Postdoctoral Fellow
University of Waterloo
As of September 2017, I am a postdoctoral fellow in the Institute of Quantum Computing at the University of Waterloo.
Prior to this, I received my PhD under Prof. K. Behrend, in the Department of Mathematics of the University of British Columbia. In my thesis, I worked on the algebro-geometric properties of moduli spaces that are envisaged to appear naturally in the framework of String Theory. Specifically, I developed a simple and geometric approach to the theory of generalized Donaldson-Thomas invariants by viewing the inertia construction of algebraic stacks as an operator on a corresponding motivic ring and studying the spectrum of it.
In the past four years, I have also served as a Research Lead in 1QBit, an R&D firm dedicated to advancements in emerging paradigms of quantum computing. I lead a group of more than ten mathematicians and computer scientists working on applicability of quantum computation to machine learning and operation research.
University of Waterloo
University of British Columbia
1QB Information Technologies
University of British Columbia
University of Pennsylvania
Ph.D. in Mathematics
University of British Columbia
M.Sc. in Mathematics
University of British Columbia
B.Sc. in Computer Science
B.Sc. in Mathematics
Sharif University of Technology
In the same manner that Newton developed the calculus of differentials and integrals as a language to express classical mechanics, physicists and mathematicians nowadays are inventing the convenient mathematics in which modern theoretical physics can flourish.
My interest in algebraic geometry steams from applicability of it in high-energy physics. Current attempts towards developing a theory of Quantum Gravity, and in particular String Theory, are strong motivations for research in algebraic geometry. I am in particular, interested in studying the moduli stacks that arise naturally in the content of String Theory.
My interest in mathematics of Quantum Mechanics, is the leading force in my involvement in research projects in quantum computation. My research in quantum computation is currently focused on applicability of the adiabatic quantum computation in machine learning and optimization theory.
According to string theory the world we live in is not the 4-dimensional time-space we see but has a minuscule 6-dimensional curled shape attached to each point of it known as a Calabi-Yau threefold. What physicists would really like from a good theory, is the capability of extracting quantitative information, simply numbers, that can be checked through experiments. According to string theory, such numbers are supposed to be independent of the minor changes that can happen to the Calabi-Yau 3-fold. It is also possible to check this fact mathematically, hence these numbers are called invariants assigned to a Calabi-Yau 3-fold in mathematics. [Deligne, et. al]
Inside a Calabi-Yau 3-fold physical objects called the BPS-branes live that encodes the states in which certain particles can exist. Physical interpretation suggests that there should be only a finite number of these BPS-states. These invariants translate mathematically to the Donaldson-Thomas (DT) invariants associated to the space of the BPS-states called moduli. Consistent definition of DT-invariants and studying their properties is a crucial ingredient of string theory, and a beautiful mathematical endeavor. The challenges for a suitable definition of these invariants are as follows: (1) beyond some special cases, the moduli are complicated objects that are called stacks. (2) The invariants one hopes to associated to these spaces are supposed to behave very much like Euler characteristics. An important feature of Euler characteristic of a geometric shape is that it remains unchanged if one deforms the shape as if it was made of rubber. (3) A potential definition of such invariants, depends on extra factors (called stability conditions) and therefore this dependence also has to be understood thoroughly. Kontsevich-Soibelman, and Joyce-Song have independently developed two approached to the definition of generalized Donaldson-Thomas invariants.
In my Ph.D. project, in collaboration with my supervisor Prof. K. Behrend we developed a third approach to define the generalized Donaldson-Thomas invariant following ideas of T. Bridgeland. We consider the Hall algebra of algebroids over the Calabi-Yau 3-fold. We view the inertia construction of algebroids as an operator on this ring. We show that the (semi-simple variant of the) inertia operator is diagonalizable on this space and the eigenvalue decomposition of it creates a graded structure on the space of stack functions. We use this decomposition to define a Lie sub-algebra of the Hall algebra, generated as a Q-vector space by those eigenvectors of the inertia that do not have eigenvalues dividing (q-1)^2. We then show that there exists a well-defined integration map that associates an invariant to each object in this smaller Lie algebra.
As a pre-requisite to developing a framework of generalized Donaldson-Thomas invariants using the inertia, in collaboration with Prof. K. Behrend I investigated properties of the inertia construction as an operator on the Grothendieck ring of algebraic stacks. We show that the Inertia operator and its variants, semisimple and unipotent inertia, are locally-finite and diagonalizable on the Grothendieck ring of Deligne-Mumford stacks. Using auxiliary operators that are defined similar to the inertia operators, we provide formulae for computation of the eigenprojections of a Deligne-Mumford stack on its eigendirections.
The situation happens to be more complicated in the case of Artin stacks. One can associate, up to stratification, a maximal torus to the inertia of an Artin stack over itself. We show that if a category of algebraic stacks, closed under taking inertia of its objects, has the property that this maximal torus of each object of it is quasi-split then the inertia operator on the ring of motivic classes of that category is Diagonalizable. This is in particular the case, for the category of algebroids over a linear algebraic stack with interesting applications in generalized Donaldson-Thomas theory.
