath majors to fullfill a mathematics requirement. Calculus
Six semesters as the main instructor, four semesters as an assistant. Class size 20100.
 PreCalculus
Two semesters as the main instructor. About 30 students per class.
 Finite Mathematics
Four semesters as the main instructor, one semester as an assistant. Class size 20200.
 Calculus

Mathematics Education
Content courses for students majoring in education. Foundations of Geometry
One semester as the main instructor. Mixed undergraduate/graduate course.
 Modern Algebra for Secondary School Teachers
One semester experience as a teaching assistant.
 Topics in Euclidean Geometry
One semester experience as a teaching assistant.
 Foundations of Geometry