# Jim Bryan

#### Relevant Thesis-Based Degree Programs

## Great Supervisor Week Mentions

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!

## Graduate Student Supervision

##### Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

This dissertation consists of two main chapters, each pertaining to the enumerative geometry of Calabi-Yau manifolds with an action:In Chapter 2 we study Euler characteristics of the G-invariant Hilbert schemes of points on an Abelian surface with a symplectic action by finite group G. One can package these Euler characteristics into generating series, whose reciprocal we prove is a holomorphic modular form for a particular congruence subgroup. For the standard involution of multiplication by -1, we prove an analogue of the Yau-Zaslow formula--that is, these Euler characteristics determine a weighted number of curves invariant under the involution and with rational quotient. Motivated by the results of Chapter 2, in Chapter 3 we develop in more generality a theory counting invariant curves in Calabi-Yau threefolds with an involution. Our theory conjecturally results in analogues of the Gopakumar-Vafa invariants which count invariant curves of genus g and with genus h quotient. We prove the conjecture and compute all invariants in the case of a local Abelian surface with involution multiplication by -1, or a local Nikulin K3 surface together with the Nikuln involution.

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This research develops a method to compute Katz's genus 0 Gopakumar-Vafa invariants of fiber curve classes of the Banana manifold, a smooth compact Calabi-Yau threefold, and its generalizations, the multi-Banana configurations. The invariants turn out to have geometric significance, being an actual count of certain local configurations of rational curves on a related threefold. In both these cases, the partition functions for the invariants exhibit a modularity and can be expressed in terms of Jacobi forms or theta functions.

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This thesis contains two chapters which reflect the two main viewpoints of modern enumerative geometry.In chapter 1 we develop a theory for stable maps to curves with divisible ramification. For a fixed integer r>0, we show that the condition of every ramification locus being divisible by r is equivalent to the existence of an r-th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to r-spin Hurwitz theory. In particular it is expected to provide a proof of the r-spin ELSV formula [SSZ'15, Conj. 1.4] when used with virtual localisation. In chapter 2 we further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a “banana configuration”. In [Bryan'19] the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this chapter we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the KKV formula and present new Gopakumar-Vafa invariants for the banana threefold.

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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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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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The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but Bryan, Cadman, and Young have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold χ. The crepant resolution conjecture of Bryan-Cadman-Young gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland.

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##### Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

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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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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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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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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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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