The goal of machine learning algorithms is training a neural network that can identify objects of interesting (e.g. cat, words, sentences, structures, patterns) given a complicated input signal (e.g. image, voice recorded speech, fluctuations of a stock price, daily transactions in a financial market).
The input signal and output information of the network may be viewed as points in a probability space. Therefore the training procedure of the network, is a gradient descent where the update schedule involves moments of a current and a target probability distribution. However, numerical simulation and finding statistical data of distributions is not in general computationally efficient.
The modern theory of machine learning works around need to such numerical simulations. For example, many machine learning frameworks have Restricted Boltzmann Machines as their building blocks. Using quantum annealing however, one can perform efficient sampling from Gibbs measure of quantum Hamiltonians. The goal of our machine learning projects is development of machine learning algorithms that take advantage of quantum sampling. Training the weights of a deep network for example, is now a problem in global optimization of the set of weights on vertices and nodes of the underlying graph of the network.
Designed on the basis of the Adiabatic Theorem in quantum mechanics, there has been several recent efforts in building a quantum mechanical processor that solves an unconstrained binary quadratic programming (UBQP) problem by finding the ground state of an associated Hamiltonian. UBQPs are theoretically enough to encode all types of integer programming problems where the objective and constraints are represented as polynomials over bounded integer domains. However in practice, such encodings are not well-suited for the mentioned quantum processors.
Together with collaborators in 1QBit, I investigate techniques for practical use of the quantum devices in optimization theory. These efforts are based mainly on solving the Lagrangian duals of primal integer programming problems. We show that outer linear approximations and subgradient descent methods provide suitable techniques for solving Lagrangian dual problems using a hybrid of a digital and a quantum processor. Then we investigate application of this duality theory in branch and bound/price frameworks for non-linear integer programming.
The following is a list of some papers, write-ups, and lecture notes of mine.
Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.
In an earlier work [14], Ronagh et al. propose a method for solving the Lagrangian dual of a constrained binary quadratic programming problem via quantum adiabatic evolution using an outer approximation method. This should be an efficient prescription for solving the Lagrangian dual problem in the presence of an ideally noise-free quantum adiabatic system. Current implementations of quantum annealing systems demand methods that are efficient at handling possible sources of noise. In this paper we consider a subgradient method for finding an optimal primal-dual pair for the Lagrangian dual of a polynomially constrained binary polynomial programming problem. We then study the quadratic stable set (QSS) problem as a case study. We see that this method applied to the QSS problem can be viewed as an instance-dependent penalty-term approach that avoids large penalty coefficients. Finally, we report our experimental results of using the D-Wave 2X quantum annealer and conclude that our approach helps this quantum processor to succeed more often in solving these problems compared to the usual penalty-term approaches.
We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We are interested in showing that the inertia operator is (locally finite and) diagonalizable over for instance the field of rational functions of the motivic class of the affine line q = [A¹]. This is proved for the Grothendieck group of Deligne-Mumford stacks and the category of quasi-split Artin stacks. Motivated by the quasi-splitness condition we then develop a theory of linear algebraic stacks and algebroids, and define a space of stack functions over a linear algebraic stack. We prove diagonalization of the semisimple inertia for the space of stack functions. A different family of operators is then defined that are closely related to the semisimple inertia. These operators are diagonalizable on the Grothendieck ring itself (i.e. without inverting polynomials in q) and their corresponding eigenvalue decompositions are used to define a graded structure on the Grothendieck ring. We then define the structure of a Hall algebra on the space of stack functions. The commutative and non-commutative products of the Hall algebra respect the graded structure defined above. Moreover, the two multiplications coincide on the associated graded algebra. This result provides a geometric way of defining a Lie subalgebra of virtually indecomposables. Finally, for any algebroid, an ε-element is defined and shown to be contained in the space of virtually indecomposables. This is a new approach to the theory of generalized Donaldson-Thomas invariants.
In this essay we will survey some of the results of Markus Reineke on geometry of the moduli spaces of stable quiver representations, and Tom Bridgeland on properties of Donaldson-Thomas invariants of Calabi-Yau threefolds. The underlying idea of these results is to assign a suitable Hall algebra to the abelian category of objects of interest in the moduli problem and translate categorical statements about this category into identities in the Hall algebra. An integration on the Hall algebra is defined such that integrating identities in the Hall algebra will then produce generating functions involving invariants that we want to study.
These notes are live-TeXed and compiled in the format of a book based on a course Prof. Jim Bryan offered in 2014 on the enumerative geometry of Calabi-Yau threefolds.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Julia Gordon.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Kalle Karu.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Sujatha Ramdora.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Donald Stanley.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Zinovy Reichstein.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are live-TeXed and compiled in the present format based on a seminar course organized by Prof. Julia Gordon.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are based on a course offered by Prof. Kai Behrend.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are based on a short-course offered by Prof. Kai Behrend.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are based on a short-course offered by Prof. Brent Doran.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
These notes are based on a short-course offered by Prof. Markus Reineke.
Any mistakes, typos, messy type-setting you may find are due to this fact and does not reflect upon the content of the lectures. I would appreciate it if you point them out to me.
Differential Calculus with Applications to Life Sciences
Integral Calculus with Applications to the Life Sciences
Linear Systems Course Lab using MATLAB