Graduate – Department of Mathematics https://math.uprrp.edu Río Piedras Tue, 03 Nov 2026 16:03:44 +0000 en-US hourly 1 https://wordpress.org/?v=7.1.4 https://math.uprrp.edu/wp-content/uploads/2017/09/gallo_red_color-150x150.png Graduate – Department of Mathematics https://math.uprrp.edu 32 32 Anthony Cristiana – “Crossingless Connections & Knot Theory” https://math.uprrp.edu/anthony-cristiana-crossingless-connections-knot-theory/ Tue, 03 Nov 2026 15:59:07 +0000 https://math.uprrp.edu/?p=7390 The elements of the Temperley-Lieb Algebra, TLn, can be realized as crossingless connections between an even number of marked boundary points in a rectangle, and multiplication in TLn is given by concatenation of these crossingless connections. This talk will begin with exposition on the Temperley-Lieb algebra—especially in its connections with knot theory—before describing a bilinear form on the Temperley-Lieb algebra defined by “counting unknots”. We will generalize this bilinear form and its corresponding Gram determinant to a set of crossingless connections in the Mobius band.

]]> Javier O. Sierra Padilla’s dissertation defense https://math.uprrp.edu/javier-o-sierra-padillas-dissertation-defense/ Fri, 16 Oct 2026 16:14:08 +0000 https://math.uprrp.edu/?p=7350 Congratulations to Javier O. Sierra Padilla for his successful Dissertation Defense. If the frame below does not scroll on your device click here.

]]> Israel A. Almodóvar-Rivera – “Kernel-Estimated Nonparametric Diagnostic for Outlier Detection in a Clustering Framework” https://math.uprrp.edu/israel-a-almodovar-rivera-kernel-estimated-nonparametric-diagnostic-for-outlier-detection-in-a-clustering-framework/ Thu, 15 Oct 2026 17:55:16 +0000 https://math.uprrp.edu/?p=7327

Finding groups in the presence of scattered or outlying observations can
be challenging. In this work, we propose a distribution-free approach for
developing a diagnostic tool to identify potential outliers in a given
clustering solution. The proposed methodology is based on a smooth
estimate of the distribution function of the normed residuals obtained
from a k-means solution. We also incorporate an adaptive threshold for
the detection of potential outliers. We study the performance of the
proposed diagnostic tool through several simulation experiments using
existing clustering algorithms that are designed to identify outliers in
homogeneous spherical groups. Finally, we apply the proposed
diagnostic tool to a functional Magnetic Resonance Imaging (fMRI) data
from a single-task finger-tapping experiment to identify potential
outlying observation as activated brain regions.

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]]> NEW DATE! José La Luz Concepción – “Introduction To Toric Topology” https://math.uprrp.edu/jose-la-luz-concepcion-introduction-to-toric-topology/ Mon, 28 Sep 2026 13:13:03 +0000 https://math.uprrp.edu/?p=7324 Toric topology is an active area at the intersection of topology, algebraic geometry,
differential topology, and combinatorics. Roughly speaking, a quasitoric manifold is
a complex manifold equipped with a torus action whose orbit space is a simple
convex polytope. This talk will introduce the basic ideas of toric topology and
explore algebraic-topological aspects of these manifolds through examples, with
emphasis on their cohomology rings and homotopy groups. I will also discuss open
problems, particularly rigidity conjectures. Time permitting, I will briefly overview
stable splittings of certain toric spaces related to quasitoric manifolds.

]]> Iván Cardona – “Topology, DNA and persistent homology” https://math.uprrp.edu/ivan-cardona-topology-dna-and-persistent-homology/ Wed, 09 Sep 2026 13:08:29 +0000 https://math.uprrp.edu/?p=7303 The Topology Seminar at the Department of Mathematics of the University of Puerto Rico, Río Piedras—affectionately known by locals as la IUPI—was launched in the summer of 2026. The seminar was created through the initiative of Professors Iván Cardona Torres and Gabriel Montoya Vega to provide a dedicated space for the discussion of research in topology and to foster a vibrant culture of mathematical research within the department. While the seminar focuses on topology, it also explores the many ways in which topological methods and ideas influence research throughout mathematics.

A bit of history:

The seminar’s inaugural speaker was Professor Józef H. Przytycki, who presented the lecture Quantum Plane and Plucking Polynomial of Rooted Trees on July 15. Designed for a broad audience, the talk brought together faculty members, graduate students, and incoming first-year students participating in the university’s summer preparatory pre-calculus program.

We look forward to welcoming more speakers and participants as the seminar continues to grow and strengthen the topology community at UPR–Río Piedras.
Feel free to reach out!

The Topology Seminar page.

]]> Alec S. Zabel-Mena’s thesis defense https://math.uprrp.edu/alec-s-zabel-menas-thesis-defense/ Wed, 19 Aug 2026 12:18:07 +0000 https://math.uprrp.edu/?p=7274

Congratulations to Alec S. Zabel-Mena for his successful thesis defense. If the frame below does not scroll on your device click here.

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]]> Rodrigo A. León Prato’s dissertation defense https://math.uprrp.edu/rodrigo-a-leon-pratos-dissertation-defense/ Thu, 30 Jul 2026 13:53:48 +0000 https://math.uprrp.edu/?p=7256 Congratulations to Rodrigo A. León Prato for his successful Dissertation Defense. If the frame below does not scroll on your device click here.

