Seminars and Colloquia by Series

Algebraization theorems in p-adic geometry

Series
Job Candidate Talk
Time
Tuesday, January 30, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Abhishek Oswal – Michigan State University –

In recent years, algebraization theorems arising from model theory, in particular o-minimality, have been a crucial ingredient in several breakthroughs in arithmetic geometry and Hodge theory. In this talk, I'll present some of my recent work on p-adic versions of these model theoretic algebraization criteria, with a focus on two different applications of this circle of ideas. The first being an algebraization theorem in the context of Shimura varieties, which are vaguely speaking moduli spaces of Hodge structures. The second being in the context of non-abelian Hodge theory, in the setting of moduli spaces of flat connections and local systems.

Zoom: https://gatech.zoom.us/j/95425627723

Measure classification problems in smooth dynamics

Series
Job Candidate Talk
Time
Thursday, January 25, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006; Zoom streaming available
Speaker
Asaf Katz – U Michigan –

Please Note: Zoom link: https://gatech.zoom.us/j/98245747313?pwd=RmFtcmlWYjBncXJTOU00NFMvSVNsZz09 Meeting ID: 982 4574 7313 Passcode: SoM

Abstract: Classifying the invariant measures for a given dynamical system represents a fundamental challenge.

In the field of homogeneous dynamics, several important theorems give us an essentially complete picture. Moving away from homogeneous dynamics — results are more difficult to come by. A recent development in Teichmuller dynamics — the celebrated magic wand theorem of Eskin–Mirzakhani, proved by their factorization technique gives one such example.
 
I will explain an implementation of the factorization technique by Eskin–Mirzakhani in smooth dynamics, aiming to classify u-Gibbs states for non-integrable Anosov actions. Moreover, I will try to explain some applications of the theorem, including a result of Avila–Crovosier–Eskin–Potrie–Wilkinson–Zhang towards Gogolev’s conjecture on actions over the 3D torus.

Topology, geometry and adaptivity in soft and living matter

Series
Job Candidate Talk
Time
Tuesday, January 23, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vishal Patil – Stanford University –

Title: Topology, geometry and adaptivity in soft and living matter

Abstract:

Topology and adaptivity play fundamental roles in controlling the dynamics of biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. How topological rules govern the self-adaptive dynamics of living matter remains poorly understood. Here we investigate the interplay between topology, geometry and reconfigurability in knotted and tangled matter. We first identify topological counting rules which predict the relative mechanical stability of human-designed knots, by developing a mapping between elastic knots and long-range ferromagnetic spin systems. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments controls their emergent topological state. To further understand how tangled matter, along with more general biological networks, adapt to their surroundings, we introduce a theory of adaptive elastic networks which can learn mechanical information. By identifying how topology and adaptivity produce stable yet responsive structures, these results have applications in understanding broad classes of adaptive, self-optimizing biological systems.

 

Zoom: https://gatech.zoom.us/j/93619173236?pwd=ZGNRZUZ2emNJbG5pRzgzMnlFL1dzQT09

 

 

Symmetry-Preserving Machine Learning: Theory and Applications

Series
Job Candidate Talk
Time
Thursday, January 18, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Wei Zhu – U Massachusetts Amherst –

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. 

Cohomology of Line Bundles in Positive Characteristic

Series
Job Candidate Talk
Time
Tuesday, January 16, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Keller VandeBogert – Notre Dame –

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.

Point counting over finite fields and the cohomology of moduli spaces of curves

Series
Job Candidate Talk
Time
Thursday, January 11, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sam Payne – UT Austin –

Algebraic geometry studies solution sets of polynomial equations. For instance, over the complex numbers, one may examine the topology of the solution set, whereas over a finite field, one may count its points. For polynomials with integer coefficients, these two fundamental invariants are intimately related via cohomological comparison theorems and trace formulas for the action of Frobenius. I will discuss the general framework relating point counting over finite fields to topology of complex algebraic varieties and also present recent applications to the cohomology of moduli spaces of curves that resolve longstanding questions in algebraic geometry and confirm more recent predictions from the Langlands program.

Krylov Subspace Methods and Matrix Functions: new directions in design, analysis, and applications

Series
Job Candidate Talk
Time
Thursday, January 11, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tyler Chen – NYU –

Krylov subspace methods (KSMs) are among the most widely used algorithms for a number of core linear algebra tasks. However, despite their ubiquity throughout the computational sciences, there are many open questions regarding the remarkable convergence of commonly used KSMs. Moreover, there is still potential for the development of new methods, particularly through the incorporation of randomness as an algorithmic tool. This talk will survey some recent work on the analysis of the well-known Lanczos method for matrix functions and the design of new KSMs for low-rank approximation of matrix functions and approximating partial traces and reduced density matrices. 

 

Metric geometric aspects of Einstein manifolds

Series
Job Candidate Talk
Time
Wednesday, January 10, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005, https://gatech.zoom.us/j/95551591205
Speaker
Ruobing Zhang – Princeton University –

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics. My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress in the field. If time permits, I will propose several important open questions.

Probability and variational methods in PDEs — optimal transport, regularity, and universality

Series
Job Candidate Talk
Time
Tuesday, December 12, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/96443370732
Speaker
Tobias Ried – Max Planck Institute for Mathematics in the Sciences, Liepzig, Germany –
In this talk I will present an overview of my research, highlighting in more detail two topics: 
1. A purely variational approach to the regularity theory of optimal transportation, which is analogous to De Giorgi’s strategy for the regularity theory of minimal surfaces. I will show some interesting connections to Wasserstein barycenters, branched transport, and pattern formation in materials science, as well as applications in density functional theory. 
2. Variational methods for a singular stochastic PDE describing the magnetization ripple, a microstructure in thin-film ferromagnets triggered by the poly-crystallinity of the sample. I will describe how the universal character of the magnetization ripple can be addressed using variational methods based on Γ-convergence.

The Poisson point process and an application to semisimple symmetric spaces

Series
Job Candidate Talk
Time
Thursday, December 7, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006; Streaming available via zoom
Speaker
Amanda Wilkens – UT Austin –

Please Note: Link to join via Zoom: https://gatech.zoom.us/j/93394018195?pwd=MGJZaWIwQUhVYW9ZZDFoWWFOc29EZz09 Meeting ID: 933 9401 8195 Passcode: SoM

We define and motivate the Poisson point process, which is, informally, a “maximally random” scattering of points in some locally compact, second countable space. We introduce the ideal Poisson--Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove the minimal number of generators of a torsion-free lattice in a higher rank, semisimple Lie group is sublinear in the co-volume of the lattice. We give some intuition for the proof. No prior knowledge on Poisson point processes or symmetric spaces will be assumed.

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