Seminars and Colloquia by Series

Symmetric nonnegative polynomials and sums of squares: mean roads to infinity

Series
Dissertation Defense
Time
Wednesday, May 24, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Jose Acevedo – Georgia Tech –
We study the limits of cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. Using some elements of the representation theory of the symmetric group we introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, which simultaneously allows us to tropicalize its dual cone. Using tropical convexity to describe the tropicalization of the dual cone to symmetric nonnegative forms we then compare both tropicalizations, which turn out to be convex polyhedral cones. We then show that the cones are different for all degrees larger than 4. For even symmetric forms we show that the cones agree up to degree $8$, and are different starting at degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares at the limit.

Some Global Relaxation Methods for Quadratic and Semidefinite Programming

Series
Dissertation Defense
Time
Tuesday, May 9, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005 and ONLINE
Speaker
Shengding Sun – Georgia Tech –

Zoom link: https://gatech.zoom.us/meeting/96948840253

Quadratic programming and semidefinite programming are vital tools in discrete and continuous optimization, with broad applications. A major challenge is to develop methodologies and algorithms to solve instances with special structures. For this purpose, we study some global relaxation techniques to quadratic and semidefinite programming, and prove theoretical properties about their qualities. In the first half we study the negative eigenvalues of $k$-locally positive semidefinite matrices, which are closely related to the sparse relaxation of semidefinite programming. In the second half we study aggregations of quadratic inequalities, a tool that can be leveraged to obtain tighter relaxation to quadratic programming than the standard Shor relaxation. In particular, our results on finiteness of aggregations can potentially lead to efficient algorithms for certain classes of quadratic programming instances with two constraints.

Frames via Unilateral Iterations of Bounded Operators

Series
Dissertation Defense
Time
Thursday, April 27, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Victor Bailey – Georgia Tech –

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, $\{f_k\}_{k \in I} \subset H$, be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, $\{T^n \varphi\}_{n=0}^{\infty} \subset H$, to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame. This is joint work with Carlos Cabrelli.

Join Zoom meeting:  https://gatech.zoom.us/j/96113517745

Two graph classes with bounded chromatic number

Series
Dissertation Defense
Time
Monday, April 17, 2023 - 09:30 for 1 hour (actually 50 minutes)
Location
Skiles 114 (or Zoom)
Speaker
Joshua Schroeder – Georgia Tech –

Please Note: Zoom: https://gatech.zoom.us/j/98256586748?pwd=SkJLZ3ZKcjZsM0JkbGdyZ1Y3Tk9udz09 Meeting ID: 982 5658 6748 Password: 929165

A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.

Counting Hamiltonian cycles in planar triangulations

Series
Dissertation Defense
Time
Thursday, April 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaonan Liu – Georgia Tech –

Whitney showed that every planar triangulation without separating triangles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $n/ \log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles.

First passage percolation: exceptional events and asymptotic behavior of invasion restricted geodesics

Series
Dissertation Defense
Time
Thursday, April 13, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Harper – Georgia Tech –

 In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results that show that if the sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if the sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when the weight of a long path is unusually small by considering an analogous construction to Kesten's incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.-K. Lam.

Finally, we discuss a result related to work of Damron-Lam-Wang ('16) that the growth of the passage time to distance $n$ ($\mathbb{E}T(0,\partial B(n))$, where $B(n) = [-n,n]^2$)  has the same order (up to a constant factor) as the sequence $\mathbb{E}T^{\text{inv}}(0,\partial B(n))$. This second passage time is the minimal total weight of any path from 0 to $\partial B(n)$ that resides in a certain embedded invasion percolation cluster. We discuss a result that claims this constant factor cannot be taken to be 1. This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structures. This was joint work with M. Damron. 

 

 

Quantum trace maps for skein algebras

Series
Dissertation Defense
Time
Friday, April 7, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tao Yu – Georgia Institute of Technology –

We study quantizations of SL_n-character varieties, which appears as moduli spaces for many geometric structures. Our main goal is to establish the existence of several quantum trace maps. In the classical limit, they reduce to the Fock-Goncharov trace maps, which are coordinate charts on moduli spaces of SL_n-local systems used in higher Teichmuller theory. In the quantized theory, the algebras are replaced with non-commutative deformations. The domains of the quantum trace maps are the SL_n-skein algebra and the reduced skein algebra, and the codomains are quantum tori, which are non-commutative analogs of Laurent polynomial algebras. In this talk, I will review the classical theory and sketch the definition of the quantum trace maps.

