Academic Positions

  • Present 2016

    Postdoctoral Fellow

    University of British Columbia

  • Present 2013

    Lead of Fundamental Research

    1QB Information Technologies

  • 2016 2010

    Graduate Teaching & Research Assistant

    University of British Columbia

  • 2010 2009

    Benjamin Franklin Graduate Fellow

    University of Pennsylvania

Education and Training

  • Ph.D. 2016

    Ph.D. in Mathematics

    University of British Columbia

  • M.Sc.2011

    M.Sc. in Mathematics

    University of British Columbia

  • B.Sc.2009

    B.Sc. in Computer Science

    B.Sc. in Mathematics

    Sharif University of Technology

Research Projects

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    Generalized Donaldson-Thomas Invariants

    An Approach Using the Semi-simple Inertia

    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.

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    Spectrum of the Inertia of Algebraic Stacks

    Local-finiteness and Diagonalization of the Inertia

    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.

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    Quantum Machine Learning

    Global Training of Deep Networks

    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.

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    Quantum Optimization

    Constrained Programming via Quantum Adiabatic Computation

    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.

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Solving constrained quadratic binary problems via quantum adiabatic evolution

Pooya Ronagh, Brad Woods, Ehsan Iranmanesh
Journal Paper Quantum Information and Computation, Volume 16, No 11&12, Sept 2016, Pages 1029-10474

Abstract

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.

A subgradient approach for constrained binary programming via quantum adiabatic evolution

Sahar Karimi, Pooya Ronagh
Submitted Paper Quantum Information and Computation

Abstract

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.

The inertia operator and Hall algebra of algebraic stacks

Pooya Ronagh
Ph.D. ThesisUniversity of British Columbia

Abstract

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.

Ringel-Hall algebras and applications to moduli

Pooya Ronagh
M.Sc. EssayUniversity of British Columbia

Abstract

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.

Quantum Invariants of Calabi-Yau threefolds

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Topics in Algebraic Geometry, Spring 2014

Source

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.

Disclaimer

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.

Lie Theory

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Lie Theory, Fall 2011

Source

These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Julia Gordon.

Disclaimer

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.

Toric Geometry

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Lie Theory, Spring 2011

Source

These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Kalle Karu.

Disclaimer

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.

Algebraic K-Theory

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Lie Theory, Fall 2011

Source

These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Sujatha Ramdora.

Disclaimer

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.

Algebraic Topology II

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Lie Theory, Spring 2011

Source

These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Donald Stanley.

Disclaimer

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.

Algebraic Geometry II

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Lie Theory, Fall 2011

Source

These notes are live-TeXed and compiled in the present format based on a course offered by Prof. Zinovy Reichstein.

Disclaimer

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.

Motivic Integration

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Lie Theory, Spring 2011

Source

These notes are live-TeXed and compiled in the present format based on a seminar course organized by Prof. Julia Gordon.

Disclaimer

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.

Geometry of Hilbert Schemes

Lecture notes by Pooya Ronagh
Lecture notes University of British Columbia, Topics in Algebraic Geometry, 2007

Source

These notes are based on a course offered by Prof. Kai Behrend.

Disclaimer

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.

Algebraic Stacks

Lecture notes by Pooya Ronagh
Lecture notes Isaac Newton Institute for Mathematical Sciences, School on Moduli Spaces, 2011

Source

These notes are based on a short-course offered by Prof. Kai Behrend.

Disclaimer

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.

Geometric Invariant Theory

Lecture notes by Pooya Ronagh
Lecture notes Isaac Newton Institute for Mathematical Sciences, School on Moduli Spaces, 2011

Source

These notes are based on a short-course offered by Prof. Brent Doran.

Disclaimer

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.

Ringel-Hall algebras and applications to moduli

Lecture notes by Pooya Ronagh
Lecture notes Isaac Newton Institute for Mathematical Sciences, School on Moduli Spaces, 2011

Source

These notes are based on a short-course offered by Prof. Markus Reineke.

Disclaimer

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.

Currrent Teaching

  • Fall 2016

    MATH 102 | University of British Columbia

    Differential Calculus with Applications to Life Sciences

Teaching History

  • Winter 2013

    MATH 103 | University of British Columbia

    Integral Calculus with Applications to the Life Sciences

  • Winter 2013

    MATH 152 | University of British Columbia

    Linear Systems Course Lab using MATLAB