Seminars and Colloquia Schedule

Gram spectrahedra

Series
Student Algebraic Geometry Seminar
Time
Monday, September 16, 2019 - 13:15 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jaewoo JungGeorgia Tech

The structure of sums-of-squares representations of (nonnegative homogeneous) polynomials is one interesting subject in real algebraic geometry. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. In this talk, I will introduce Gram spectrahedron, connection to toric variety, a new result that if a variety $X$ is arithmetically Cohen-Macaulay and a linearly normal variety of almost minimal degree (i.e. $\deg(X)=\text{codim}(X)+2$), then every sum of squares on $X$ is a sum of $\dim(X)+2$ squares.

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 16, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andre WibisonoGeorgia Tech
Sampling is a fundamental algorithmic task. Many modern applications require sampling from complicated probability distributions in high-dimensional spaces. While the setting of logconcave target distribution is well-studied, it is important to understand sampling beyond the logconcavity assumption. We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution on R^n under isoperimetry conditions. We show a convergence guarantee in Kullback-Leibler (KL) divergence assuming the target distribution satisfies log-Sobolev inequality and the log density has bounded Hessian. Notably, we do not assume convexity or bounds on higher derivatives. We also show convergence guarantees in Rényi divergence assuming the limit of ULA satisfies either log-Sobolev or Poincaré inequality. Joint work with Santosh Vempala (arXiv:1903.08568).

The “generating function” of configuration spaces, as a source for explicit formulas and representation stability

Series
Geometry Topology Seminar
Time
Monday, September 16, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nir GadishMassachusetts Institute of Technology

As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.

Continuing the Fraction

Series
Undergraduate Seminar
Time
Monday, September 16, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Doron LubinskyGeorgia Tech

Continued fractions play a key role in number theory, especially in understanding how well we can approximate irrational numbers by rational numbers. They also play an important role in function theory, in understanding how well we can approximate analytic functions by rational functions. We discuss a few of the main achievements of the theory.

Periodic Dynamics of a Local Perturbation in the Isotropic XY Model

Series
Math Physics Seminar
Time
Monday, September 16, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Livia CorsiUniversita' di Roma 3

I will consider the isotropic XY chain with a transverse magnetic field acting on a single site, and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. I will show that, under some conditions, the state approaches a periodic orbit synchronized with the forcing. Moreover I will provide the explicit rate of convergence to the asymptotics. This is a joint work with G. Genovese.

The energy conservation of inhomogeneous Euler equations

Series
PDE Seminar
Time
Tuesday, September 17, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cheng YuUniversity of Florida

In this talk, I will discuss from a mathematical viewpoint some sufficient conditions that guarantee the energy equality for weak solutions. I will mainly focus on a fluid equation example, namely the inhomogeneous Euler equations. The main tools are the commutator Lemmas.  This is a joint work with Ming Chen.

Species network inference under the NMSC

Series
Mathematical Biology Seminar
Time
Wednesday, September 18, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hector BanosGeorgia Tech

When hybridization plays a role in evolution, networks are necessary to describe species-level relationships. In this talk, we show that most topological features of a level-1 species network (networks with no interlocking cycles) are identifiable from gene tree topologies under the network multispecies coalescent model (NMSC). We also present the theory behind NANUQ, a new practical method for the inference of level-1 networks under the NMSC.

A complex analytic approach to mixed spectral problems

Series
Analysis Seminar
Time
Wednesday, September 18, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Speaker
Burak HatinoğluTexas A&M

This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional regular Schroedinger operators, we will discuss complex analytic methods for the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential?

Surface bundles in topology, algebraic geometry, and group theory

Series
Geometry Topology Student Seminar
Time
Wednesday, September 18, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin LanierGeorgia Tech

I will give an introduction to surface bundles and will discuss several places where they arise naturally. A surface bundle is a fiber bundle where the fiber is a surface. A first example is the mapping torus construction for 3-manifolds, which is a surface bundle over the circle. Topics will include a construction of 4-manifolds as well as section problems related to surface bundles. The talk will be based on a forthcoming Notices survey article by Salter and Tshishiku.

John’s ellipsoid is not good for approximation

Series
High Dimensional Seminar
Time
Wednesday, September 18, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Han HuangGeorgia Tech

We study the subject of approximation of convex bodies by polytopes in high dimension.  

For a convex set K in R^n, we say that K can be approximated by a polytope of m facets by a distance R>1 if there exists a polytope of P m facets such that K contains P and RP contains K. 

When K is symmetric, the maximal volume ellipsoid of K is used heavily on how to construct such polytope of poly(n) facets to approximate K. In this talk, we will discuss why the situation is entirely different for non-symmetric convex bodies.

Deep Generative Models in the Diffusion Limit

Series
Stochastics Seminar
Time
Thursday, September 19, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Maxim RaginskyECE Department, University of Illinois at Urbana-Champaign

In deep generative models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. In this talk, based on joint work with Belinda Tzen, I will discuss the diffusion limit of such models, where we increase the number of layers while sending the step size and the noise variance to zero. The resulting object is described by a stochastic differential equation in the sense of Ito. I will first show that sampling in such generative models can be phrased as a stochastic control problem (revisiting the classic results of Föllmer and Dai Pra) and then build on this formulation to quantify the expressive power of these models. Specifically, I will prove that one can efficiently sample from a wide class of terminal target distributions by choosing the drift of the latent diffusion from the class of multilayer feedforward neural nets, with the accuracy of sampling measured by the Kullback-Leibler divergence to the target distribution.

Bounds on Ramsey Games via Alterations

Series
ACO Student Seminar
Time
Friday, September 20, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
He GuoMath, Georgia Tech

In this talk we introduce a refined alteration approach for constructing $H$-free graphs: we show that removing all edges in $H$-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdös and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers that extend results of Conlon–Fox–Grinshpun–He and Fox–He–Wigderson.
Based on joint work with Lutz Warnke.