Jim Bryan

Professor

Relevant Degree Programs

 

Great Supervisor Week Mentions

Each year graduate students are encouraged to give kudos to their supervisors through social media and our website as part of #GreatSupervisorWeek. Below are students who mentioned this supervisor since the initiative was started in 2017.

 

My supervisor is not just a #GreatSupervisor but also the #BestSupervisor because he is very enthusiastic about math and makes me feel super excited about my project. He has boundless energy and happily meets with each of his students several hours every week, making us all feel special. He is full of fruitful ideas on how to proceed in my project, so I never get down feeling stuck for long. He is not judgemental or hyper critical so I feel comfortable asking dumb questions. He is the #BestSupervisor because he is invaluable in helping me become a mathematician. Thank you Prof. Bryan!

Nina Morishige (2019)

 

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Computing motivic Donaldson-Thomas invariants (2012)

This thesis develops a method (dimensional reduction) to compute motivic Donaldson--Thomas invariants. The method is then employed to compute these invariants in several different cases.Given a moduli scheme with a symmetric obstruction theory a Donaldson--Thomas type invariant can be defined by integrating Behrend's function over the scheme. Motivic Donaldson--Thomas theory aims to provide a more refined invariant associated to each such moduli space - a virtual motive.From the modern point of view motivic Donaldson-Thomas invariants should be defined for a three dimensional Calabi--Yau category. These categories often arise in a geometric context as the derived category of representations of a quiver with potential.Provided the potential has a linear factor we are able to reduce the problem of computing the corresponding virtual motives to a much simpler one. This includes geometric examples coming from local curves which we compute explicitly.

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Counting hyperelliptic curves in Abelian surfaces with quasi-modular forms (2012)

In this thesis we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of P. A. MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms.

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Curve-Counting Invariants and Crepant Resolutions of Calabi-Yan Threefolds (2012)

No abstract available.

Master's Student Supervision (2010 - 2018)
Emergent geometry through holomorphic matrix models (2017)

Over the years, deep insights into string theory and supersymmetric gauge theorieshave come from studying geometry emerging from matrix models. In this thesis,I study the ℕ = 1* and ℕ = 2* theories from which an elliptic curve withmodular parameter τ is known to emerge, alongside an elliptic function called thegeneralized resolvent into which the physics is encoded. This is indicative of thecommon origin of the two theories in ℕ = 4 SYM. The ℕ = 1* Dijkgraaf-Vafamatrix model is intrinsically holomorphic with parameter space corresponding tothe upper-half plane ℍ. The Dijkgraaf-Vafa matrix model ’t Hooft coupling S(τ)has been previously shown to be holomorphic on ℍ and quasi-modular with respectto SL(2,ℤ). The allowed ℕ = 2* coupling is constrained to a Hermitianslice through the enlarged moduli space of the holomorphic ℕ = 1* model.After explicitly constructing the map from the elliptic curve to the eigenvalueplane, I argue that the ℕ = 1* coupling S(τ) encodes data reminiscent of ℕ = 2*.A collection of extrema (saddle-points) of S(τ) behave curiously like the quantumcritical points of ℕ = 2* theory. For the first critical point, the match is exact. Thiscollection of points lie on the line of degeneration which behaves in a sense, like aboundary at infinityI also show explicitly that the emergent elliptic curve along with the generalizedresolvent allow one to recover exact eigenvalue densities. At weak coupling, mymethod reproduces the inverse square root of ℕ = 2* as well as the Wigner semicirclein ℕ = 1*. At strong coupling in ℕ = 1*, I provide encouraging evidenceof the parabolic density arising in the neighborhood of the line of degeneration.To my knowledge, the parabolic density has only been observed asymptotically. Itis interesting to see evidence that it may be exactly encoded in the other form ofemergent geometry: the elliptic curve with the generalized resolvent.

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Emergent geometry through holomorphic matrix models (2017)

Over the years, deep insights into string theory and supersymmetric gauge theorieshave come from studying geometry emerging from matrix models. In this thesis,I study the ℕ = 1* and ℕ = 2* theories from which an elliptic curve withmodular parameter τ is known to emerge, alongside an elliptic function called thegeneralized resolvent into which the physics is encoded. This is indicative of thecommon origin of the two theories in ℕ = 4 SYM. The ℕ = 1* Dijkgraaf-Vafamatrix model is intrinsically holomorphic with parameter space corresponding tothe upper-half plane ℍ. The Dijkgraaf-Vafa matrix model ’t Hooft coupling S(τ)has been previously shown to be holomorphic on ℍ and quasi-modular with respectto SL(2,ℤ). The allowed ℕ = 2* coupling is constrained to a Hermitianslice through the enlarged moduli space of the holomorphic ℕ = 1* model.After explicitly constructing the map from the elliptic curve to the eigenvalueplane, I argue that the ℕ = 1* coupling S(τ) encodes data reminiscent of ℕ = 2*.A collection of extrema (saddle-points) of S(τ) behave curiously like the quantumcritical points of ℕ = 2* theory. For the first critical point, the match is exact. Thiscollection of points lie on the line of degeneration which behaves in a sense, like aboundary at infinityI also show explicitly that the emergent elliptic curve along with the generalizedresolvent allow one to recover exact eigenvalue densities. At weak coupling, mymethod reproduces the inverse square root of ℕ = 2* as well as the Wigner semicirclein ℕ = 1*. At strong coupling in ℕ = 1*, I provide encouraging evidenceof the parabolic density arising in the neighborhood of the line of degeneration.To my knowledge, the parabolic density has only been observed asymptotically. Itis interesting to see evidence that it may be exactly encoded in the other form ofemergent geometry: the elliptic curve with the generalized resolvent.

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The behavior of the Hilbert scheme of points under the derived McKay correspondence (2013)

In this thesis, we completely determine the image of structure sheaves of zero-dimensional, torus invariant, closed subschemes on the minimal, crepant resolution Y of the Kleinian quotient singularity C²/Z/n under the Fourier-Mukai equivalence of categories, between derived category of coherent sheaves on Y and Z/n-equivariant derived category of coherent sheaves on C². As a consequence, we obtain a combinatorial correspondence between partitions and Z/n-colored skew partitions.

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Formal-local structure of the Hilbert scheme of points on three-dimensional complex affine space around special monomial ideals (2016)

We show that the formal completion of the Hilbert scheme of points in ℂ³ at subschemes carved out by powers of the maximal ideal corresponding to the origin is given as the critical locus of a homogeneous cubic function. In particular, the Hilbert scheme is formal-locally a cone around these distinguished points.

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The behavior of the Hilbert scheme of points under the derived McKay correspondence (2013)

In this thesis, we completely determine the image of structure sheaves of zero-dimensional, torus invariant, closed subschemes on the minimal, crepant resolution Y of the Kleinian quotient singularity C²/Z/n under the Fourier-Mukai equivalence of categories, between derived category of coherent sheaves on Y and Z/n-equivariant derived category of coherent sheaves on C². As a consequence, we obtain a combinatorial correspondence between partitions and Z/n-colored skew partitions.

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