Seminars and Colloquia by Series

Rational values of the weak saturation limit

Series
Combinatorics Seminar
Time
Friday, April 18, 2025 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruben AscoliGeorgia Institute of Technology

Given a graph $F$, a graph $G$ is weakly $F$-saturated if all non-edges of $G$ can be added in some order so that each new edge introduces a copy of $F$. The weak saturation number $wsat(n,F)$ is the minimum number of edges in a weakly $F$-saturated graph on $n$ vertices. Bollobás initiated the study of weak saturation in 1968 to study percolation processes, which originated in biology and have applications in physics and computer science. It was shown by Alon that for each $F$, there is a constant $w_F$ such that $wsat(n,F) = w_F n + o(n)$. We characterize all possible rational values of $w_F$, proving in particular that $w_F$ can equal any rational number at least $3/2$. The techniques involve a combination of random and deterministic constructions and structural methods. Joint work with Xiaoyu He.

Programmable Matter and Emergent Computation

Series
Math Physics Seminar
Time
Friday, April 18, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
005
Speaker
Dana RandallGeorgia Tech

Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior.  We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.  By analyzing the Gibbs distribution of various fixed-magnetization models from equilibrium statistical mechanics, we show that particles moving stochastically according to local affinities can solve various useful collective tasks. Finally, we will briefly introduce new tools that may prove fruitful in nonequilibrium settings as well.

Random matrices and logarithmically correlated fields

Series
School of Mathematics Colloquium
Time
Thursday, April 17, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Paul BourgadeNYU

The Liouville quantum gravity measure is a properly normalized exponential of 2d log-correlated fields, such as the Gaussian free field. It is the volume form for the scaling limit of random planar maps and numerous statistical physics models. I will explain how this random measure naturally appears in random matrix theory either in space time from random matrix dynamics, or in space from the characteristic polynomial of random normal matrices. A 3d log-correlated field also naturally emerges in random matrix theory, from dynamics on non-Hermitian matrices.

Relations between rational functions and an analog of the Tits alternative

Series
Number Theory
Time
Wednesday, April 16, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tom TuckerRochester University

Work of Levin and Przytycki shows that if two non-special rational
functions f and g of degree $> 1 $over $\mathbb{C}$ share the same set of
preperiodic points, there are $m$, $n$, and $r$ such that $f^m g^n = f^r$.
In other words, $f$ and $g$ nearly commute.  One might ask if there are
other sorts of relations non-special rational functions $f$ and $g$ over $\mathbb{C}$
might satisfy when they do not share the same set of preperiodic
points.  We will present a recent proof of Beaumont that shows that
they may not, that if f and g do not share the same set of preperiodic
points, then they generate a free semi-group under composition.  The
proof builds on work of Bell, Huang, Peng, and the speaker, and uses a
ping-pong lemma similar to the one used by Tits in his proof of the
Tits alternative for finitely generated linear groups.

Spherical maximal functions and fractal dimensions

Series
Analysis Seminar
Time
Wednesday, April 16, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joris RoosUniversity of Massachusetts Lowell

The talk will be about spherical maximal functions with a supremum restricted to a given set $E$. The sharp $L^p$ improving regions of these operators depend on various fractal dimensions of the set $E$ such as the Minkowski dimension, quasi-Assouad dimension and certain intermediate dimensions.

A surprising aspect is that the sharp exponent regions need not be polygons; instead their boundary may follow an arbitrary convex curve in some critical region.

The talk will be about some old and some new results.

If time allows, we will also discuss a related fractal variant of the local smoothing problem for the wave equation.

Strong parity edge colorings of graphs (Peter Bradshaw, UIUC)

Series
Graph Theory Seminar
Time
Tuesday, April 15, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Peter BradshawUIUC

We consider the strong parity edge coloring problem, which aims to color the edges of a graph G so that in each open walk on G, some color appears an odd number of times.  We show that this problem is equivalent to the problem of embedding a graph in a vector space over F2 so that the number of difference vectors attained at the edges is minimized. Using this equivalence, we achieve the following:

1. We characterize graphs on n vertices that can be embedded with ceil(log_2 n) difference vectors, answering a question of Bunde, Milans, West, and Wu.

2. We show that the number of colors needed for a strong parity edge coloring of K_{s,t} is given by the Hopf-Stiefel function, confirming a conjecture of Bunde, Milans, West, and Wu.

3. We find an asymptotically optimal embedding for the power of a path.

This talk is based on joint work with Sergey Norin and Doug West.

Sharp late-time asymptotics for quasilinear wave equations satisfying a weak null condition

Series
PDE Seminar
Time
Tuesday, April 15, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sung-Jin OhUC Berkeley

We study the sharp asymptotics for a class of quasilinear wave equations satisfying a weak null condition but not the classical null condition in three spatial dimensions. We prove that the asymptotics are very different from those for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates.

Moreover, we show that any solution that decays faster than expected in a compact spatial region must vanish identically. The talk is based on joint work in progress with Jonathan Luk and Dongxiao Yu. 

Cosmetic surgeries and Chern-Simons invariants

Series
Geometry Topology Seminar
Time
Monday, April 14, 2025 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tye LidmanNorth Carolina State University

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

Contact type hypersurfaces in small symplectic 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, April 14, 2025 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tom MarkUniversity of Virginia

A codimension-1 submanifold embedded in a symplectic manifold is called “contact type” if it satisfies a certain convexity condition with respect to the symplectic structure. Given a symplectic manifold X it is natural to ask which manifolds Y can arise as contact type hypersurfaces. We consider this question in dimension 4, which appears much more constrained than higher dimensions; in particular we review evidence that no homology 3-sphere can arise as a contact type hypersurface in R^4 except the 3-sphere. We exhibit an obstruction for a contact 3-manifold to embed in certain closed symplectic 4-manifolds as the boundary of a Liouville domain---a slightly stronger condition than contact type---and explore consequences for the symplectic topology of small rational surfaces and potential applications to smooth 4-dimensional topology.

Optimal Approximation and Generalization Analysis for Deep Neural Networks for Solving Partial Differential Equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 14, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/94954654170
Speaker
Yahong YangPenn State

Neural networks have become powerful tools for solving Partial Differential Equations (PDEs), with wide-ranging applications in engineering, physics, and biology. In this talk, we explore the performance of deep neural networks in solving PDEs, focusing on two primary sources of error: approximation error, and generalization error. The approximation error captures the gap between the exact PDE solution and the neural network’s hypothesis space. Generalization error arises from the challenges of learning from finite samples. We begin by analyzing the approximation capabilities of deep neural networks, particularly under Sobolev norms, and discuss strategies to overcome the curse of dimensionality. We then present generalization error bounds, offering insight into when and why deep networks can outperform shallow ones in solving PDEs.

Pages