Seminars and Colloquia by Series

Max-Intersection Completeness of Neural Codes and the Neural Ideal

Series
Algebra Seminar
Time
Monday, January 22, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alexander Ruys de PerezGeorgia Tech

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:30 am to noon in Skiles 005.

A neural code C on n neurons is a collection of subsets of {1,2,...,n} which is used to encode the intersections of subsets U_1, U_2,...,U_n of some topological space. The study of neural codes reveals the ways in which geometric or topological properties can be encoded combinatorially. A prominent example is the property of max-intersection completeness: if a code C contains every possible intersection of its maximal codewords, then one can always find a collection of open convex U_1, U_2,..., U_n for which C is the code. In this talk I will answer a question posed by Curto et al. (2018), which asks if there is a way of determining max-intersection completeness from examination of the neural ideal, an algebraic counterpart to the neural code.

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 ZhuU 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. 

Hidden Convexity, Rotation Matrices, and Algebraic Topology

Series
Algebra Student Seminar
Time
Thursday, January 18, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Clough 262
Speaker
Kevin ShuGeorgia Tech

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.

Finite Generation of the Terms of the Johnson Filtration

Series
Geometry Topology Student Seminar
Time
Wednesday, January 17, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan MinahanGeorgia Tech

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.

Global Solutions For Systems of Quadratic Nonlinear Schrödinger Equations in 3D

Series
PDE Seminar
Time
Tuesday, January 16, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Boyang SuUniversity of Chicago


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.
 

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 VandeBogertNotre 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 PayneUT 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 ChenNYU

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 ZhangPrinceton 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.

Geometric Structures for the G_2’ Hitchin Component

Series
Geometry Topology Seminar
Time
Monday, January 8, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Parker EvansRice University

Abstract: Fundamental to our understanding of Teichm\"uller space T(S) of a closed oriented genus $g \geq 2$ surface S are two different perspectives: one as connected  component in the  PSL(2,\R) character variety  \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a higher rank generalization of T(S). In this talk, we discuss a new geometric structures (i.e., (G,X)-structures) interpretation of Hit(S, G_2'), where G_2' is the split real form of the exceptional complex simple Lie group G_2.


After discussing the motivation and background, we will present some of the main ideas of the theorem, including a family of almost-complex curves
that serve as bridge between the geometric structures and representations.

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