in this defense i will talk about two lines of research during my phd. i will develop some scan statistics and derive their asymptotic distributions, using moderate deviation, large deviation, kolmogorov's theorem and a poisson approximation. i will briefly mention their applications in global testing and multiple testing, resulting from the work before advancement. next i will talk about another project on causal inference, where the causal effects of etanercept on birth defects are investigated, in the presence of missing outcomes not at random, left truncation, observational nature, and rare events.

in this talk we will introduce a generalization of babuska-lax-milgram theorem that is particularly well-suited for the study of global existence of solutions to certain parabolic pdes on compact manifolds.

i will give a light introduction to the theory of quantum graphs. quantum graphs are non-commutative generalizations of finite graphs that arise naturally in many areas, including the study of zero-error capacities of quantum channels, quantum symmetry groups of graphs, and in the theory of non-local games. i will give an overview of some of these connections and explain some ongoing joint work with kari eifler, christian voigt and moritz weber, where we generalize the well-known graph $c^\ast$-algebra construction to the setting of quantum graphs.

as part of a general theory for the isomorphism problem for actions of amenable groups, ornstein and weiss proved that any two poisson point processes are isomorphic as measure-preserving actions. we give an elementary construction of an isomorphism between poisson point processes that is finitary. this is joint work with terry soo.

the laplacian is a canonical second order elliptic operator defined on any riemannian manifold. the study of upper bounds for its eigenvalues under the volume constraint is a classical problem of spectral geometry going back to j. hersch, p. li and s.-t. yau. the particular interest to this problem stems from a surprising connection to the theory of harmonic maps to spheres. in the present talk we will survey some recent results in the area with an emphasis on the role played by the energy index of harmonic maps. in particular, we will discuss some recent applications, including a min-max proof of the existence of optimisers and an optimal upper bound for laplacian eigenvalues on the projective plane.

in the first part of the talk (based on joint work with l.coburn, j. sj"ostrand, and f. white) we discuss continuity conditions for toeplitz operators acting on spaces of entire functions with quadratic exponential weights (bargmann spaces), in connection with a conjecture by c. berger and l. coburn, relating toeplitz and weyl quantizations. in the second part of the talk (based on joint work in progress with j. sj"ostrand), we discuss elements of a semiglobal approach to analytic second microlocalization with respect to a hypersurface, in the semiclassical case, based on the study of the heat evolution semigroup for large times. we describe properties of the associated exponentially weighted spaces of holomorphic functions with
($h$--dependent) plurisubharmonic exponents and construct asymptotic bergman projections in such spaces.

we show how to express the solution to an ordinary differential
equation as a sum over rooted trees, discover a group structure on the set
of rooted trees, extend it to a hopf algebra related to quantum
renormalization, and finally bring ourselves back down to earth and derive
the coefficients used in runge-kutta integrators.

modeling and tractable computation form two fundamental but competing pillars of data science; indeed, fitting good models to data is often computationally challenging in modern applications. focusing on the canonical tasks of ranking and regression, i introduce problems where this tension is immediately apparent, and present methodological solutions that are both statistically sound and computationally tractable.

i begin by describing a class of ``permutation-based'' models as a flexible alternative to parametric modeling in a host of inference problems including ranking from ordinal data. i introduce procedures that narrow a conjectured statistical-computational gap, demonstrating that carefully chosen non-parametric structure can significantly improve robustness to mis-specification while maintaining interpretability. next, i address a complementary question in the context of convex regression, where i show that the curse of dimensionality inherent to non-parametric modeling can be mitigated via parametric approximation. our provably optimal methodology demonstrates that it is often possible to enhance the interpretability of non-parametric models while maintaining important aspects of their flexibility.

in his work on modularity theorems, wiles proved a numerical criterion for a map of
rings r $\rightarrow$ t to be an isomorphism of complete intersections. he used this to show
that certain deformation rings and hecke algebras associated to a mod p galois
representation at non-minimal level were isomorphic and complete intersections,
provided the same was true at minimal level. in addition to proving modularity
theorems, this numerical criterion also implies a connection between the order of a
certain selmer group and a special value of an l-function. in this talk i will
consider the case of a hecke algebra acting on the cohomology a shimura curve
associated to a quaternion algebra. in this case, one has an analogous map of ring
r $\rightarrow$ t which is known to be an isomorphism, but in many cases the rings r and t
fail to be complete intersections. this means that wiles' numerical criterion will fail
to hold. i will describe a method for precisely computing the extent to which the
numerical criterion fails (i.e. the ``wiles defect''), which will turn out to be
determined entirely by local information at the primes dividing the discriminant of
the quaternion algebra. this is joint work with gebhard bockle and chandrashekhar
khare.

we present one aspect of bergman kernels, which includes the
following topics:
fourier integral operators with complex phase and projections for certain scalar equations,
bergman and szeg"o kernels for strictly pseudoconvex domains,
projections for exponentially weighted spaces and powers of complex line bundles.

oprea and pandharipande studied the virtual euler characteristic of quot schemes on surfaces. based on the calculations in several cases, they conjectured the rationality of the generating series of the virtual invariants. in this talk, i will explain the virtual $\chi_{-y}$-genera of quot schemes, thus refining the work of [op]. the main result expresses the quot scheme invariants universally in terms of seiberg-witten invariants of d"urr, kabanov, and okonek. based on the calculations in several cases and the blow up formula, we resolve the $\chi_{-y}$-genus analogue of the rationality conjecture of [op], for all surfaces with $p_g>0$. in addition, the reduced quot scheme invariants for $k3$ surfaces will be discussed with connections to the kawai-yoshioka formula.