Seminars and Colloquia by Series

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. 

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.

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.

Springer fibers and Richardson varieties

Series
Algebra Seminar
Time
Monday, April 14, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Steven KarpUniversity of Notre Dame

Please Note: There will be a pre-seminar from 10:55 to 11:15 in Skiles 005.

A Springer fiber is the set of complete flags in Cn which are fixed by a given nilpotent matrix. It is a fundamental object of study in geometric representation theory and algebraic combinatorics. The irreducible components of a Springer fiber are indexed by combinatorial objects called standard Young tableaux. It is an open problem to describe geometric properties of these components (such as their singular loci and cohomology classes) in terms of the combinatorics of tableaux. We initiate a new approach to this problem by characterizing which irreducible components are equal to Richardson varieties, which are comparatively much better understood. Another motivation comes from Lusztig's recent study of the cell decomposition of the totally nonnegative part of a Springer fiber into totally positive Richardson cells. This is joint work in progress with Martha Precup.

Fractional Brownian motions, Kerov's CLT, and semiclassical dynamics of Gaussian wavepackets

Series
CDSNS Colloquium
Time
Friday, April 11, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Alexander MollReed College

Please Note: Zoom link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs.  In work in progress with Robert Chang (Rhodes College), we introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with positive Hurst index s -1/2.  For a special Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams.  The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp.  The global Gaussian fluctuations depend on s and, if we continue s to negative values, coincides with the process in Kerov's CLT for s = - 1/2.  Although it might be possible to give a direct explanation for this coincidence by regularization, in this talk we give an indirect dynamical explanation by combining (i) results of Eliashberg and Dubrovin for a specific Hamiltonian QFT and (ii) the fact that in Hamiltonian systems, at short time scales, the quantum evolution of pure Gaussian wavepacket initial data agrees statistically with the classical evolution of mixed Gaussian random initial data.

Hypergraph Random Turán Problems and Sidorenko conjecture

Series
Combinatorics Seminar
Time
Friday, April 11, 2025 - 15:15 for 1 hour (actually 50 minutes)
Location
Skilles 005
Speaker
Jiaxi NieGeorgia Institute of Technology

Given an $r$-uniform hypergraph $H$, the random Turán number $\mathrm{ex}(G^r_{n,p},H)$ is the maximum number of edges in an $H$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the Erdős-Rényi random hypergraph with parameter $p$. In the case when $H$ is not r-partite, the problem has been essentially solved independently by Conlon-Gowers and Schacht. In the case when $H$ is $r$-partite, the degenerate case, only some sporadic results are known.

The Sidorenko conjecture is a notorious problem in extremal combinatorics. It is known that its hypergraph analog is not true. Recently, Conlon, Lee, and Sidorenko discovered a relation between the Sidorenko conjecture and the Turán problem. 

 In this talk, we introduce some recent results on the degenerate random Turan problem and its relation to the hypergraph analog of the Sidorenko conjecture.

Solvability of Some Integro-Differential Equations with Transport and Concentrated Sources

Series
Math Physics Seminar
Time
Friday, April 11, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Vitali VougalterUniversity of Toronto

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.

https://gatech.zoom.us/j/94295986362?pwd=8euEJ3ojkWl5c3Y3hLyXTiKBts3Rrq.1

The Airy-beta line ensemble

Series
Stochastics Seminar
Time
Thursday, April 10, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vadim GorinUniversity of California, Berkeley

Beta-ensembles generalize the eigenvalue distributions of self-adjoint
real, complex, and quaternion matrices for beta=1,2, and 4,
respectively. These ensembles naturally extend to two dimensions by
introducing operations such as corner truncation, addition, or
multiplication of matrices. In this talk, we will explore the edge
asymptotics of the resulting two-dimensional ensembles. I will present
the Airy-beta line ensemble, a universal object that governs the
asymptotics of time-evolving largest eigenvalues. This ensemble
consists of an infinite collection of continuous random curves,
parameterized by beta. I will share recent progress in developing a
framework to describe this remarkable structure.

Pages