Georgia Institute of Technology College of Sciences

In this talk I will discuss a particular convex relaxation of quadratic programs called the Shor relaxation. We study the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. I will discuss conditions under which this region is invariant under the choice of generators defining the feasible set as well as how this region reflects the symmetry in the feasible region. Finally, I will discuss applications of these results to quadratic binary programs. This is joint work with Jose Rodriguez.

The Heegaard Floer d-invariant is a numerical invariant of rational homology spheres which is analogous to the Frøyshov h-invariant from Instanton theory. In this talk, we use Zemke’s recent isomorphism between lattice Floer and Heegaard Floer homology to compute the d-invariant for all rational homology spheres which arise as negative definite plumbed manifolds, verifying a 20 year old conjecture of Némethi. 

Deep neural network–based surrogate models have recently gained traction for solving fluid-flow partial differential equations, but their reliance on global interpolation often demands large, computationally expensive architectures and extensive training data. Neural networks with local converging inputs (NNLCI) offer a contrasting strategy. By restricting attention to the local domain of dependence and using converging coarse-grid solutions as inputs, NNLCI dramatically reduces computational cost and data requirements while achieving strong generalization. In this work, we extend the NNLCI framework to the three‑dimensional Stokes equations and introduce a new subdomain data generation methodology specifically tailored for NNLCI, enabling high‑fidelity prediction while completely eliminating the need to compute fine‑grid numerical solutions on the full domain at any stage of the computing process. This innovation eliminates the most computationally intensive component of 3D simulations at its root.

 A central theme in algebraic geometry is understanding how much of the well-behaved theory of smooth varieties survives in the presence of singularities. Du Bois and rational singularities are among the most important classes studied in algebraic geometry due to their nice cohomological behavior. For instance, they preserve features like the Hodge decomposition and vanishing theorems one expects for smooth varieties.
Recently, motivated by developments in Hodge theoretic methods, there has been substantial interest in studying their higher analogs. This talk will survey recent developments connecting these notions to deformation properties and moduli theory, and applications to Calabi-Yau varieties. I will also report on recent joint work with Brian Nugent extending the theory of higher Du Bois and higher rational singularities to pairs — a framework that is both essential in modern birational geometry and a natural setting for studying Hodge theory for open varieties.

Isometric embeddings between a domain manifold and a target manifold are differentiable maps f such that the pullback of the target metric h coincides with the metric g in the domain manifold. This problem can also be formulated as a non-linear PDE via $\nabla f^{\top} h \nabla f = g$. In the case of contact manifolds, it is additionally required that the embedding preserves a certain restriction on the tangent bundle.

We prove that the Nash iteration scheme can be quantified in order to construct infinitely many $C^{1,\alpha}$-isometric embeddings for contact manifolds. In this way, we extend an existing result regarding non-uniqueness for $C^1$ regularity. The strategy of the proof follows a paper by Conti, De Lellis and Szekelyhidi Jr. on the Riemannian case, which is built on the Nash-Kuiper scheme. The main difficulty in our case is to keep the additional linear constraint coming from the contact setting along the iteration procedure.

In the larger program of a quantitative analysis of isometric embeddings between sub-Riemannian manifolds, our result can be seen as an important first step. Another aspect is the flexibility of this convex integration method: the geometric constraint coming from the contact condition is just one special case of a (potentially large) class of admissible constraints, under which this scheme can still be applied.

An $r$-sunflower is a collection of $r$ sets such that the intersection of any two sets in the collection is identical. We analyze a random process which constructs a $w$-uniform $r$-sunflower free family starting with an empty family, and at each step, adding a set chosen uniformly at random from all choices that could be added without creating an $r$-sunflower with the previously chosen sets. To analyze this process, we extend results of Bennett and Bohman (arXiv:1308.3732v5 [math.CO]) who analyzed a general random process which adds one object at a time chosen uniformly at random from all objects that can be added without creating certain forbidden subsets. This talk is based on joint work with Professor Patrick Bennett.

In many applications of mathematical optimization, one may wish to optimize an objective function without access to its derivatives. These situations call for derivative-free optimization (DFO) methods. Among the most successful approaches in practice are model-based trust-region methods, such as those pioneered by M.J.D Powell. These methods rely on function approximations via low degree polynomials and carefully adapt the local geometry of interpolation points to balance exploration and exploitation.  While relatively complex to implement, these methods are now available in standard scientific computing platforms, including MATLAB and SciPy. However, theoretical analysis of their computational complexity lags behind practice. In particular, it is important to bound the number of function evaluations required to achieve a desired level of accuracy. Using concepts from Lagrangian interpolation and linear algebra we systematically derive complexity bounds for classical model-based trust-region methods and their modern variations. We establish, for the first time, that these methods can have the same worst case complexity than any other known DFO method.