Past

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.

I will exposit some recent joint work with Paul Larson and Grigor Sargsyan that uses higher models of the Axiom of Determinacy---models with nontrivial structure above $\Theta$, the least ordinal which is not the surjective image of the reals---to show that instances of the fundamental problem of inner model theory, the iterability conjecture, consistently fail.

Let $S$ be a set of rational places of odd cardinality containing infinity and a rational prime $p$. We can associate to $S$ a Shimura curve $X$ defined over $\mathbb{Q}$. The Gross--Kohnen--Zagier theorem states that certain generating series of Heegner points of $X$ are modular forms of weight $3/2$ valued in the Jacobian of $X$. We will state this theorem and outline a new approach to proving it using the theory of $p$-adic uniformization and $p$-adic families of modular forms of half-integral weight. This is joint work with Lea Beneish, Henri Darmon, and Lennart Gehrmann.

We extend stochastic optimal transport to path-dependent settings. The problem is to find a semimartingale measure that satisfies general path-dependent constraints, while minimising a cost function on the drift and diffusion coefficients. Duality is established and expressed via non-linear path-dependent partial differential equations (PPDEs). The technique has applications in volatility calibration, including the calibration of path-dependent derivatives, LSV models, and joint SPX-VIX models. It produces a non-parametric volatility model that localises to the features of the derivatives. Another application is in the robust pricing and hedging of American options in continuous time. This is achieved by establishing duality in a space enlarged by the stopping decisions, and showing that the extremal points of martingale measures on the enlarged space are in fact martingale measures on the original space coupled with stopping times.

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $O((mn)^{1-1/(d+1)}+m+n)$. This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers.

In this talk, we discuss a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. This approach can also be used to study point-variety incidences and unit-distance problem in finite fields, giving tight bounds for both problems under a similar non-degeneracy assumption. Joint work with A. Milojevic and I. Tomon.

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.
 

This seminar will introduce 2D-MoSub, a derivative-free optimization method based on the subspace method and quadratic models, specifically tackling large-scale derivative-free problems. 2D-MoSub combines 2-dimensional quadratic interpolation models and trust-region techniques to update the points and explore the 2-dimensional subspace iteratively. Its framework includes constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. Computation details and theoretical properties will be discussed. Numerical results demonstrate the advantage of 2D-MoSub.

Short Bio:
Pengcheng Xie, PhD (Chinese Academy of Sciences), is joining Lawrence Berkeley National Laboratory as a postdoctoral scholar specializing in mathematical optimization and numerical analysis. He has developed optimization methods, including 2D-MoSub and SUSD-TR. Pengcheng has published in major journals and presented at ISMP 2024 (upcoming), ICIAM 2023, and CSIAM 2022. He received the Hua Loo-keng scholarship in 2019 and the CAS-AMSS Presidential scholarship in 2023.
 

Even though most of us tie our shoelaces "wrongly," knots in ropes and filaments have been used as functional structures for millennia, from sailing and climbing to dewing and surgery. However, knowledge of the mechanics of physical knots is largely empirical, and there is much need for physics-based predictive models. Tight knots exhibit highly nonlinear and coupled behavior due to their intricate 3D geometry, large deformations, self-contact, friction, and even elasto-plasticity. Additionally, tight knots do not show separation of the relevant length scales, preventing the use of centerline-based rod models. In this talk, I will present an overview of recent work from our research group, combining precision experiments, Finite Element simulations, and theoretical analyses. First, we study the mechanics of two elastic fibers in frictional contact. Second, we explore several different knotted structures, including the overhand, figure-8, clove-hitch, and bowline knots. These knots serve various functions in practical settings, from shoelaces to climbing and sailing. Lastly, we focus on surgical knots, with a particularly high risk of failure in clinical settings, including complications such as massive bleeding or the unraveling of high-tension closures. Our research reveals a striking and robust power law, with a general exponent, between the mechanical strength of surgical knots, the applied pre-tension, and the number of throws, providing new insights into their operational and safety limits. These findings could have potential applications in the training of surgeons and enhanced control of robotic-assisted surgical devices.

Solovay defined the inner model $L(\mathbb{R}, \mu)$ in the context of $\mathsf{AD}_{\mathbb{R}}$ by using it to define the supercompactness measure $\mu$ on $\mathcal{P}_{\omega_1}(\mathbb{R})$ naturally given by $\mathsf{AD}_{\mathbb{R}}$. Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure $\mu$. Woodin, in his unpublished work, defined $\mu_{\infty}$ which is arguably the ultimate version of the supercompactness measure $\mu$ that Solovay had defined. I will talk about $\mu_{\infty}$ in the context of $\mathsf{AD}^+$ and the axiom $\mathsf{V} = \mathsf{Ultimate\ L}$.

https://woloszyn.org/