coat-of-arms Sanjeevi Krishnan me research papers teaching

sanjeevi@math.upenn.edu

DRL Office 3C3
209 South 33rd Street
Philadelphia, PA 19104
(215) 898-5988
I'm a postdoc, interested in algebraic topology and its applications, in the Math Dept at the University of Pennsylvania. Previously, I was a postdoc at the Naval Research Lab in DC and the Commissariat à l'Énergie Atomique Saclay and École Polytechnique (France). I obtained my Ph.D. under Peter May at the University of Chicago.

I'm interested in refinements of algebraic topology for spaces equipped with direction (like phases spaces), sheaves of semigroups (like probability distributions), and other naturally occurring algebro-topological structures. The main motivation is to apply ideas from algebraic topology to make local-to-global inferences in optimization, data analysis, and engineering.
I'm interested in refining and applying local-to-global methods from algebraic topology to problems in optimization and data analysis. To that end, I am interested in algebraic invariants on topological structures naturally occurring in the real-world but are not amenable to standard homological algebra.

Directed Algebraic Topology Directed Topology

Spaces often come equipped with directionality, like vector fields, partial orders, and simplicial orientations. Directed algebraic topology studies algebraic invariants on such spaces that can capture information about the topology and directionality.
Directed (co)homology refines classical (co)homology to detect directed structure on a directed space. Examples of directed homology and cohomology respectively include the flows and Poincare sections of dynamical systems. Recent interests of mine include showing how such theories exhibit Alexander and Poincare Dualities. Together with Eric Goubault and Emmanuel Haucourt, I've used directed homology in the past to study the local order structure of locally preordered spaces that arise in computer science [3].
Directed homotopy theory studies directed spaces up to deformations respecting directionality. I'm currently interested in unifying certain aspects of higher category theory and dynamics as a formal equivalence between combinatorial and topological directed homotopy theories, extending previous results on simplicial and cubical approximation [6]. I've also worked in the past on formally connecting the theory of computation with directed homotopy [2] and developing adequate categorical foundations [1].
Directed Algebraic Topology Applied Topology