Algebraic and semi-algebraic invariants on quadrics

Series
Dissertation Defense
Time
Friday, July 22, 2022 - 08:30 for 2 hours
Location
Skiles 006 and Zoom meeting (https://gatech.zoom.us/j/96755126860)
Speaker
Jaewoo Jung – Georgia Institute of Technology –

Dissertation defense information

Date and Time: July 22, 2022, 08:30 am - 10:30 am (EST)

Location:

  • Skiles 006 (In-person)
  • Zoom meeting (Online): https://gatech.zoom.us/j/96755126860

 

Summary

This dissertation consists of two topics concerning algebraic and semi-algebraic invariants on quadrics.

 The ranks of the minimal graded free resolution of square-free quadratic monomial ideals can be investigated combinatorially. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the quadratic ideals in terms of properties on the corresponding simple graphs. Our main theorem is the graph decomposition theorem that provides a bound on the regularity of a quadratic monomial ideal. By combining the main theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of square-free quadratic monomial ideals.

 The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. This project is motivated by an intriguing (lower) bound of the Hankel index of a variety by an algebraic invariant, the Green-Lazarsfeld index, of the variety. We study the Hankel index of the image of the projection of rational normal curves away from a point. As a result, we found a new rank of the center of the projection which detects the Hankel index of the rational curves. It turns out that the rational curves are the first class of examples that the lower bound of the Hankel index by the Green-Lazarsfeld index is strict.

 

Advisor: Dr. Grigoriy Blekherman, School of Mathematics, Georgia Institute of Technology

Committee:

  • Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
  • Dr. Anton Leykin, School of Mathematics, Georgia Institute of Technology
  • Dr. Rainer Sinn, Institute of Mathematics, Universität Leipzig
  • Dr. Josephine Yu, School of Mathematics, Georgia Institute of Technology

 

Application of Circle Method in Five Number Theory Problems

Series
Dissertation Defense
Time
Friday, July 15, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hamed Mousavi – Georgia Institute of Technology –

This thesis consists of three applications of the circle method in number theory problems. In the first part, we study the $p-$divisibility of the central binomial coefficients. For a certain set of large prime numbers, we prove that there are infinitely many integers $n$, which $\binom{2n}{n}$ has these primes with unexpectedly small multiplicity in its prime factorization. This result is related to an open problem conjectured by Graham, stating that there are infinitely many integers $n$ which the binomial coefficients $\binom{2n}{n}$ is coprime with $105$. The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes.

In the second part, we discover an unexpected cancellation on the sums involving the exponential functions. Applying this theorem on the first terms of the Ramanujan-Hardy-Rademacher expansion gives us a natural proof of a ``weak" pentagonal number theorem. We find several similar upper bounds for many different partition functions. Additionally, we prove another set of ``weak" pentagonal number theorems for the primes, which allows us to count the number of primes in certain intervals with small error. Finally, we show an approximate solution to the Prouhet-Tarry-Escott problem using a similar technique. The core of the proofs is an involved circle method argument.

The third part of this thesis is about finding an endpoint $\ell^p-$improving inequality for an ergodic sum involving the primes. As the set of the prime is almost full-dimensional, the question on the endpoint becomes more interesting, because we want to bound $\ell^{\infty}$ to $\ell^{1}$ operator. The weak-type inequality we propose depends on the assumption of the Generalized Riemann Hypothesis. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a $\log $ factor.  The proof is based on a circle method argument and careful use of the Ramanujan sums.

Factorization theorems and canonical representations for generating functions of special sums

Series
Dissertation Defense
Time
Wednesday, July 6, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Hybrid - Skiles 006 and Zoom
Speaker
Maxie Dion Schmidt – Georgia Tech –
ABSTRACT: This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. These results are a main focus of the author's publications from 2017-2021. The last primary (non-appendix) section of the thesis explores the topic of how to best rigorously define a so-termed "canonically best" matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type. COMMITTEE:
  • Dr. Josephine Yu, Georgia Tech
  • Dr. Matthew Baker, Georgia Tech
  • Dr. Rafael de la Llave, Georgia Tech
  • Dr. Jayadev Athreya, University of Washington
  • Dr. Bruce Berndt, University of Illinois at Urbana-Champaign
HYBRID FORMAT LOCATIONS: LINKS:

 

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