Georgia Institute of Technology College of Sciences

Proving conjectures is an essential part of our job as mathematicians. Another essential part is to come up with plausible conjectures. In the talk, we focus on this part. We present a new twist to an old method from computer algebra for detecting recurrence equations of infinite sequences of which only the first few terms are known. By applying this new version systematically to all the entries of the Online Encyclopedia of Integer Sequences, we detected a number of potential recurrence equations that could not be found by the classical methods. Some of these have meanwhile been proven. This is joint work with Christoph Koutschan. 

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Titile: Lattice Reduction 
                                                                                                           
Abstract: It is well known how to go from an exact number (e.g. 1/3) into an approximation (e.g. 0.333). But how can we get back? At first glance, this seems impossible, because some information got lost during the approximation. However, there are techniques for doing this and similar seemingly magic tricks. We will discuss some such tricks that rely on an algorithm for finding short vectors in integer lattices.          

Inspired by colored Khovanov homology, for any knot K in the 3-sphere, we define n-colored knot Floer homology as the limit of the cobordism maps from the (full) link Floer homology of the (n,mn)-cable of K to the (full) link Floer homology of  (n,(m+1)n)-cable as m goes to infinity. Colored knot Floer homology is graded by Alexander multi-grading and Maslov grading and it is finite dimensional at each fixed degree. We discuss the module structure of this invariant and overview some examples. This is a joint work with Eugene Gorsky and Beibei Liu.

This talk presents two recent advances in Neural Network with Local Converging Inputs (NNLCI) —a novel surrogate model for efficiently resolving nonlinear flow dynamics at modest computational cost

First, a powerful and efficient technique is introduced to extend NNLCI to unstructured computational fluid dynamics. The framework is validated on two-dimensional inviscid supersonic flow in channels with varying bump geometries and positions. The NNLCI model accurately captures key flowfield structures and dynamics, including regions with highly nonlinear shock interactions while achieving a speedup of more than two orders of magnitude.

Second, we conduct a comprehensive benchmark study to compare our method with current state-of-the-art AI-based PDE solvers. Across representative hyperbolic conservation law problems, NNLCI consistently deliver superior accuracy, efficiency and robustness in resolving challenging sharp discontinuities and wave interactions. The work provides practical guidance for model selection in scientific machine learning applications

In this talk we will review past and recent results pertaining to the emerging field of integrable space-time nonlocal nonlinear evolution equations. In particular, we will discuss blow-up in finite time of soliton solutions as well as the physical derivations of many integrable nonlocal systems.

An $S_k$-set is a subset of a group whose $k$-tuples have distinct products. We give explicit constructions of large $S_k$-sets in the groups $\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)$ and of large $S_2$-sets in $\mathrm{Sym}(n)\times\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)\times\mathrm{Alt}(n)$, as well as some probabilistic constructions for 'nice' groups. We show that $k$ is even and $\Gamma$ has a normal abelian subgroup with bounded index then any $S_k$-set has size at most $(1-\varepsilon)|\Gamma|^{1/k}$. We describe some connections between $S_k$-sets and extremal graph theory. We determine up to a constant factor the largest possible minimum outdegree in a digraph with no subgraph in $\{C_{2,2},\ldots,C_{k,k}\}$, where $C_{\ell,\ell}$ is the orientation of $C_{2\ell}$ with two maximal directed $\ell$-paths. Contrasting with undirected cycles, the extremal minimum outdegree for $\{C_{2,2},\ldots,C_{k,k}\}$ is much smaller than that for any $C_{\ell,\ell}$. We count the directed Hamilton cycles in one of our constructions to improve the upper bound for a problem on Hamilton paths introduced by Cohen, Fachini, and Körner. Joint work with Michael Tait.

I will discuss several classical problems in number theory about representation of numbers and forms by quadratic forms and related counting results. We will describe several applications such as pair correlations and eigenvalue statistics for quantum systems.
Then we will move to related problems about irrational forms, such as the Oppenheim conjecture and explain how homogeneous dynamics can help to tackle such problems.
The talk will be self contained and hopefully accessible.

The theory of dynamical frames arose from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of future states of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call central frame representations,  whose  frame generators are unique (up to equivalence). In this talk, we will address the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han and Larson in 2000 for group representation frames, but also proves that the frame vectors for any system of the form $\{A_1^{n_1}\cdots A_k^{n_k}: n_j\geq 0\}$, where  $A_1,...,A_k \in B(H)$ commute,  are equivalent. This is joint work with Deguang Han, Keri Kornelson, David Larson, and Rui Liu.

Dehn surgery (with a fixed slope p/q) associates, to a knot in S^3, a 3-manifold M with first homology isomorphic to the integers mod p. One might wonder if this function is one-to-one or onto; Cameron Gordon (1978) conjectured that it is never injective nor surjective. The surjectivity case was established a decade later, while the injectivity case was only recently proven by Hayden, Piccirillo, and Wakelin. We will survey this latter result and its proof. 

