Four years ago, I entered the world of remote work, marking a significant shift in my professional life. It all began as a response to the COVID-19 pandemic while I was working on my master’s thesis from home. Little did I know that this initial adaptation would transform into a lifestyle.
In this journey, I found myself exploring new routines, seeking motivation, and discovering ways to collaborate in this evolving work dynamic. Along the way, I stumbled upon a few realizations.
For instance, beyond the unexpected expenditure of my excessive coffee consumption habits over these years, the years spent working from home felt short and somewhat blurry. The virtual environment, with its inherent monotony, has the peculiar ability to distort our perception of time over weeks.
In response to this realization, I proactively introduced measures to break the monotony, such as scheduling in-person social interactions two to three times a week. This deliberate effort helped disrupt the relentless pace and inject a sense of variety into my daily routine.
Reflecting on these changes, I decided to write this post about some of the lessons I have learned along the way while working remotely from home over the past four years.
In the world of remote work, making sure others see and recognize your efforts takes some thoughtful steps. I learned that staying visible in remote work involves using digital tools, being active in virtual meetings, and keeping everyone updated on your work. Since we’re not physically present, it’s essential to proactively communicate through digital tools like Slack and emails. Regularly updating colleagues and managers about what you’re working on helps them stay in the loop.
Being part of virtual meetings is crucial for staying visible. It’s not just about showing up but actively participating— turning the camera on, sharing ideas and contributing to discussions. This not only shows your dedication but also strengthens connections within the remote team. These interactions provide insights into everyone’s work, creating a sense of unity and progress.
Alongside digital communication and virtual meetings, keeping everyone posted on your project’s progress is vital. Clearly communicating milestones, challenges, and achievements ensures everyone is aware of what’s happening. This transparency not only makes you more visible but also contributes to a well-informed and connected remote work environment.
Making sure work doesn’t creep into personal time (and vice versa) is essential for a balanced remote work life. It’s like putting up invisible walls to avoid unnecessary interruptions. Here’s how I am trying to do it:
Define Work Hours: Being clear about when you’re on the clock and when you’re off. Set specific work hours and stick to them. This helps create a routine and ensures you have dedicated time for work.
Create Dedicated Workspace: Having a specific area for work helps mentally separate your job from your personal life. It could be a desk, a corner of a room, or any spot where you ‘go to work.’ When you’re there, it’s work time.
Communicate Your Availability: Let the team know if you are off for a couple of hours. Let your colleagues (and family) know when you’re available and when you need focused time. This way, they’ll understand when it’s okay to interrupt and when you need uninterrupted work.
Setting boundaries boils down to creating a clear distinction between your work hours and personal time. It’s about ensuring you have the space and time you need to be productive while still enjoying a healthy work-life balance.
I realised that building strong connections within a virtual team takes intentional effort. Nurturing collaborative relationships is all about creating a friendly and supportive team atmosphere. Regular interactions, both work-related and casual, contribute to better teamwork and make your virtual work experience more enjoyable.
Just because you’re not in the same office doesn’t mean you can’t have coffee breaks together. Schedule a virtual coffee chat with a colleague. It’s a casual way to catch up and build a friendly connection.
You can also take the time for informal check-ins. It’s not always about work. Ask your colleagues how they’re doing or share a bit about your day. These small conversations go a long way in building camaraderie.
Juggling tasks effectively becomes even more crucial when working remotely.
Task Breakdowns: When you’re not in a physical office, it’s essential to know and communicate what needs your attention first. Prioritizing tasks helps you focus on what matters most, ensuring you stay on track.
Remote work often involves tackling projects more independently. Breaking down bigger tasks into smaller, more manageable parts makes the workload less daunting.
In the absence of face-to-face collaboration, project management tools become even more important. They help communicate, keep everything organized, from to-do lists to project timelines, ensuring everyone is on the same page.
Setting Realistic Expectations: Remote work allows for autonomy, but it’s vital to understand your own capabilities. Setting realistic expectations involves recognizing what you can achieve within a given timeframe and aligning your goals accordingly, and the ability to communicate it well. Setting expectations is a two-way street and mismatched expectations is even more dangerous while working remotely. Communicating transparently with stakeholders, whether it’s your team or clients, ensures everyone is on the same page.
One of the perks of remote work is the ability to set your own hours. Finding the hours when you’re most productive contributes to better work outcomes.
Leveraging Asynchronous Communication: Remote work transcends traditional office hours, making asynchronous communication a powerful tool. Embrace the freedom to communicate on your own time, allowing for thoughtful responses and reducing the pressure of immediate replies.
