| Krishanu Sankar
I'm a mathematician who enjoys solving puzzles, building abstract structures, and using technology. I am driven by a deep desire to be a part of a cause -- to find solutions that positively impact the community around me and the world at large. I'm currently a postdoctoral fellow in the Mathematics Department at the University of British Columbia in Vancouver, BC.
Office: ESB 4118 (PIMS)
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I am excited by automation and the prospect of self-modifying machines which can learn from data
and solve problems we can't (or solve ones we can, more efficiently). For example, using machine learning
to speed up chemistry and materials research (I learned about this through friends working 1
and 2). On the artistic side, there's the fact that learning
algorithms are speaking back to us through computer-generated stories and art (see
Learning algorithms fundamentally relies on large amounts of data, thus privileging a few industry giants -- I worry about whether cutting-edge research results will scale down to smaller players. I like the idea of building technology which can be used by everyone and by communities. A lot of computer power is necessary as well, which makes training very energy-intensive, and I worry about the carbon footprint of this work.
To arrive later - expository notes and my own (beginner) code.
Building energy-efficient, super-powerful computers by harnessing the fundamental laws of physics and
mathematics is the coolest thing there is. There's interesting math that goes into algorithm design,
quantum error-correction, designing circuit and chip architecture, and so on. The current application
which excites me the most is applications to chemistry research and drug discovery, by molecular
modeling (for example,
My understanding is that we're still some distance from producing anything useful, and nowhere close to a universal gate-model quantum computer (and may never be able to reach one!). I will be very curious to see what new problems can be solved efficiently by D-Wave's quantum annealer. I am thrilled by how open-source the community is, and I've been able to download software packages and start learning how it's used.
To arrive - notes on quantum annealing, gate model vs. adiabatic QC, variational quantum eigensolver, some basic algorithms and ideas I like. Links for experimentation.
Representations of these two groups are ubiquitous throughout mathematics and physics. For example, they are used to produce power operations. I've been thinking about a project concerning the relation between these in homotopy theory. notes
Knot polynomials seem to be related to questions in theoretical physics, and have gained attention recently with connection to quantum computing. I have been reading about TQFTs and topological phases of matter, and Cihan Okay has been organizing a seminar in the UBC physics department on this topic.
I am interested in the simultaneous similarity problem for matrices, and more generally the space of homomorphisms from a given group into the unitary group (or into another Lie group G). When the source group is a cosimplicial group, these spaces of homomorphisms piece together into a simplicial space, whose geometric realization is an exotic variant of the classifying space of G. I am interested in these homotopy types for some specific source groups, such as nilpotent groups and the braid groups.
My PhD work was about computing the C_p-equivariant dual Steenrod algebra at odd primes p. The motivation for doing so
was to detect classes in the odd-primary stable homotopy groups of spheres. I've worked out how to compute the module
structure, in joint work with Dylan Wilson. We can also identify ring generators which, on underlying points, descend to
the conjugate Milnor generators. But the coproduct structure is still a mystery.
When p=2 and the ambient group G is a 2-group, there is another way to get at the ring structure by filtering HF_2 via Thom spectra on equivariant classifying spaces. I'm working on writing up what I know so far, but there is still work to be done, and I would love new ideas.
I have spent a fair amount of time thinking about G-equivariant classifying spaces and G-equivariant Eilenberg-MacLane
spaces, particularly when G is a cyclic group of prime order. These spaces have cell structures built from representations
of G (i.e., RO(G)-graded cell structures), and their homology can be computed explicitly. These showed up in my study
of equivariant Steenrod operations.
I am curious about computing the C_2-equivariant homotopy groups of the (unstable) unitary group U(n). Nonequivariantly these can be computed in a range using the homology (coming from Schubert cell decomposition) combined with the Hurewicz theorem. I am curious about whether this can be done equivariantly. A followup question would be how to adapt these techniques to the unstable motivic setting to study GL_n.
I have always been fascinated by local-to-global principles in number theory and elsewhere. For example, the
Hasse-Minkowski principle for quadratic forms says that quadratic forms over the rational numbers may be studied
one prime at a time. These principles can be interpreted using cohomology.