We explore the modular properties of generating functions for Hurwitz class numbers endowed with level structure. Our work is based on an inspection of the weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$, extending the work of Duke, Imamo\={g}lu and T\'{o}th in the level $4$ setting. We construct a higher level analogue of Zagier's Eisenstein series and a preimage under the $\xi_{\frac{1}{2}}$-operator.  We deduce a linear relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers, giving rise to a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$ for $N > 1$ odd and square-free. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla--Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo\={g}lu. I wil lbe discussing joint work with Andreas Mono and Ngoc Trinh Le.

We will start with a 15-minute presentation by Noah Solomon and continue with a free discussion.

One of the foundational ideas in modern machine learning is the scaling hypothesis: that machine learning models will improve in a predictable manner, with each doubling of resources leading to a commensurate improvement in abilities.  These were formalized for large language models in the Kaplan et al. scaling laws.

This is an almost entirely empirically observed law, which motivates the development probabilistic models that can explain these laws and to ultimately inform how to answer fundamental questions, such as: what can improve these laws? Or what causes them to break?

In this talk I’ll focus on a simple random matrix model of these scaling laws, the power law random features model, which motivates new iteration of stochastic algorithms which have the potential to change these scaling laws.  This random matrix model is not fully solved, and there are many open questions, both in pure probability and machine learning that rise in this study.

Abstract:

Low-rank adaptation (LoRA) has become the de-facto state-of-the-art method for parameter efficient fine-tuning of large-scale, pre-trained neural networks.  Similarly, low-rank compression of pre-trained networks has become a widely adopted technique to reduce the parameter count of networks for fast inference on resource constraint devices.  The idea of low-rank methods is based upon the assumption that the weight matrices of overparametrized neural networks are of low-rank.  Thus, a factorization of the weight layers based on truncated singular value decompositions can be employed to reduce the memory footprint of the network.  However, LoRA and its extensions face several challenges in practice, including the need for rank adaptivity, robustness, and computational efficiency during the fine-tuning process.  In this talk, Dr. Schotthoefer investigates mathematical concepts of low-rank training and uses the gained insights to design efficient and robust low-rank training algorithms.

                                                                                        

Speaker’s Bio:

Dr. Steffen Schotthoefer is the current Householder Fellow in the Mathematics in Computation Section at the Oak Ridge National Laboratory (ORNL), affiliated with the Multiscale Methods and Dynamics Group.  Steffen's work centers on creating efficient numerical methods for training and fine-tuning artificial intelligence models in environments with limited resources and at large scales.  He investigates low-rank methods for model compression to minimize the computational cost of neural network training and inference.  In addition, Steffen develops neural network-based surrogate models for scientific domains such as radiation transport and plasma dynamics.  His research aims to tackle the challenges posed by memory and communication bottlenecks in large-scale simulations.  Prior to joining ORNL, Steffen completed his Ph.D. in Applied Mathematics at Karlsruhe Institute of Technology, Germany, focusing on neural network-based surrogate modeling for radiation transport.  During his doctoral studies, he devised numerical methods for the simulation of kinetic partial differential equations and neural network training, establishing the foundation for his current research.

 

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Depending on how long it takes to discuss these items, I will also attempt to include discussions on:

• Biran-Giroux decompositions of symplectic manifolds • Boothby-Wang bundles and contact plumbings of these • Milnor's fibration theorem for isolated singularities and connections to open book decompositions and Lefschetz fibrations • Open questions and interesting avenues of research

Most of our discussion will, as a side effect, outline the topological structure behind Type IIA String theory (the "topological A-model") which requires a 6-dimensional Calabi-Yau (Kahler) background.

 

In a recent joint work with J. Buzzi, Y. Shi, and J. Yang, given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property.

We then prove that for any measure of maximal $\mathcal F$-entropy, its conditional measures on each leaf must be equivalent to the reference measures.

When the reference measures are equivalent to the leafwise Lebesgue measure, we prove that the log-determinant of $f$ must be cohomologous to a constant.

We will consider several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature. We will also discuss several conjectures on the unique ergodicity and (exponential) equidistribution for the strong unstable foliations of Anosov systems. 

Zoom link: https://gatech.zoom.us/j/92005780980?pwd=ptlx7KdBAbHI3DTvv6V7fjFn27LDaE.1

Meeting ID: 920 0578 0980
Passcode: 604975

A geometric graph consists of a set of points in the plane together with line 

segments between some pairs of points. A convex geometric graph is a geometric graph whose 

points are in convex position. We present some old and new extremal results and applications, 

and their extension to geometric hypergraphs, together with a variety of open problems. 

 

Partly joint work with Zoltan Furedi, Tao Jiang, Sasha Kostochka and Dhruv Mubayi