]]> Minicourses: Algebraic Structures in Topology 2026 https://math.uprrp.edu/minicourses/ Tue, 30 Jun 2026 13:35:25 +0000 https://math.uprrp.edu/?p=7222 The Algebraic Structures in Topology is a conference series that meets every two years in San Juan, Puerto Rico and this year is partially supported by the National Science Foundation, Purdue University, the Institute of the Mathematical Sciences of the Americas at the University of Miami and the K-Theory Foundation. The conferences will feature a variety of events focusing on recent developments in algebraic topology and their applications to geometry, physics, and data science. This iteration is an official satellite event of the International Congress of Mathematicians 2026 (July 23-30, 2026).

This year’s iteration of the series will occur on July 10 – July 17, 2026. From July 10th to July 12th it will feature three introductory mini-courses on the topics of topological quantum computation, stable homotopy theory, and the cohomology of the Torelli group accessible to graduate students and researchers in related fields outside algebraic topology.

Mini-courses (at Julio García Díaz building JGD-123)

July 10–12

Conference schedule here

Friday · July 10

Time Activity
10:00–11:00 Invited Lecture – I. Levy
11:30–12:30 Invited Lecture – A. Putman
12:30–2:30 Lunch
2:30–3:30 Invited Lecture – S. Cui
4:00–4:30 Contributed Talk – A. Waugh
4:30–5:00 Contributed Talk – A. Watkins

Saturday · July 11

Time Activity
10:00–11:00 Invited Lecture – I. Levy
11:30–12:30 Invited Lecture – A. Putman
12:30–2:30 Lunch
2:30–3:30 Invited Lecture – S. Cui
4:00–4:30 Contributed Talk – S. Kim
4:30–5:00 Contributed Talk – U. Pal

Sunday · July 12

Time Activity
10:00–11:00 Invited Lecture – I. Levy
11:30–12:30 Invited Lecture – A. Putman
12:30–2:30 Lunch
2:30–3:30 Invited Lecture – S. Cui
4:00–4:30 Contributed Talk – J. Kaufmann
4:30–5:00 Contributed Talk – M. Ji

Ishan Levy (Institute for Advanced Study)

The Chromatic View of Stable Homotopy Theory

Lecture 1

The goal of this talk is to give an overview of the category of spectra and some
of its basic properties. We will take the point of view that the category of
spectra is a universal place in which algebra can be done.

Lecture 2

Work of Quillen and Novikov gives a deep connection between the category of
spectra and the theory of formal groups. In this talk, I will explain this
connection, and discuss some relevant structural results about formal groups,
such as the height filtration.

Lecture 3

I will explain how the height filtration of formal groups can be incarnated at
the level of spectra in two ways: as the chromatic filtration as well as a
finitary version of it. In doing so I will also explain some of Ravenel’s
conjectures, what is still unknown, and their connection to the homotopy groups
of spheres.

Andrew Putman (University of Notre Dame)

The Cohomology of the Torelli Group

Lecture 1: The Mapping Class Group and Its Finiteness Properties

This talk will introduce the mapping class group of a surface and discuss its
finiteness properties.

Lecture 2: The Torelli Group

The Torelli group is the kernel of the action of the mapping class group on the
first homology group of the surface. This talk will introduce the Torelli group
and explain some of its structure. In particular, we will give an introduction
to the Johnson homomorphism. One of the themes of this talk will be finiteness
and non-finiteness properties.

Lecture 3: Homological Finiteness of the Torelli Group

Recent work of myself and Dan Minahan shows that for surfaces of sufficiently
large genus, H2 of the Torelli group is finite-dimensional and
spanned by cup products of classes in H1. A very recent (June 11)
paper of Gaifullin gives a spectacular generalization of this and shows that
each Hk is finitely generated for g large (relative to k). I will
explain Gaifullin’s proof, specialized to the case of H2.

Shawn Cui (Purdue University)

An Introduction to Topological Quantum Computation

Lecture 1: From Quantum Mechanics to Topological Quantum Computation

We will review the basic ideas of quantum mechanics and quantum computation,
explain the challenge of protecting quantum information from noise, and motivate
the topological approach. We will then introduce the general framework of TQC,
anyons as its fundamental objects, and several important computational models.

Lecture 2: The Algebraic Theory of Anyons

We will develop the algebraic description of anyons using representation theory
and category theory. Topics will include fusion rules, braid group
representations, and the categorical structures that encode fusion and braiding.

Lecture 3: Universality, Simulation, and Topological Applications

We will discuss several central problems in TQC, including computational
universality and simulations between different anyon models. Time permitting, we
will also explore applications of topological quantum computation to topological
quantum field theory and low-dimensional topology.

]]> Daniel Anglada Marrero’s thesis defense https://math.uprrp.edu/daniel-anglada-marreros-thesis-defense/ Tue, 30 Jun 2026 12:03:27 +0000 https://math.uprrp.edu/?p=7216 Congratulations to Daniel Anglada Marrero for his successful thesis defense. If the frame below does not scroll on your device click here.

]]> Desiré Vázquez Ortiz’s thesis defense https://math.uprrp.edu/desire-vazquez-ortizs-thesis-defense/ Tue, 19 May 2026 19:46:24 +0000 https://math.uprrp.edu/?p=7185 Congratulations to Desiré Vázquez Ortiz for her successful Thesis Defense. If the frame below does not scroll on your device click here.

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