Georgia Institute of Technology College of Sciences

The complete flag variety is a fundamental object at the confluence of algebraic geometry, representation theory, and algebra. It is defined to be the space parametrizing certain chains of vector subspaces, and is intimately linked to Grassmannians, incidence varieties, and other important geometric objects of a representation-theoretic flavor. The problem of computing the cohomology of any line bundle on a flag variety in characteristic 0 was solved in the 1950's, culminating in the celebrated Borel--Weil--Bott theorem. The situation in positive characteristic is wildly different, and remains a wide-open problem despite many decades of study. After surveying this topic, I will speak about recent progress on a characteristic-free analogue of the Borel--Weil--Bott theorem through the lens of representation stability and the theory of polynomial functors. This "stabilization" of cohomology, combined with certain universal categorifications of the Jacobi-Trudi identity, has opened the door to concrete computational techniques whose applications include effective vanishing results for Koszul modules, yielding an algebraic counterpart for the failure of Green's conjecture for generic curves in arbitrary characteristic.

The existence of global solutions for the Schrödinger equation
 i\partial_t u + \Delta u = P_d(u),
with nonlinearity $P_d$ homogeneous of degree $d$, has been extensively studied. Most results focus on the case with gauge invariant nonlinearity, where the solution satisfies several conservation laws. However, the problem becomes more complicated as we consider a general nonlinearity $P_d$. So far, global well-posedness for small data is known for $d$ strictly greater than the Strauss exponent. In dimension $3$, this Strauss exponent is $2$, making NLS with quadratic nonlinearity an interesting topic.

In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized data. To tackle the challenge presented by the $u\Bar{u}$-type nonlinearity, we require an $\epsilon$ regularization for the terms of this type in the system.
 

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov—He and Church—Ershov—Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.

Symmetry is prevalent in a variety of machine learning and scientific computing tasks, including computer vision and computational modeling of physical and engineering systems. Empirical studies have demonstrated that machine learning models designed to integrate the intrinsic symmetry of their tasks often exhibit substantially improved performance. Despite extensive theoretical and engineering advancements in symmetry-preserving machine learning, several critical questions remain unaddressed, presenting unique challenges and opportunities for applied mathematicians.

Firstly, real-world symmetries rarely manifest perfectly and are typically subject to various deformations. Therefore, a pivotal question arises: Can we effectively quantify and enhance the robustness of models to maintain an “approximate” symmetry, even under imperfect symmetry transformations? Secondly, although empirical evidence suggests that symmetry-preserving models require fewer training data to achieve equivalent accuracy, there is a need for more precise and rigorous quantification of this reduction in sample complexity attributable to symmetry preservation. Lastly, considering the non-convex nature of optimization in modern machine learning, can we ascertain whether algorithms like gradient descent can guide symmetry-preserving models to indeed converge to objectively better solutions compared to their generic counterparts, and if so, to what degree?

In this talk, I will provide an overview of my research addressing these intriguing questions. Surprisingly, the answers are not as straightforward as one might assume and, in some cases, are counterintuitive. My approach employs an interesting blend of applied probability, harmonic analysis, differential geometry, and optimization. However, specialized knowledge in these areas is not required. 

This talk will describe connections between algebraic geometry, convex geometry and algebraic topology. We will be discussing linear projections of the special orthogonal  group and when they are convex (in the sense that every pair of points in the image of the projection are connected by a line segment contained in the projection). In particular, I'll give a proof of the fact that the image of SO(n) under any linear map to R^2 is convex using some elementary homotopy theory. These kinds of question are not only geometrically interesting but are also useful in solving some optimization problems involved in space travel.