Past

We discuss the trace problem and stable rank for tracialy complete C*-algebras

In the category of operator systems, identification comes via complete order isomorphisms, and so an operator system can be identified with a C*-algebra without itself being an algebra. So, when is an operator system a C*-algebra? This question has floated around the community for some time. From Choi and Effros, we know that injectivity is sufficient, but certainly not necessary outside of the finite-dimensional setting. In this talk, I will give a characterization in the separable nuclear setting coming from C*-encoding systems. This comes from joint work with Galke, van Lujik, and Stottmeister.

The "curse of dimensionality" refers to the host of difficulties that occur when we attempt to extend our intuition about what happens in low dimensions (i.e. when there are only a few features or variables)  to very high dimensions (when there are hundreds or thousands of features, such as in genomics or imaging).  With very high-dimensional data, there is often an intuition that although the data is nominally very high dimensional, it is typically concentrated around a much lower dimensional, although non-linear set. There are many approaches to identifying and representing these subsets.  We will discuss topological approaches, which represent non-linear sets with graphs and simplicial complexes, and permit the "measuring of the shape of the data" as a tool for identifying useful lower dimensional representations.

Since Gromov's celebrated polynomial growth theorem, the understanding of nilpotent groups has become a cornerstone of geometric group theory. An interesting aspect is the conjectural quasiisometry classification of nilpotent groups. One important quasiisometry invariant that plays a significant role in the pursuit of classifying these groups is the Dehn function, which quantifies the solvability of the world problem of a finitely presented group. Notably, Gersten, Holt, and Riley's work established that the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$.  

In this talk, I will explain recent results that allow us to compute Dehn functions for extensive families of nilpotent groups arising as central products. Consequently, we obtain a large collection of pairs of nilpotent groups with bilipschitz equivalent asymptotic cones but with different Dehn functions.

This talk is based on joint work with Claudio Llosa Isenrich and Gabriel Pallier.

Since Gromov's celebrated polynomial growth theorem, the understanding of nilpotent groups has become a cornerstone of geometric group theory. An interesting aspect is the conjectural quasiisometry classification of nilpotent groups. One important quasiisometry invariant that plays a significant role in the pursuit of classifying these groups is the Dehn function, which quantifies the solvability of the world problem of a finitely presented group. Notably, Gersten, Holt, and Riley's work established that the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$.  

In this talk, I will explain recent results that allow us to compute Dehn functions for extensive families of nilpotent groups arising as central products. Consequently, we obtain a large collection of pairs of nilpotent groups with bilipschitz equivalent asymptotic cones but with different Dehn functions.

This talk is based on joint work with Claudio Llosa Isenrich and Gabriel Pallier.

The Cuntz semigroup of a C*-algebra is an ordered monoid consisting of equivalence classes of positive elements in the stabilization of the algebra.  It can be thought of as a generalization of the Murray-von Neumann semigroup, and records substantial information about the structure of the algebra.  Here we examine the set of positive elements having a fixed equivalence class in the Cuntz semigroup of a simple, separable, exact and Z-stable C*-algebra and show that this set is path connected when the class is non-compact, i.e., does not correspond to the class of a projection in the C*-algebra.  This generalizes a known result from the setting of real rank zero C*-algebras.

The classification of simple, unital, nuclear UCT C*-algebras with finite nuclear dimension can be achieved using an invariant derived from K-theory and tracial information. In this talk, I will present a classification theorem for certain classes of C*-algebras that rely solely on tracial deformations.  

 Structural properties of C*-Algebras such as Stable Rank One, Real Rank Zero, and radius of comparison have played an important role in classification.  Crossed product C*-Algebras are useful examples to study because knowledge of the base Algebra can be leveraged to determine properties of the crossed product.  In this talk we will discuss the permanence of various structural properties when taking crossed products of several types.  Crossed products considered will include the usual C* crossed product by a group action along with generalizations such as crossed products by a partial automorphism.  

This talk is based on joint work with Julian Buck and N. Christopher Phillips and on joint work with Maria Stella Adamo, Marzieh Forough, Magdalena Georgescu, Ja A Jeong, Karen Strung, and Maria Grazia Viola.

In recent work, Aganagic proposed a categorification of quantum link invariants based on a category of A-branes. The theory is a generalization of Heegaard–Floer theory from gl(1|1) to arbitrary Lie algebras. It turns out that this theory is solvable explicitly and can be used to compute homological link invariants associated to any minuscule representation of a simple Lie algebra. This invariant coincides with Khovanov–Rozansky homology for type A and gives a new invariant for other types. In this talk, I will introduce the relevant category of A-branes, explain the explicit algorithm used to compute the link invariants, and give a sketch of the proof of invariance. This talk is based on 2305.13480 with Mina Aganagic and Miroslav Rapcak and work in progress with Mina Aganagic and Ivan Danilenko.

The Zappa–Szép (ZS) product of two groupoids is a generalization of the semi-direct product: instead of encoding one groupoid action by homomorphisms, the ZS product groupoid encodes two (non-homomorphic, but “compatible”) actions of the groupoids on each other. I will show how to construct the ZS product of two twists over such groupoids and give an example using Weyl twists from Cartan pairs arising from Kumjian--Renault theory.

