Seminars and Colloquia by Series

Matchings in hypergraphs defined by groups

Series
Graph Theory Seminar
Time
Tuesday, February 14, 2023 - 03:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alp MuyesserUniversity College London

When can we find perfect matchings in hypergraphs whose vertices represent group elements and whose edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups.  Joint work with Alexey Pokrosvkiy

Generalized square knots, homotopy 4-spheres, and balanced presentations

Series
Geometry Topology Seminar
Time
Monday, February 13, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 303)
Speaker
Jeff MeierWestern Washington University

We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.

Handle numbers of nearly fibered knots

Series
Geometry Topology Seminar
Time
Monday, February 13, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
University of Georgia & Zoom
Speaker
Ken BakerUniversity of Miami

In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4.  Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work in progress with Fabiola Manjarrez-Gutierrez.

Embedded solitary internal waves

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Paul MilewskiUniversity of Bath, UK/Courant Institute NYU

Please Note: We expect to have an online option available: https://gatech.zoom.us/j/98355006347

The ocean and atmosphere are density stratified fluids. A wide variety of gravity waves propagate in their interior, redistributing energy and mixing the fluid, affecting global climate balances.  Stratified fluids with narrow regions of rapid density variation with respect to depth (pycnoclines) are often modelled as layered flows. We shall adopt this model and examine horizontally propagating internal waves within a three-layer fluid, with a focus on mode-2 waves which have oscillatory vertical structure and whose observations and modelling have only recently started. Mode-2 waves (typically) occur within the linear spectrum of mode-1 waves (i.e. they travel at lower speeds than mode-1 waves), and thus mode-2 solitary waves are  generically associated with an unphysical resonant mode-1 infinite oscillatory tail. We will show that these tail oscillations can be found to have zero amplitude, thus resulting in families of localised solutions (so called embedded solitary waves) in the Euler equations. This is the first example we know of embedded solitary waves in the Euler equations, and would imply that these waves are longer lived that previously thought.

Excluding a line from complex-representable matroids

Series
Algebra Seminar
Time
Monday, February 13, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Zach WalshGeorgia Institute of Technology

The extremal function of a class of matroids maps each positive integer n to the maximum number of elements of a simple matroid in the class with rank at most n. We will present a result concerning the role of finite groups in minor-closed classes of matroids, and then use it to determine the extremal function for several natural classes of representable matroids. We will assume no knowledge of matroid theory. This is joint work with Jim Geelen and Peter Nelson.

Computation with sequences of neural assemblies

Series
ACO Student Seminar
Time
Friday, February 10, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Max DabagiaGeorgia Tech CS

Assemblies are subsets of neurons whose coordinated excitation is hypothesized to represent the subject's thinking of an object, idea, episode, or word. Consequently, they provide a promising basis for a theory of how neurons and synapses give rise to higher-level cognitive phenomena. The existence (and pivotal role) of assemblies was first proposed by Hebb, and has since been experimentally confirmed, as well as rigorously proven to emerge in the model of computation in the brain recently developed by Papadimitriou & Vempala. In light of contemporary studies which have documented the creation and activation of sequences of assemblies of neurons following training on tasks with sequential decisions, we study here the brain's mechanisms for working with sequences in the assemblies model of Papadimitriou & Vempala.  We show that (1) repeated presentation of a sequence of stimuli leads to the creation of a sequence of corresponding assemblies -- upon future presentation of any contiguous sub-sequence of stimuli, the corresponding assemblies are activated and continue until the end of the sequence; (2) when the stimulus sequence is projected to two brain areas in a "scaffold", both memorization and recall are more efficient, giving rigorous backing to the cognitive phenomenon that memorization and recall are easier with scaffolded memories; and (3) existing assemblies can be quite easily linked to simulate an arbitrary finite state machine (FSM), thereby capturing the brain's ability to memorize algorithms. This also makes the assemblies model capable of arbitrary computation simply in response to presentation of a suitable stimulus sequence, without explicit control commands. These findings provide a rigorous, theoretical explanation at the neuronal level of complex phenomena such as sequence memorization in rats and algorithm learning in humans, as well as a concrete hypothesis as to how the brain's remarkable computing and learning abilities could be realized.

 

Joint work with Christos Papadimitriou and Santosh Vempala.

On Extremal Polynomials: 5. Upper Estimates and Irregularity of Widom Factors

Series
Mathematical Physics and Analysis Working Seminar
Time
Friday, February 10, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Burak HatinogluGeorgia Institute of Technology

We will continue to focus on Cantor-type sets we introduced last week. Using them we will consider maximal growth rate and irregular behavior of Widom factors (nth Chebyshev number divided by nth power of logarithmic capacity). We will also discuss a recent result of Jacob Christiansen, Barry Simon and Maxim Zinchenko, which shows that Widom factors of Parreau-Widom sets are uniformly bounded.

Effective deep neural network architectures for learning high-dimensional Banach-valued functions from limited data

Series
Applied and Computational Mathematics Seminar
Time
Friday, February 10, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and https://gatech.zoom.us/j/98355006347
Speaker
Nick DexterFlorida State University

In the past few decades the problem of reconstructing high-dimensional functions taking values in abstract spaces from limited samples has received increasing attention, largely due to its relevance to uncertainty quantification (UQ) for computational science and engineering. These UQ problems are often posed in terms of parameterized partial differential equations whose solutions take values in Hilbert or Banach spaces. Impressive results have been achieved on such problems with deep learning (DL), i.e. machine learning with deep neural networks (DNN). This work focuses on approximating high-dimensional smooth functions taking values in reflexive and typically infinite-dimensional Banach spaces. Our novel approach to this problem is fully algorithmic, combining DL, compressed sensing, orthogonal polynomials, and finite element discretization. We present a full theoretical analysis for DNN approximation with explicit guarantees on the error and sample complexity, and a clear accounting of all sources of error. We also provide numerical experiments demonstrating the efficiency of DL at approximating such high-dimensional functions from limited data in UQ applications.
 

The controllability function method and the feedback synthesis problem for a robust linear system

Series
CDSNS Colloquium
Time
Friday, February 10, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Tetiana RevinaV. N. KARAZIN KHARKIV NATIONAL UNIVERSITY

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

The talk is about controllability for uncertain linear systems. Our approach is 
based on the Controllability Function (CF) method proposed by V.I. Korobov in 
1979. The CF method is a development of the Lyapunov function method and the 
dynamic programming method. The CF includes both approaches at a certain values 
of parameters. The main advance of the CF method is finiteness of the time of motion 
(settling-time function). 
In the talk the feedback synthesis problem for a chain of integrators system 
with continuous bounded unknown perturbations is considered. This problem consist 
in constructing a control in explicit form which depends on phase coordinates and 
steers an arbitrary initial point from a neighborhood of the origin to the origin in a 
finite time (settling-time function). Besides the control is satisfies some preassigned 
constrains. The range of the unknown perturbations such that the control solving the 
synthesis problem for the system without the perturbations also solves the synthesis 
problem for the perturbed system are found. This study shows the relations between 
the range of perturbations and the bounds of the settling-time function.
In particular the feedback synthesis problem for the motion of a material 
point with allowance for friction is solved. 
Keywords: chain of integrators, finite-time stability, robust control, settling 
time estimation, uncertain systems, unknown bounded perturbation

Groups, Extensions, and Cohomology

Series
Algebra Student Seminar
Time
Friday, February 10, 2023 - 10:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Akash NarayananGeorgia Tech

Group extensions are a natural way of building complicated groups out of simpler ones. We will develop techniques used to study group extensions. Through these techniques, we will motivate and discuss connections to the cohomology of groups. 

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