Finding the right balance between structure and adaptability is the key to a successful remote work experience. Embracing flexibility doesn’t mean sacrificing organization; rather, it’s about creating a work environment that aligns with the team and your lifestyle - allowing for seamless adjustments as needed.
Navigating Time Zones and Global Collaboration: For those engaged in global collaboration, navigating different time zones becomes an integral part of the balancing act. Setting clear expectations regarding response times, and fostering a global mindset contribute to seamless collaboration across borders.
I’ve come to realize the vital role social interactions play in maintaining good mental health, especially when faced with the challenges of working remotely. Despite the inherently solitary nature of working from home, I’ve learned the value of actively pursuing social connections beyond the confines of work hours.
The lack of a commute and the comfort of my home office provided me with the flexibility to make spontaneous plans and engage in social activities that might not have been feasible otherwise. This newfound freedom extends to exploring and working from new cities, as well as enjoying more quality time with friends and family.
While remote work fosters a social post-work routine, it’s essential to strike a balance. Setting boundaries ensures that work commitments don’t encroach on personal time, and vice versa. This balance allows me to enjoy the best of both worlds—productive workdays and vibrant social interactions.
Remote work, with all its conveniences, still falls short in replicating the vibrant atmosphere of traditional office interactions in some situations. The impromptu chats, coffee break banter, and chance encounters with colleagues contribute to a unique sense of camaraderie that virtual platforms struggle to fully emulate. Admittedly, I find myself missing these moments from time to time.
Substituting with Virtual Connection: While virtual platforms can’t entirely replace face-to-face interactions, there are creative ways to foster a sense of connection in the remote landscape. Scheduled virtual coffee breaks, casual team check-ins, or dedicated channels for non-work-related conversations can provide a space for informal bonding.
Team Onsites: To address the yearning for office interactions team onsites provide valuable opportunities for face-to-face connections, team-building activities, and the spontaneous interactions that make traditional office life so memorable.
As the work landscape continues to evolve, adaptability remains crucial. Looking forward to the future of remote work, there are several aspects I aim to incorporate into my routine. One key focus is having a more structured physical activity routine, and actively being involved in a in-person hobby group.
Looking forward!
]]>Spectral geometry may sound like an abstract concept reserved for mathematicians, but its potential applications in various fields make it a fascinating topic for scientists and technology experts alike. At its core, spectral geometry explores the relationship between a manifold’s geometry and its spectrum - an idea that has found relevance in classical and quantum mechanics, analysis, and even the study of the universe itself. As a computer scientist, I became particularly intrigued by how these concepts could be extended to shape analysis applications, such as correspondence matching, style transfer, and interpolation. To dive deeper into this intriguing field, let’s first explore the basic concepts of a Riemannian manifold and the Laplace-Beltrami operator.
Flat earthers are right. The surface of the earth is a 2D manifold. Every region around it looks like a region of the Euclidean plane.
A manifold is defined as a topological space that locally resembles Euclidean space near each point. This notion allows complicated systems to be explained in simpler terms of local topological properties of Euclidean space.
In computer vision, the notion a manifold is used quite frequently – in rotation averaging \( SO3 \), structure and motion (“Essential” manifold), to capture the shape of an object (“Shape” manifolds), to model a set of images (“Grassman” manifolds) or to simply represent a sphere \( S^n \).
A “Hilbert” space is an abstract vector space with an inner product which has to be complete under convergence (i.e. to allow calculus to be used). This generalises the notion of Euclidean space by extending the methods of vector algebra and calculus from 2D Euclidean plane and 3D spaces to higher dimensions.
Thus, we can define norms in Hilbert space to allow us to measure distances, and inner products to allow us to measure angles.
On a Riemannian manifold, you get to measure lengths of the curves. A Riemannian manifold is a manifold with an inner product defined in the tangent space at each point. Riemannian geometry is a field by itself.
A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, length of a curve, area of a surface and higher-dimensional analogues (volume, etc.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself. It also allows us to define “geodesic distance” on the manifold.
Geodesics are locally shortest curves. They preserve a direction on a surface and have many interesting properties. In a plane, the geodesics are straight lines. On a sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration. Again, metric geometry is a field by itself.
Equivalently in other areas, it can be defined as a path that a particle which is not accelerating would follow.
In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true.
Without going into too much details on metrics and (pseudo-) Riemannian manifolds, an isometry of a manifold can be defined in simpler terms as any (smooth) mapping of that manifold (into itself, or into another manifold) that preserves the notion of distance between points.
