Past

to follow

Oxford University Innovation, the University’s commercialisation team, will explain the support they can give to Maths researchers who want to generate commercial impact from their work and expertise. In addition to an overview of consulting, this talk will explain how mathematical techniques and software can be protected and commercialised.

In this possibly speculative talk I will try to outline a way to define analytic D-modules, using (higher) category theory and the ``six operations" on quasicoherent sheaves as the main tools. The aim is to follow the successful approach of Andy Jiang in the algebraic setting, who obtained such a theory without using stacks or formal schemes (as in Gaitsgory-Rozenblyum's approach). By using local cohomology, Jiang was able to avoid enlarging the category of algebras beyond the usual ones. We believe that an analytic variant of local cohomology can be used to recover the Ardakov-Wadsley theory of D-cap modules ``on the nose". (Work in progress).

Kaufman constructed a family of Fourier-decaying measures on the set of badly approximable numbers. Pollington and Velani used these to show that Littlewood’s conjecture holds for a full-dimensional set of pairs of badly approximable numbers. We construct analogous measures that have implications for inhomogeneous diophantine approximation. In joint work with Agamemnon Zafeiropoulos and Evgeniy Zorin, our idea is to shift the continued fraction and Ostrowski expansions simultaneously.

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.

Global scale ocean currents are strongly constrained by the Earth’s rotation, while this effect is generally negligible at small scales. In between, motions with scales from 1-10km are marginally affected by the Earth’s rotation. These intermediate scales, collectively termed the ocean submesoscale, have been hidden from view until recent years. Evidence from field measurements, numerical models, and satellite data have shown that submesoscales play a particularly important role in the upper ocean where they help to control the transport of material between the ocean surface and interior. In this talk I will review some recent work on submesoscale dynamics and their influence on biogeochemistry and accumulation of microplastics in the surface waters.

This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.

For a hyperbolic knot there are two types of invariants, the hyperbolic invariants coming from the geometric structure and the classical invariants coming from the topology or combinatorics. It has been observed in many different cases that these seemingly different types of invariants are in fact related. I will give examples of these relationships and discuss in particular a link by Stoimenow between the determinant and volume.  

We model a tumor as an incompressible flow considering two antagonistic effects: repulsion of cells when the tumor grows (they push each other when they divide) and cell-cell adhesion which creates surface tension. To take into account these two effects, we use a 4th-order parabolic equation: the Cahn-Hilliard equation. The combination of these two effects creates a discontinuity at the boundary of the tumor that we call the pressure jump.  To compute this pressure jump, we include an external force and consider stationary radial solutions of the Cahn-Hilliard equation. We also characterize completely the stationary solutions in the incompressible case, prove the incompressible limit and prove convergence of the parabolic problems to stationary states.

I will give an introduction to quasidiagonality of group actions wherein an action on a C^*-algebra is approximated by actions on matrix algebras.  This has implications for crossed product C^*-algebras, especially as pertains to finite dimensional approximation.  I'll sketch the proof that all isometric actions are quasidiagonal, which we can view as a dynamical Petr-Weyl theorem.  Then I will discuss an interplay between quasidiagonal actions and semiprojectivity of C^*-algebras, a property that allows "almost representations" to be perturbed to honest ones.  

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

Fourth-order elliptic problems arise in a variety of applications from thin plates to phase separation to liquid crystals. A conforming Galerkin discretization requires a finite dimensional subspace of $H^2$, which in turn means that conforming finite element subspaces are $C^1$-continuous. In contrast to standard $H^1$-conforming $C^0$ elements, $C^1$ elements, particularly those of high order, are less understood from a theoretical perspective and are not implemented in many existing finite element codes. In this talk, we address the implementation of the elements. In particular, we present algorithms that compute $C^1$ finite element approximations to fourth-order elliptic problems and which only require elements with at most $C^0$-continuity. We also discuss solvers for the resulting subproblems and illustrate the method on a number of representative test problems.

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. Rödl and Ruciński, and Friedgut, Rödl and Schacht proved a random version of Rado’s theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$.

In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold for $S_{n,p}$ being $(A,r)$-Rado.

Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

This is joint work with Andrea Freschi and Andrew Treglown (both University of Birmingham).

We present new high-order finite elements discretizing the $L^2$ de Rham complex on triangular and tetrahedral meshes. The finite elements discretize the same spaces as usual, but with different basis functions. They allow for fast linear solvers based on static condensation and space decomposition methods.

The new elements build upon the definition of degrees of freedom given by (Demkowicz et al., De Rham diagram for $hp$ finite element spaces. Comput.~Math.~Appl., 39(7-8):29--38, 2000.), and consist of integral moments on a symmetric reference simplex with respect to a numerically computed polynomial basis that is orthogonal in both the $L^2$- and $H(\mathrm{d})$-inner products ($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\}$).

