We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend. Our goal for this meeting will be to prove Kim’s lemma for dividing, the equivalence of forking and dividing, and the symmetry of forking-independence, all over models in simple theories.

It was observed very recently that the classical SOP_n hierarchy, a family of approximations of the strict order property which Shelah introduced with the goal of classifying non-simple first-order theories, extends to a hierarchy of properties SOP_r for r any real number at least 3. However, it remains open whether the real-valued NSOP_r hierarchy is distinct from the original integer-valued NSOP_n hierarchy. To make this question more tractable, we can ask it at the quantifier-free level, obtaining a real-valued quantity, of independent combinatorial interest, associated with any hereditary class of finite structures. While it is also open whether this quantity can have non-integer values, we can show that, in the case of a hereditary class defined by finitely many omitted weak substructures, it is an integer. We will discuss, and aim to prove, this result.

The Twin Prime Conjecture predicts that there are infinitely many primes p for which p+2 is also prime (in this case, we call the pair (p,p+2) a pair of twin primes). Despite a lot of recent progress on this conjecture, it remains unsolved today. In this talk, I will explain a heuristic argument leading to a precise form of the conjecture predicting an asymptotic formula as x goes to infinity for the number of twin prime pairs (p,p+2) with p ≤ x. If time permits, I will also discuss the Hardy-Littlewood Prime k-tuples Conjecture, which may be viewed as a generalization of the Twin Prime Conjecture.

For a broad class of convergence groups, we construct free subsemigroups having critical exponents arbitrarily close to, but strictly less than, the critical exponent of the ambient convergence group. I'll explain how to apply this result to a particular class of discrete subgroups of Lie groups known as transverse groups, and also explain how our result implies a counterpart to a classical gap theorem of Kevin Corlette concerning discrete subgroups of the isometry groups of quaternionic hyperbolic space and the Cayley (octonionic) hyperbolic plane. I will also briefly mention work in progress (to appear hopefully by the end of the month on the arXiv) which shows analogous results for any Zariski dense discrete subgroup of a semisimple real Lie group G. I'll introduce these notions from scratch and try to focus on their relations to (hopefully more familiar) concepts in real hyperbolic geometry. Time permitting, I'll discuss the general framework in which we prove these results.

What do water waves have to do with mirror symmetry, or even with Kovalevskaya tops? All of them are, or are closely related to, integrable systems, in particular to a class of algebraically completely integrable systems known as Hitchin systems. In this talk, I will explore how Hitchin systems provide a unifying framework connecting seemingly disparate areas of mathematics and physics. The presentation will be accessible to a general mathematical audience, while offering concrete connections for specialists in geometry, topology, and algebraic geometry.

I will discuss a gauge theoretic approach to the construction of the moduli space of Higgs bundles in 2-dimensionswhere the complex structure of the underlying surface also varies. This "joint" moduli space fibers over Teichmueller spacewith fiber the usual moduli space of Higgs bundles. I will explain why indefinite Hermitian structures arise naturally on the joint moduli space, and I will indicate the existence of pseudo-Kaehler metrics in a number of cases of Higgs bundles with special holonomy. I will also discuss the relationship between complex tangencies of isomonodromicl eaves and the strict plurisubharmonicity of the energy function. This recovers and extends several recent constructions of various authors. This work is part of a collaboration with Brian Collier and Jeremy Toulisse.

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A holomorphic symplectic manifold is a complex manifold $X$ together with a closed, non-degenerate holomorphic 2-form $\Omega$. The top power of $\Omega$ gives a trivialisation of the canonical bundle so that $X$ has trivial first Chern class. In the context of K\"ahler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact K\"ahler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperkahler manifolds in the K\"ahler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present new examples of compact holomorphic symplectic manifolds. These manifolds correspond to moduli spaces of sheaves on Kodaira surfaces and are non-K\"ahler. This is work in progress with Tom Baird and Eric Boulter.

The exact WKB method originated in the study of linear differential equations, e.g. the Schrodinger equation in one variable. Recently it has turned out that exact WKB has connections to various topics in modern geometry, topology and quantum field theory. I will explain what the exact WKB method is, and briefly survey some of its connections to other fields.

