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.

E-mail:  ksankar[at]math[dot]ubc[dot]ca
Office:  ESB 4118 (PIMS)
Resume

Linkedin Facebook Github Reddit
Here I will discuss some interests, and if applicable give notes describing the projects I am thinking about. If you have insights, corrections, or just find any of it cool and want to talk, please reach out!

Artificial Intelligence

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 1 2).
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.

Quantum Computers

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, 1).
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.

The Braid group and the Symmetric group

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 Theory and Quantum Computation

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.

K-theory and Classifying Spaces

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.

The Steenrod Algebra

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.

Equivariant Homotopy Theory

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.

Local to Global Principles

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.

  • The Hasse Principle for Quadratic Forms , my minor thesis in graduate school. All graduate students in the math department at Harvard are required to write an expository work on a topic outside of their main field of study. This was a very useful exercise for me in reading papers and learning to see the forest for the trees!

    Assorted Expository Talk Notes


  • The Steenrod Algebra , notes from a talk I gave in the Harvard graduate student seminar in 2017.
  • Power Operations and some Classical Questions in Topology from the first talk I gave in our 2019 group on power operations.
  • Characters of GL_n and Adams Operations from the second and third talks I gave in our 2019 group on power operations.

  • Published papers


  • Achievable Sets in Z^n (2012) paper, journal link . I wrote this paper as an undergraduate at the 2010 Duluth REU.
  • Equipopularity Classes of 132 avoiding permutations (2014), journal link . Joint work in 2013 with Lynn Chua at the Duluth REU.

    Preprints


  • Symmetric Powers and Eilenberg-Maclane Spectra draft
  • Equivariant Steinberg Summands draft

    PhD Thesis


  • Symmetric Powers and the Dual Steenrod Algebra final version
  • University teaching


    Math 220: Mathematical Proof (Spring 2019, UBC)
    Math 102: Differential Calculus with Applications to the Life Sciences (Fall 2018, UBC)
    Math 220: Mathematical Proof (Spring 2018, UBC)
    Math 102: Differential Calculus with Applications to the Life Sciences (Fall 2017, UBC)
    Math 21b: Linear Algebra and Differential Equations (Spring 2017, Harvard). I've prepared worksheets and solutions for this course, available here.
    I upload my current projects to Github. At the moment I'm just getting started, and appreciate feedback or comments!
    Rubik's cube solver - I've written a sequence of scripts to solve a Rubik's cube.
    Over the years, I have been involved in a number of programs, both as a student and as a teacher. These programs have shaped me in a positive way.

    PROMYS

    A program in rigorous mathematical inquiry, taught through number theory. It's for high school students and takes place in the summer at BU university. I was a student in the program in 2006, and then returned as a mentor for the Research Labs program in 2016.

    American Math Competition (MAA)

    A highly popular contest series for middle and high school students in the USA. These contest questions ignited my passion for math and honed my problem solving skills. It was also through the Math Olympiad Summer program (2011 and 2012) and subsequently International Math Olympiad in 2012 that I met many close friends. In the subsequent year, I continued to develop training materials for the program, and returned as a grader.

    The Duluth REU

    A summer research program for undergraduates. I was a student in the program in 2010, and then returned as a counselor in 2013.

    Enroot

    An after-school program for immigrant youth, based in Cambridge and Somerville, MA. I mentored a student in the program during my last year of graduate school.
    I run and race as part of VFAC, a running club in Vancouver, BC. I enjoy road and track races varying from the 1500m to the half marathon.

    I also like cycling, trail running, rock climbing, and hiking. I've done some exciting trips in the past, and have too many photos to count. I documented a six week trip to South America in 2017 through a travelogue. Here are some assorted photos:

    Summer 2017, Bolivia