On the reference symmetric simplex, the resulting stiffness matrix has diagonal interior block, and does not couple together the interior and interface degrees of freedom. Thus, on the reference simplex, the Schur complement resulting from elimination of interior degrees of freedom is simply the interface block itself.

This sparsity is not preserved on arbitrary cells mapped from the reference cell. Nevertheless, the interior-interface coupling is weak because it is only induced by the geometric transformation. We devise a preconditioning strategy by neglecting the interior-interface coupling. We precondition the interface Schur complement with the interface block, and simply apply point-Jacobi to precondition the interior block.

The combination of this approach with a space decomposition method on small subdomains constructed around vertices, edges, and faces allows us to efficiently solve the canonical Riesz maps in $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$, at very high order. We empirically demonstrate iteration counts that are robust with respect to the polynomial degree.

I will then explain that hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will explain some of our recent results as hybrid quantum algorithms that exploit so-called classical shadows (random unitary protocols) in order to extract and post-process a large amount of information from the quantum computer [PRX 12, 041022 (2022)] and [arXiv:2212.11036]. I will finally identify the most likely areas where quantum computers may deliver a true advantage in the near term.

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov-Hausdorff distance, by finite dimensional Banach spaces. Time permitting, we will discuss recent joint work with Hyde and Schul that provides quantitative analogues of this statement.
 

TBC

Bicommutant categories, initially invented for the purposes of Chern-Simons theory and 2d CFT, seem to also appear in other domains of math with examples related to group theory, and dynamical systems.

One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.

Based on works with Andrew Ledvina and with Emre Parmaksiz.

Learning to process and analyze the raw weight matrices of neural networks is an emerging research area with intriguing potential applications like editing and analyzing Implicit Neural Representations (INRs), weight pruning/quantization, and function editing. However, weight spaces have inherent permutation symmetries – permutations can be applied to the weights of an architecture, yielding new weights that represent the same function. As with other data types like graphs and point clouds, these symmetries make learning in weight spaces challenging.

This talk will overview recent advances in designing architectures that can effectively operate on weight spaces while respecting their underlying symmetries. First, we will discuss our ICML 2023 paper which introduces novel equivariant architectures for learning on multilayer perceptron weight spaces. We first characterize all linear equivariant layers for their symmetries and then construct networks composed of these layers. We then turn to our ICLR 2024 work, which generalizes the approach to diverse network architectures using what we term Graph Metanetworks (GMN). This is done by representing input networks as graphs and processing them with graph neural networks. We show the resulting metanetworks are expressive and equivariant to weight space symmetries of the architecture being processed. Our graph metanetworks are applicable to CNNs, attention layers, normalization layers, and more. Together, these works make promising steps toward versatile and principled architectures for weight-space learning.

Join us for a session with Keith Macksoud, Executive Director at global recruitment consultant Options Group in London and who previously has > 20 years’ experience in Prime Brokerage Sales at Morgan Stanley, Citi, and Deutsche Bank.  Keith will discuss the recruitment process for financial institutions, and how to increase your chances of a successful application. 

Keith will detail his finance background in Prime Brokerage and provide students with an exclusive look behind the scenes of executive search and strategic consulting firm Options Group. We will look at what Options Group does, how executive search firms work and the Firm’s 30-year track record of placing individuals at many of the industries’ largest and most successful global investment banks, investment managers and other financial services-related organisations. 

About Options Group

Options Group is a leading global executive search and strategic consulting firm specializing in financial services including capital markets, global markets, alternative investments, hedge funds, and private banking/wealth management.

https://www.optionsgroup.com/

In this talk I will describe a new model for time-varying graphs (TVGs) based on persistent topology and cosheaves. In its simplest form, this model presents TVGs as matrices with entries in the semi-ring of subsets of time; applying the classic Kleene star construction yields novel summary statistics for space networks (such as STARLINK) called "lifetime curves." In its more complex form, this model leads to a natural featurization and discrimination of certain Earth-Moon-Mars communication scenarios using zig-zag persistent homology. Finally, and if time allows, I will describe recent work with David Spivak and NASA, which provides a complete description of delay tolerant networking (DTN) in terms of an enriched double category.

Malignant transformation of somatic tissues is an evolutionary process, driven by selection for oncogenic mutations. Understanding when these mutations occur, and how fast mutant cell clones expand can improve diagnostic schemes and therapeutic intervention. However, clonal dynamics are not directly accessible in humans, posing a need for inference approaches to reconstruct the division history in normal and malignant cell clones, and to predict their future evolution. Inspired from population genetics theory, we develop mathematical models to detect imprints of clonal selection in the variant allele frequency distribution measured in a single tissue sample of a homeostatic tissue. I will present the theoretical basis of our approach and inference results for the tissue dynamics in physiological and clonal hematopoiesis, obtained from variant allele frequencies measured by snapshot bulk whole genome sequencing of human bone marrow samples.

