Vincent Guingona's Mathematics Webpage

Contact Information:

Name: Vincent Guingona
Office: 121 Deichmann Bldg. (#58)
Office Hours: By appointment
Email: (last name) (at) math (dot) bgu (dot) ac (dot) il

Introduction:

Greetings! My name is Vincent Guingona and I am a mathematician. Starting Fall 2015, I will be a postdoc at Wesleyan University. Prior to that, I was a postdoc at Ben-Gurion University and a postdoc at the University of Notre Dame. I received my Ph.D. in mathematics from the University of Maryland College Park under the direction of Chris Laskowski. My research interests include Model Theory, specifically VC-minimal theories, VC-density, NIP theories, and definability of types. I am also interested in applications of model theory to algebra, combinatorics, and computational learning theory. I did my undergraduate work at the University of Chicago, and I am originally from Western Massachusetts.

Employment History:

Education:

Papers:

• On VC-density in VC-minimal theories, submitted.
Abstract: We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acleq = dcleq, the VC-codensity of a formula is at most the number of free variables.
(Modnet Preprint 778, arXiv 1409.8060)

• A Local Characterization of VC-Minimality, Joint with: Uri Andrews - Proceedings of the American Mathematical Society, to appear.
Abstract: We show VC-minimality is Π04-complete. In particular, we give a local characterization of VC-minimality. We also show dp-smallness is Π11-complete.

• On a Common Generalization of Shelah's 2-Rank, dp-Rank, and o-Minimal Dimension, Joint with: Cameron Donnay Hill - Annals of Pure and Applied Logic, 166 (2015), 502-525.
Abstract: In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multiorder property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
(Modnet Preprint 605, arXiv 1307.4113, Annals of Pure and Applied Logic)

• On VC-Density Over Indiscernible Sequences, Joint with: Cameron Donnay Hill - Mathematical Logic Quarterly, 60 (2014), 59-65.
Abstract: In this paper, we study VC-density over indiscernible sequences (denoted VCind-density). We answer an open question in [1], showing that VCind-density is always integer valued. We also show that VCind-density and dp-rank coincide in the natural way.
(Modnet Preprint 363, arXiv 1108.2554, Mathematical Logic Quarterly)

• On VC-Minimal Fields and dp-Smallness, Archive for Mathematical Logic, 53 (2014), 503-517.
Abstract: In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
(Modnet Preprint 609, arXiv 1307.8004, Archive for Mathematical Logic)

• Convexly Orderable Groups and Valued Fields, Joint with: Joseph Flenner - Journal of Symbolic Logic, 79 (2014), 154-170.
Abstract:We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
(Modnet Preprint 505, arXiv 1210.0404, JSL on Cambridge Journals)

• Canonical Forest in Directed Families, Joint with: Joseph Flenner - Proceedings of the American Mathematical Society, 142 (2014), 1849-1860.
Abstract: Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
(Modnet Preprint 379, arXiv 1111.2843, Proceedings of the AMS)

• On VC-Minimal Theories and Variants, Joint with: Michael C. Laskowski - Archive for Mathematical Logic, 52, (2013), 743-758.
Abstract: In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
(Modnet Preprint 364, arXiv 1110.4274, Archive for Mathematical Logic)

• On Uniform Definability of Types over Finite Sets - Journal of Symbolic Logic, 77 (2012), 499-514.
Abstract: In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
(Modnet Preprint 250, arXiv 1005.4924, JSL on Project Euclid)

• Dependence and Isolated Extensions - Proceedings of the American Mathematical Society, 139 (2011), 3349-3357
Abstract: In this paper, we show that φ is a dependent formula if and only if all φ-types have an extension to a φ-isolated φ-type that is an "elementary φ-extension" (see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of φ new elements to the domain of the original φ-type. We give corollaries to this theorem and discuss parallels to the stable setting.
(Modnet Preprint 212, arXiv 0911.1361, Proceedings of the AMS)

Invited Talks:

Other Research:

Teaching:

Current Classes:
• Currently researching.
Previous Classes:
• I gave a three-part lecture series titled "Model Theory and Computational Learning Theory" at Ben-Gurion, Fall 2014.
• I was the lecturer for Beginning Logic, MATH 10130, at Notre Dame, Fall 2013.
• I was the lecturer for Topics in Mathematical Logic, "NIP Theories and Computational Learning Theory," MATH 80510, at Notre Dame, Fall 2013.
• I was the lecturer for Calculus III, MATH 20550, at Notre Dame, Spring 2013.
• I was the lecturer for Calculus III, MATH 20550, at Notre Dame, Fall 2012.
• I was the lecturer for Calculus B, MATH 10360, at Notre Dame, Spring 2012.
• I was the lecturer for Calculus A, MATH 10350, at Notre Dame, Fall 2011.
• I substituted for MATH 713 at Maryland, three weeks of Spring 2011.
• I was a TA for MATH 220 at Maryland, Fall 2006 and Fall 2009.
• I was a TA for MATH 141 at Maryland, Spring 2007.
• I was an advisor for the undergraduate math club at Maryland, Fall 2007.
• I was a grader for MATH 405 at Maryland, Spring 2008.