 Based on joint work with Boyu Li, New Mexico State University

The set $M_n(\mathbb{R})$ of all $n \times n$ matrices over the real numbers is an example of an algebraic structure called a $C^*$-algebra. The subalgebra $D$ of diagonal matrices has special properties and is called a \emph{Cartan subalgebra} of $M_n(\mathbb{R})$. Given an arbitrary $C^*$-algebra, it can be very hard (but also very rewarding) to find a Cartan subalgebra, if one exists at all. However, if the $C^*$-algebra is generated by a cocycle $c$ and a group (or groupoid) $G$, then it is natural to look within $G$ for a subgroup (or subgroupoid) $S$ that may give rise to a Cartan subalgebra. In this talk, we identify sufficient conditions on $S$ and $c$ so that the subalgebra generated by $(S,c)$ is indeed a Cartan subalgebra of the $C^*$-algebra generated by $(G,c)$. We then apply our theorem to $C^*$-algebras generated by $k$-graphs, which are directed graphs in higher dimensions. This is joint work with J. Briones Torres, A. Duwenig, L. Gallagher, E. Gillaspy, S. Reznikoff, H. Vu, and S. Wright.

A self-similar k-graph is a pair consisting of a (discrete countable) group and a k-graph, such that the group acts on the k-graph self-similarly. For such a pair, one can associate it with a universal C*-algebra, called the self-similar k-graph C*-algebra. This class of C*-algebras embraces many important and interesting C*-algebras,  such as the higher rank graph C*-algebras of Kumjian-Pask, the Katsura algebras,  the Nekrashevych algebras constructed from self-similar groups, and the Exel-Pardo algebra. 

In this talk, we will survey some results on self-similar k-graph C*-algebras. 

Dimensions of mapping class groups of orientable and non-orientable surfaces

1:30pm

Luis Jorge Sánchez Saldaña (UNAM)

Mapping class groups have been studied extensively for several decades. Still in these days these groups keep being studied from several point of views. In this talk I will talk about several notions of dimension that have been computed (and some that are not yet known) for mapping class groups of both orientable and non-orientable manifolds. Among the dimensions that I will mention are the virtual cohomological dimension, the proper geometric dimension, the virtually cyclic dimension and the virtually abelian dimension. Some of the results presented are in collaboration with several colleagues: Trujillo-Negrete, Hidber, León Álvarez and Jimaénez Rolland.

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Diffeomorphisms of reducible 3-manifolds

2:45pm

Rachael Boyd (Glasgow)

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

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Nonunique ergodicity in strata of geodesic laminations and the boundary of Outer space

4:00pm

Mladen Bestvina (Utah)

It follows from the work of Gabai and Lenzhen-Masur that the maximal number of projectively distinct ergodic transverse measures on a filling geodesic lamination on a hyperbolic surface is equal to the number of curves in a pants decomposition. In a joint work with Jon Chaika and Sebastian Hensel, we answer the analogous question when the lamination is restricted to have specified polygons as complementary components. If there is enough time, I will also talk about the joint work with Elizabeth Field and Sanghoon Kwak where we consider the question of the maximal number of projectively distinct ergodic length functions on a given arational tree on the boundary of Culler-Vogtmann's Outer space of a free group.
 

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

In this talk, I will describe some of the ways in which mathematical modelling contributed to the Covid-19 pandemic response in New Zealand. New Zealand adopted an elimination strategy at the beginning of the pandemic and used a combination of public health measures and border restrictions to keep incidence of Covid-19 low until high vaccination rates were achieved. The low or zero prevalence for first 18 months of the pandemic called for a different set of modelling tools compared to high-prevalence settings. It also generated some unique data that can give valuable insights into epidemiological characteristics and dynamics. As well as describing some of the modelling approaches used, I will reflect on the value modelling can add to decision making and some of the challenges and opportunities in working with stakeholders in government and public health.        

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

To neural activity one may associate a space of correlations and a space of population vectors. These can provide complementary information. Assume the goal is to infer properties of a covariate space, represented by ochestrated activity of the recorded neurons. Then the correlation space is better suited if multiple neural modules are present, while the population vector space is preferable if neurons have non-convex receptive fields. In this talk I will explain how to coherently combine both pieces of information in a bifiltration using Dowker complexes and their total weights. The construction motivates an interesting extension of Dowker’s duality theorem to simplicial categories associated with two composable relations, I will explain the basic idea behind it’s proof.

In this presentation, we will review how the field of Mechanics of Materials is generally framed and see how it can benefit from and be of benefit to the current progress in AI. We will approach this problematic in the particular context of Brain mechanics with an application to traumatic brain injury in police investigations. Finally we will briefly show how our group is currently applying the same methodology to a range of engineering challenges.

It is well-known that the set of irreducible (finite-dimensional) representations of a semisimiple complex Lie algebra g can be indexed by the dominant weights. The Borel-Weil theorem asserts that they can be seen geometrically as the global sections of line bundles over the flag variety. The Borel-Weil-Bott theorem computes the higher sheaf cohomology groups. There are several ways to prove the Borel-Weil-Bott theorem, which we will discuss. The classical idea is to study how the Casimir operator acts on the sheaf of sections of line bundles. Instead of this, the geometric idea is trying to compute the Doubeault cohomology, transferring the sheaf cohomology to the Lie algebra cohomology. The algebraic idea is to realize that the sheaf cohomology group can be computed by the derived functor of the induction, by using the Peter-Weyl the Borel-Weil theorem can be shown immediately.