The Monster group M is the largest sporadic simple group. It is also the group of automorphisms of 196, 884-dimensional Fischer-Norton-Griess algebra V_M. In 2009, A. A. Ivanov offered an axiomatic approach to studying the structure of V_M by introducing the notions of Majorana algebra and Majorana representation. Later, the theory developed, and Majorana representations of several groups were constructed. Our talk is dedicated to the existence of standard Majorana representations of 3-transposition groups for the Fischer list. The main result is that the groups from the Fischer list which admit a standard Majorana representation can be embedded into the Monster group.

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP1 theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

Diffusion models, which transform noise into new data instances by reversing a Markov diffusion process, have become a cornerstone in modern generative models. A key component of these models is to learn the score function through score matching. While the practical power of diffusion models has now been widely recognized, the theoretical developments remain far from mature. Notably, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. In this talk, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models and the accuracy of score estimation. Our analysis covers both the optimization and the generalization aspects of the learning procedure, which also builds a novel connection to supervised learning and neural tangent kernels.

This is based on joint work with Yinbin Han and Meisam Razaviyayn (USC).

A recent conjecture proposed by Harris and Venkatesh relates the action of derived Hecke operators on the space of weight one modular forms to certain Stark units. In this talk, I will explain how this can be rephrased as a conjecture about "tame" analogues of triple product periods for a triple of mod p eigenforms of weights (2,1,1). I will then present an elliptic counterpart to this conjecture relating a tame triple product period to a regulator for global points of elliptic curves in rank 2. This conjecture can be proved in some special cases for CM weight 1 forms, with techniques resonating with the so-called Jochnowitz congruences. This is joint work in preparation with Henri Darmon. 

To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. 

Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin. 

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.

We give a strongly polynomial algorithm for minimum cost generalized flow, and as a consequence, for all linear programs with at most two nonzero entries per row, or at most two nonzero entries per column. While strongly polynomial algorithms for the primal and dual feasibility problems have been known for a long time, various combinatorial approaches used for those problems did not seem to carry over to the minimum-cost variant.

Our approach is to show that the ‘subspace layered least squares’ interior point method, an earlier joint work with Allamigeon, Dadush, Loho and Natura requires only a strongly polynomial number of iterations for minimum cost generalized flow. We achieve this by bounding the straight line complexity, introduced in the same paper. The talk will give an overview of the interior point method as well as the combinatorial straight-line complexity analysis for the particular setting. This is joint work with Daniel Dadush, Zhuan Khye Koh, Bento Natura, and Neil Olver.

Let q be a prime power and let C be a smooth curve defined over F_q. The number of points of C over the finite extensions of F_q are determined by the Zeta function of C, which can be written in the form P_C(t)/((1-t)(1-qt)), where P_C(t) is a polynomial of degree 2g and g is the genus of C; this is often called the L-polynomial of C. We use a Chebotarev-like statement (over function fields instead of Z) due to Katz in order to study the distribution, as C varies, of the coefficients of P_C(t) in a non-archimedean setting.

In the 1960s Shanks conjectured that the  ζ'(ρ), where ρ is a non-trivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so far failed. Building on the work of Keating and Snaith using characteristic polynomials from Random Matrix Theory, the Hybrid model of Gonek, Hughes and Keating, and the Ratios Conjecture of Conrey, Farmer, and Zirnbauer, we have been able to produce new conjectures for the full asymptotics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.

The (finite) Hilbert matrix is arguably one of the single most well-known matrices in mathematics. The infinite Hilbert matrix H was introduced by David Hilbert around 120 years ago in connection with his double series theorem. It can be interpreted as a linear operator on spaces of analytic functions by its action on their Taylor coefficients. The boundedness of H on the Hardy spaces H^p for 1 < p < ∞ and Bergman spaces A^p for 2 < p < ∞ was established by Diamantopoulos and Siskakis. The exact value of the operator norm of H acting on the Bergman spaces A^p for 4 ≤ p < ∞ was shown to be π /sin(2π/p) by Dostanic, Jevtic and Vukotic in 2008. The case 2 < p < 4 was an open problem until in 2018 it was shown by Bozin and Karapetrovic that the norm has the same value also on the scale2 < p < 4. In this talk, we introduce some background, review some of the old results, and consider the still partly open problem regarding the value of the norm on weighted Bergman spaces. We also consider a generalised Hilbert matrix operator and its (essential) norm. The talk is partly based on a joint work with Mikael Lindström, David Norrbo, and Niklas Wikman (Åbo Akademi University).
 

Let C,D be classes of finitely presented groups. The epimorphism problem from C to D is the following decision problem:

Input: Finite descriptions (presentation, multiplication table, other) for groups  G in C and H in D

Question: Is there an epimorphism from G to H?

