# Second week

### Course 1

Diagrammatic methods for quantum symmetries
Scott Morrison
Australian National University

### Course 2

Hopf Algebras and Their Generalizations from a Categorical Point of View
Gabriella Böhm
Wigner Research Centre for Physics of the Hungarian Academy of Sciences

### Course 3

Topological Quantum Field Theory
Noah Snyder
Indiana University, Bloomington

### Research talk

The Yoneda algebra of some Nichols algebras
Andrea Solotar
Universidad de Buenos Aires.

Abstract: Nichols algebras are an important tool for the classification of Hopf algebras. Within those with finite Gelfand-Kirillov dimension, we study homological invariants of the Jordan and super Jordan plane, that is, the Nichols algebras $\mathcal{B}(V(1, 2))$ and $\mathcal{B}(V(−1, 2))$. We prove that the Yoneda algebra of the bosonization of the Jordan and super Jordan plane are finitely generated.

Given a finite dimensional Hopf algebra $H$, there is a conjecture by Etingof and Ostrik stating that the cohomology algebra $H^∗(H, k)$ (i.e. the Yoneda algebra), is finitely generated. Even if these Hopf algebras are not finite dimensional, their Gelfand-Kirillov dimension is finite, and our results are in line with previous results on this subject. Notice that in our cases the braiding group is infinite and the braiding is not diagonal.

## Professional development activity:

Panel session about the different career paths and applying to positions:

Abstract: In this panel, we will discuss different possibilities for career path for mathematicians (academia, industry, teaching positions, etc). We will focus on the application process for academic jobs in different places (where to look for positions, timeline, materials, etc). We will also comment on some resources that we know are useful. Questions from the participants are highly encouraged.

## Poster Session:

Name University Title
Fabio Calderón Universidad Nacional de Colombia. Colombia Some interactions between Hopf Galois Extensions and noncommutative rings
Angie Tatiana Suárez Romero Federal University of Goiás. Brazil Conley index and periodic trajectories.
Marcen Laguna Universidad de Sao Paulo. Brazil Introduction to classical quantum logic.
Paolo Saracco Université Libre de Bruxelles. Belgium On Frobenius and Hopf
Jhoan Sebastian Sierra Vargas Pontificia Universidad Javeriana. Colombia Funciones multisimétricas y politopos de transporte
Qing Zhang Texas A&M University. USA Braid Group Representations from Twisted Tensor Products of Group Algebras
Danilo José Polo Ojito Universidad del Atlántico. Colombia Frames in Krein C-modules over krein C-algebras.
Camilo Rengifo Gutiérrez Universidad de La Sabana. Colombia Transgression for Courant Algebroids
Pablo Sanchez Ocal Texas A&M University. USA Hochschild Cohomology of Twisted Tensor Product Algebras
Roberto Hernandez Palomares The Ohio State University. USA Classifying Module Categories for Generalized Temperley-Lieb-Jones ∗-2-Categories

# First week

### Course 1

The Mathematics of Topological Quantum Computing
Eric Rowell
Texas A&M University

### Course 2

Tensor categories
Victor Ostrik
University of Oregon

### Course 3

On finite-dimensional Hopf algebras and their representations
Siu-Hung Ng
Louisiana State University

## Research talk

Tensor Algebras in Finite Tensor Categories
Chelsea Walton
The University of Illinois at Urbana-Champaign.

Abstract: I will discuss a method for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $$T_B(V)$$ where $$B$$ is a semisimple $\mathbb{k}$-algebra and $V$ is a $B$-bimodule. This is done by working within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, parameterizing tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. This is joint work with Pavel Etingof and Ryan Kinser, arxiv:1906.02828.

See a handout for the talk.

## Public lecture

Teoría de nudos y física: de Lord Kelvin a la información cuántica
Eric Rowell
Texas A&M University

Resumen: Las matemáticas y la física tienen una relación simbiótica. Por ejemplo, el desarrollo de la teoría de nudos y eslabonamientos debe mucho a la física: la primera clasificación de nudos con pocos cruces fue lograda por el físico escocés Peter Tait. Él fue motivado por un sueño de Lord Kelvin que elementos químicos diferentes correspondían a nudos distintos.

Mendeléyev aplastó éste sueño en 1869 con su tabla periódica de los elementos. Pero el trabajo de Tait no fue desperdiciado: les dio ímpetu a los matemáticos para estudiar esta teoría con más profundidad. En los últimos años, hemos tenido la oportunidad de devolverle el favor a los físicos: la teoría de nudos es útil para modelar ciertas formas exóticas de materia conocidas como fases topológica de la materia. Vamos a explorar un poco esta rama de la topología y su aplicación a la física y computación cuántica.

## Professional development talk:

Navigating Collaboration
Chelsea Walton
The University of Illinois at Urbana-Champaign.

Abstract: I have listened to many, many math talks, and most of the presentations have been on someone’s accomplishments (either the speaker’s or others’). But rarely have I seen a discussion of how these theorems and articles and programs developed. How do mathematical ideas originate and evolve? For collaborative projects, in particular, how do people do ‘groupwork’ in mathematics? In this talk, I aim to pull back the curtains and show that this process isn’t magic, both by sharing some behind-the-scenes stories of my own work, and by providing some concrete tips for working on (collaborative) research projects.

Titles and abstracts - César Galindo