Data analysis, which often takes the form of global geometric inferences from sample points whose local metric structure only is known, naturally lends itself to sheaf-theoretic methods. Together with Rob Ghrist, Dave Lipsky, Michael Robinson, Hank Owen, and Mike Stein, I'm working on software that analyzes video imagery by means of homological invariants on sheaves.
Optimization dualities sometimes make seemingly intractable computations simple and fast. Recently, I've been working on interpreting and generalizing flow-cut dualities as special cases of a Poincare Duality for sheaves on directed graphs [8]. Together with Rob Ghrist and Greg Henselman, I'm interested in applying such a generalized flow-cut duality to handle logical, stochastic, and multicommodity constraints on directed graphs [7] and higher dimensional spaces.
Semantics and Homotopy The use of directed spaces to model computation dates back to early work on the Lambda Calculus. Recently, I've become interested in directed extensions of homotopy type theory. Together with Eric Goubault and Emmanuel Haucourt, I've worked in the past on using directed homotopy theory to automate the formal verification of large, concurrent programs [4].
[13] A homological approach to target-tracking
R. Ghrist, S. Krishnan, D. Lipsky, M. Robinson, M. Stein, "A homological approach to target-tracking," in progress, 2015.
[12] Ten Examples of Topological Flow-Cut Duality
R. Ghrist, G. Henselman and S. Krishnan, "Ten Examples of Topological Flow-Cut Duality ," in progress, 2015.
[11] Dynamic Sensor Networks
R. Ghrist and S. Krishnan, "Dynamic Sensor Networks," in progress, 2015.
[10] Directed Homotopy of Directed Spheres
S. Krishnan, "Directed homotopy of directed spheres," preprint available upon request, 2014.
[9] Directed Poincare Duality
S. Krishnan, "Directed Poincare Duality," preprint available upon request, 2014.
[8] Flow-cut dualities for sheaves on graphs
S. Krishnan, "Flow-cut dualities for sheaves on graphs," arXiv:1409.6712, 2014.
[pdf] [arXiv]
[7] A Topological Max-Flow Min-Cut Theorem
Abstract: This note generalizes the Max-Flow Min-Cut (MFMC) theorem from numerical edge capacities to cellular semimodule-valued sheaves on directed graphs. Examples of such sheaves include probability distributions, multicommodity constraints, and logical propositions.
R. Ghrist, S. Krishnan, "A Topological Max-Flow Min-Cut Theorem," Proceedings of Global Signals. Inf., (2013).
[pdf]
[6] Cubical approximation for directed topology I
The homotopy theory of locally preordered state spaces can detect machine behavior unseen by the classical homotopy theory of spaces. The geometric realizations of simplicial sets and cubical sets admit "local preorders" encoding the orientations of simplices and 1-cubes, respectively. We prove simplicial and cubical approximation theorems appropriate for this homotopy theory. Along the way, we discover criteria under which two alternative definitions of a homotopy relation on locally monotone maps coincide.
S. Krishnan, "Cubical approximation for directed topology I," Applied Categorical Structures, Springer Netherlands, (2013), doi: 10.1007/s10485-013-9330-y, pp 1-38.
[pdf] [arXiv]
[5] A free object in quantum information theory
The homotopy theory of locally preordered state spaces can detect machine behavior unseen by the classical homotopy theory of spaces. The geometric realizations of simplicial sets and cubical sets admit "local preorders" encoding the orientations of simplices and 1-cubes, respectively. We prove simplicial and cubical approximation theorems appropriate for this homotopy theory. Along the way, we discover criteria under which two alternative definitions of a homotopy relation on locally monotone maps coincide.
J. Feng, K. Martin, S. Krishnan, "A free object in quantum information theory," Electronic Notes in Theoretical Computer Science, (2010), vol. 265, pp 35-47.
[pdf]
[4] Future path-components in directed topology
The global states of complex systems often form pospaces, topological spaces equipped with compatible partial orders reflecting causal relationships between the states. The calculation of tractable invariants on such pospaces can reveal critical system behavior unseen by ordinary invariants on the underlying spaces, thereby sometimes cirumventing the state space problem bedevilling static analysis. We introduce a practical technique for calculating future path-components, algebraic invariants on pospaces of states and hence tractable descriptions of the qualitative behavior of concurrent processes.
E. Goubault, E. Haucourt, S. Krishnan, "Future path-components in directed topology," Electronic Notes in Theoretical Computer Science, (2010), vol. 265, pp 325-335.
[pdf]
[3] Covering space theory for directed topology
The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a "local preorder" encoding control flow. In the case where time does not loop, the "locally preordered" state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a "locally monotone" covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes.
E. Goubault, E. Haucourt, S. Krishnan, "Covering space theory for directed topology," Theory and Application of Categories, (2009), vol. 22, pp. 252-268.
[pdf] [arXiv]
[2] Criteria for homotopic maps to be so along monotone homotopies The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homoAt, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.
S. Krishnan, "Criteria for homoAt maps to be so along monotone homotopies," Electronic Notes in Theoretical Computer Science (2009), pp. 141-148, vol. 230, pp 141–148.
[pdf] [arXiv]
[1] A convenient category of locally preordered spaces
As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.
S. Krishnan, "A convenient category of locally preordered spaces," Applied Categorical Structures, vol. 17 (5), pp 445-466.
[arXiv]
I'm not teaching this semester.

At the University of Chicago, I've taught single-variable calculus and multi-variable calculus and TA'd honors calculus and complex analysis. At Ecole Polytechnique, I've TA'd courses on concurrency and numerical approximation. At the University of Pennsylvania, I've taught multi-variable calculus (Math 114) and graduate algebraic topology (Math 619).