I will discuss some cases where it is decidable and where it is NP-complete. Spoiler alert: it is undecidable for C=D=the class of 2-step nilpotent groups (Remeslennikov).

This is joint work with Jerry Shen (UTS) and Armin Weiss (Stuttgart).

For integers $k \ge 1$ and $n \ge 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph $K(5,2)$. This problem received considerable attention in the literature, including a recent solution for the sparsest case $n=2k+1$. The main contribution of our work is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph $J(n,k,s)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two sets whose intersection has size exactly $s$. Clearly, we have $K(n,k)=J(n,k,0)$, i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász' conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.

This is joint work with my students Arturo Merino (TU Berlin) and Namrata (Warwick).

Recent joint work with Konstantin Ardakov has been devoted to classifying equivariant line bundles with flat connection on the Drinfeld p-adic half-plane defined over F, a finite extension of Q_p, and proving that their global sections yield admissible locally analytic representations of GL_2(F) of finite length. In this talk we will discuss this work and invite reflection on how it might be extended to equivariant vector bundles with connection on the p-adic half-plane and, if time permits, to higher dimensional analogues of the half-plane.

We study a McKean–Vlasov control problem with killing and common noise. The particles in this control model live on the real line and are killed at a positive intensity whenever they are in the negative half-line. Accordingly, the interaction between particles occurs through the subprobability distribution of the living particles. We establish the existence of an optimal semiclosed-loop control that only depends on the particles’ location and not their cumulative intensity. This problem cannot be addressed through classical mimicking arguments, because the particles’ subprobability distribution cannot be reconstructed from their location alone. Instead, we represent optimal controls in terms of the solutions to semilinear BSPDEs and show those solutions do not depend on the intensity variable.

We consider one-dimensional nonlinear Schrodinger equations around a traveling wave. We prove its asymptotic stability for general nonlinearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show long-range scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman and, and of Krieger and Schlag, and on precise renormalizations, computations, and estimates of space-time resonances to handle its interaction with the soliton. This is joint work with Pierre Germain.

The Liouville function $\lambda(n)$ is defined to be $+1$ if $n$ is a product of an even number of primes, and $-1$ otherwise. The statistical behaviour of $\lambda$ is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of $+1$'s and $-1$'s. For example, the two-point Chowla conjecture predicts that the average of $\lambda(n)\lambda(n+1)$ over $n < x$ tends to zero as $x$ goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.

Fluctuating hydrodynamics provides a framework for approximating density fluctuations in interacting particle systems by suitable SPDEs. The Dean-Kawasaki equation - a strongly singular SPDE - is perhaps the most basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N diffusing weakly interacting particles in the regime of large particle numbers N. The strongly singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by approaches like regularity structures, it has recently been shown to not even admit nontrivial martingale solutions.

In this talk, we give an overview of recent quantitative results for the justification of fluctuating hydrodynamics models. In particular, we give an interpretation of the Dean-Kawasaki equation as a "recipe" for accurate and efficient numerical simulations of the density fluctuations for weakly interacting diffusing particles, allowing for an error that is of arbitarily high order in the inverse particle number. 

Based on joint works with Federico Cornalba, Jonas Ingmanns, and Claudia Raithel

In this joint work with Filip Zivanovic, we construct symplectic cohomology for a class of symplectic manifolds that admit ${\mathbb C}^*$-actions and which project equivariantly and properly to a convex symplectic manifold. The motivation for studying these is a large class of examples known as Conical Symplectic Resolutions, which includes quiver varieties, resolutions of Slodowy varieties, and hypertoric varieties. These spaces are highly non-exact at infinity, so along the way we develop foundational results to be able to apply Floer theory. Motivated by joint work with Mark McLean on the Cohomological McKay Correspondence, our goal is to describe the ordinary cohomology of the resolution in terms of a Morse-Bott spectral sequence for positive symplectic cohomology. These spectral sequences turn out to be quite computable in many examples. We obtain a filtration on ordinary cohomology by cup-product ideals, and interestingly the filtration can be dependent on the choice of circle action.

The COVID-19 pandemic has brought a spotlight to the field of infectious disease modelling, prompting widespread public awareness and understanding of its intricacies. As a result, many individuals now possess a basic familiarity with the principles and methodologies involved in studying the spread of diseases. In this presentation, I aim to deliver a somewhat comprehensive (and hopefully engaging) overview of the methods employed in infectious disease modelling, placing them within the broader context of their significance for government and public health policy.

I will navigate through applications of Spatial Statistics, Branching Processes, and Binary Trees in modelling infectious diseases, with a particular emphasis on integrating machine learning methods into these areas. The goal of this presentation is to take you on a broad tour of methods and their applications, offering a personal perspective by highlighting examples from my recent work.