In this talk we consider compact holomorphic symplectic manifolds (aka hyperkahler manifolds), particular those fibred by Lagrangian submanifolds. The general fibre must be an abelian variety. Focusing on dimension four, we will describe how to construct examples whose fibres are abelian surfaces that are Jacobians of genus two curves or Prym varieties of double covers of curves. We will also describe some classification results for these kinds of fibrations.

I will give a tour of some places in modern mathematical physics where classical combinatorial objects of discrete math arise. More specifically, I will explore how counting points (of various moduli spaces) and surfaces (holomorphic curves) in symplectic topology can be related to the enumeration of graph colorings (chromatic polynomial), triangulations (Catalan numbers) and quiver structures (clusters).

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Spaces of quantum theories are the fundamental objects in modern mathematical physics. Outside of some basic examples, little is known about the geometry and topology of these spaces. In this talk, I will begin with spaces built from quantum mechanics. We will find that twisted equivariant K-theory encodes the homotopy type of the space of (supersymmetric) quantum mechanical systems. Viewing quantum systems as 1-dimensional quantum field theories, a natural generalization suggests a connection between 2-dimensional (supersymmetric) quantum field theories and twisted equivariant elliptic cohomology. This builds on ideas of Segal, Stolz and Teichner, and is intended as an introduction to their program.

The geometry of propagating strings in Conformal Field Theory (i.e. string theory), naturally gives rise to an algebraic structure called a vertex operator algebra describing the interactions of fields. The graded traces and pseudo-traces of families of linear operators corresponding to components of the fields (vertex operators) of interacting particles exhibit certain invariance properties with respect to action of the modular group SL(2,Z). We discuss aspects of modular symmetries that appear in graded trace and pseudo-trace functions for various classes of algebras of fields (vertex algebras) and their modules.

The nonlinear Schrödinger equation (NLSE) arises from various applications in quantum physics and chemistry, nonlinear optics, plasma physics, Bose-Einstein Condensates, etc. In these applications, it is necessary to incorporate low-regularity or singularity into the NLSE, which may arise from the potential, nonlinearity, and/or initial data. Typical examples include the discontinuous square-well potential, the singular Coulomb potential, the non-integer power nonlinearity, the logarithmic nonlinearity, and initial data that are ground states of the Schrödinger operator with such potential. Such low regularity and singularity pose significant challenges in the analysis of standard numerical methods and the development of novel accurate, efficient, and structure-preserving numerical schemes. In this talk, I will introduce several new analysis techniques to establish optimal error bounds for some widely used numerical methods under optimally weak regularity assumptions. Based on the analysis, we also propose novel temporal and spatial discretizations to handle the low regularity and singularity more effectively.

We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend. We will aim to finish proving Kim’s lemma for dividing, the equivalence of forking and dividing, and the symmetry of forking-independence, all over models in simple theories. The goal will be to encourage a graduate student to speak, continuing the discussion from the previous meeting.

The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for vector bundles and higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. In this talk, we will study Brill-Noether loci for vector bundles on the projective plane in the case where the number of sections is close to the largest possible number. When the number of sections is very large, Brill-Noether problems are all "trivial"--the Brill-Noether loci are either empty or the entire moduli space. As the number of sections decreases, we find that there is a "first" nontrivial Brill-Noether locus, and we discuss its geometry.

Testing composite null hypotheses is fundamental to many scientific applications, including mediation and replicability analyses, and becomes particularly challenging in high-throughput settings involving tens of thousands of features. Existing high-dimensional composite null hypotheses testing often ignores the dependence structure among features, leading to overly conservative or liberal results. To address this limitation, we develop a four-state hidden Markov model (HMM) for bivariate $p$-value sequences arising from two-study replicability analysis. This model captures local dependence among features and accommodates study-specific heterogeneity. Based on the HMM, we propose a multiple testing procedure that asymptotically controls the false discovery rate (FDR). Extending this framework to more than two studies is computationally intensive, with complexity growing exponentially in the number of studies $n.$ To address this scalability issue, we introduce a novel e-value framework that reduces computational complexity to quadratic in $n,$ while preserving asymptotic FDR control. Extensive simulations demonstrate that our method achieves higher power than existing approaches at comparable FDR levels. When applied to genome-wide association studies (GWAS), the proposed approach identifies novel biological findings that are missed by current methods.

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