When such a mapping is on smooth manifolds (and is a diffeomorphism: the mapping is a bijection and its inverse is differentiable), it is called isometry, and provides a notion of isomorphism (“sameness”).
Terminology : Let \( M \) be a smooth manifold. Denote the tangent space at \( x \in M \) by \( T_x{M} \).
If \( f: M \rightarrow N \) is a smooth map between smooth manifolds, denote the associated map on \( T_x{M} \) by \( (Df)_x : T_x{M} \rightarrow T_fN \). If \( I \) is an open interval in \( \mathbb{R} \) and \( \alpha : I \rightarrow M \) is a smooth path, then for \( t \in I, \alpha (t) \) denotes, \( (D \alpha )_t (I) \in T_aM \).
Definition : A Riemannian metric on a smooth manifold \( M \) is a choice at each point \( x \in M \) of a positive definite inner product \( , \) on \( T_xM \), the inner products varying smoothly with \( x \). Then \( M \) is known as a Riemannian manifold.
A local isometry between two Riemannian manifolds \( M \) and \( N \) is a local diffeomorphism \( h : M \rightarrow N \), such that, for all points \( x \in M \) and all vectors \( v \) and \( w \) in \( T_xM \):
\( v, w = (Dh)_x(v), (Dh)_x(w) \).
A (Riemannian) isometry is a local isometry that is also a diffeomorphism.
Now, that we have a basic idea about the Riemannian manifolds, let’s have a very quick look at the Laplace-Beltrami operator.
The Laplace operator is a fundamental concept in calculus, given by the divergence of the gradient of a function on Euclidean space. The laplacian of a function \( f \) at point \( p \) is the rate at which the average value of \( f \) over spheres centered at \( p \) deviates from \( f(p) \) as the radius of the sphere shrinks towards 0.
Initially introduced in the study of celestial mechanics, solutions of the equation Δf=0, now called Laplace’s equation, are the so-called harmonic functions and represent the possible gravitational fields in regions of vacuum.
The Laplacian generalized to the Riemannian manifold \( (M,g) \) by the Laplace-Beltrami operator (△g).
Equivalently other areas (esp. in diffusion processes) utilize this idea - Fluid mechanics (the Navier-stokes equation), potential theory (Poisson equation), heat diffusion (heat equation), wave equation, quantum physics (Schrodinger equation) and so on.
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators.
Given a compact Riemannian manifold, we can associate to it the (linear unbounded) Laplace-Beltrami operator. This operator is self-adjoint and its spectrum is discrete : namely the spectrum consists of a increasing sequence of real eigenvalues with finite multiplicity.
Two closed Riemannian manifolds are said to be isospectral if the eigenvalues of their Laplace–Beltrami operator (Laplacians), counted multiplicities, coincide.
Thus, spectral geometry is the connection between the spectrum \( Spec(M,g) \) and the geometry of the manifold \( (M,g) \) . This fundamentally deals with two kinds of problems:
A problem that arises a lot in physics, the analysis of PDEs, probabilty etc is to compute the spectrum of Laplacian (or other operators). The main idea is to find a lower bound estimate on the eigenvalues of the spectrum on a Riemannian manifold. This is we,
compute (exactly or not) the spectrum \( Spec(M,g) \)? And (or) find properties on the spectrum \( Spec(M,g) \)
Direct problems attempt to infer the behavior of the eigenvalues of a Riemannian manifold from knowledge of the geometry.
The problem we’ll look more into detail in the next post, is the inverse problem. One of fundamental problems in spectral geometry is to ask to what extent the eigenvalues determine the geometry of a given manifold.
If the notion that a Riemannian invariant is true: if two Riemannian manifolds \( (M,g) \) and \( (M’,g’) \) are isometric, then they are isospectral i.e., \( Spec(M,g) == Spec(M’,g’) \).
which geometric information of the manifold can we determine from the spectrum? does the data of the spectrum \( Spec(M,g) \) determine the “shape” of the manifold \( (M,g) \)?
This question of isospectrality (spectral alignment) in Riemannian geometry may be traced back to H. Weyl in 1911–1912 and became popularized thanks to M. Kac’s article of 1966. The famous sentence of Kac “Can one hear the shape of a drum?” refers to this type of isospectral problem.
Spoiler: The answer is “theoretically” false. There exists isospectral manifolds that are nonisometric.
A formulation of this inverse problem in terms of computer vision and shape analysis, is isospectralization, a numerical optimisation procedure in relation with interpolation, correspondence matching and style transfer. More details of this can be found in my presentation for “Recent Advances in the Analysis of 3D Shapes” seminar at TU Munich.
Checkout my isospectralization pytorch implementation notebooks.