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We study conformal field theories (CFTs) on curved spaces including both orientable and unorientable manifolds possibly with boundaries. We first review conformal transformations on curved manifolds. We then compute the identity components…
Counterfactual reasoning -- the practice of asking ``what if'' by varying inputs and observing changes in model behavior -- has become central to interpretable and fair AI. This thesis develops frameworks that use counterfactuals to…
We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can `lift'. The lifting can be computed by a path integral…
In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice $\Z^d$,…
The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several…
For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a…
We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the…
Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this article we lay out general foundations for studying…
It is shown that a recently conjectured form for the critical scaling function for planar self-avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem,…
We study two dimensional conformal field theories in the semiclassical limit. In this limit, the four-point function is dominated by intermediate primaries of particular weights along with their descendants, and the crossing equations…
We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF…
We develop a classification theory for real-analytic hypersurfaces in $\mathbb C^2$ in the case when the hypersurface is of {\em infinite type} at the reference point. This is the remaining, not yet understood case in $\mathbb C^2$ in the…
Within the exact renormalisation group, the scaling solutions for O(N) symmetric scalar field theories are studied to leading order in the derivative expansion. The Gaussian fixed point is examined for d>2 dimensions and arbitrary infrared…
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…
An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\geq0$ such that…
Let $X$ be a smooth hypersurface $X$ of degree $d\geq4$ in a projective space $\mathbb P^{n+1}$. We consider a projection of $X$ from $p\in\mathbb P^{n+1}$ to a plane $H\cong\mathbb P^n$. This projection induces an extension of function…
Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can…
We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results of Kolar, Meylan and Zaitsev, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear…
We lay the foundations of convex hypersurface theory in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one.…
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the…