## Seminars and Colloquia by Series

### Combinatorial Surgery Graphs on Unicellular Maps by Abdoul Karim Sane

Series
Geometry Topology Seminar
Time
Monday, August 29, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Abdoul Karim SaneGeorgia Tech

A map (respectively, a unicellular map) on a genus g surface Sg is the Homeo+(Sg)-orbit of a graph G embedded on Sg such that Sg-G is a collection of finitely many disks (respectively, a single disk). The study of maps was initiated by W. Tutte, who was interested in counting the number of planar maps. However, we will consider maps from a more graph theoretic perspective in this talk. We will introduce a topological operation called surgery, which turns one unicellular map into another. Then, we will address natural questions (such as connectedness and diameter) about surgery graphs on unicellular maps, which are graphs whose vertices are unicellular maps and where two vertices share an edge if they are related by a single surgery. We will see that these problems translate to a well-known combinatorial problem: the card shuffling problem.

### Convergence of denoising diffusion models

Series
Applied and Computational Mathematics Seminar
Time
Monday, August 29, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Valentin DE BORTOLICNRS and ENS Ulm
Generative modeling is the task of drawing new samples from an underlying distribution known only via an empirical measure. There exists a myriad of models to tackle this problem with applications in image and speech processing, medical imaging, forecasting and protein modeling to cite a few.  Among these methods score-based generative models (or diffusion models) are a  new powerful class of generative models that exhibit remarkable empirical performance. They consist of a noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a denoising'' process defined by approximating the time-reversal of the diffusion.

In this talk I will present some of their theoretical guarantees with an emphasis on their behavior under the so-called manifold hypothesis. Such theoretical guarantees are non-vacuous and provide insight on the empirical behavior of these models. I will show how these results imply generalization bounds on denoising diffusion models. This presentation is based on https://arxiv.org/abs/2208.05314

### Weights and Automorphisms of Cyclic Subspace Codes.

Series
Algebra Seminar
Time
Monday, August 29, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 classroom
Speaker
Hunter LehmannGeorgia Institute of Technology

Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup F_{q^n}^* on an F_q-subspace U of F_{q^n}. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference generator of the code. We will investigate the weight distribution for a few categories of cyclic orbit codes, including optimal codes. Further, we want to know when two cyclic orbit codes with the same weight distribution are isometric. To answer this question, we determine the possible automorphism groups for cyclic orbit codes.

### Mapping Class Groups of Sliced Loch Ness Monsters by Ryan Dickmann

Series
Geometry Topology Seminar
Time
Monday, August 22, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Ryan DickmannGeorgia Tech

This talk will focus on surfaces (orientable connected 2-manifolds) with noncompact boundary. Since a general surface with noncompact boundary can be extremely complicated, we will first consider a particular class called Sliced Loch Ness Monsters. We will discuss how to show the mapping class group of any Sliced Loch Ness Monster is uniformly perfect and automatically continuous. Depending on the time remaining, we will also discuss the classification of surfaces with noncompact boundary due to Brown and Messer, and how Sliced Loch Ness Monsters are used to prove results about the mapping class groups of general surfaces.

### Rank inequalities for the knot Floer homology of (1,1)-satellites

Series
Other Talks
Time
Thursday, August 18, 2022 - 09:30 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Weizhe ShenGeorgia Tech

One application of the immersed-curve technique, introduced by Hanselman-Rasmussen-Watson, is to study rank inequalities for Heegaard Floer homology in the presence of certain degree-one maps. Another application, discovered by Chen, is to describe the knot Floer homology of satellite knots with (1,1)-patterns. We will discuss similar rank inequalities for the knot Floer homology of (1,1)-satellites.

### Trellis Decoding And Applications For Quantum Error Correction

Series
Time
Tuesday, August 2, 2022 - 09:45 for
Location
Online
Speaker
Eric SaboSchool Of Math

Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost.

Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products.

Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.

--------------------------------------------------------------------------------------------------

Advisor: Dr. Evans Harrell, School of Mathematics, Georgia Institute of Technology

Committee:
Dr. Evans Harrell, School of Mathematics, Georgia Institute of Technology
Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
Dr. Martin Short, School of Mathematics, Georgia Institute of Technology
Dr. Moinuddin Qureshi, School of Computer Science, Georgia Institute of Technology
Dr. Kenneth Brown, Pratt School of Engineering, Duke University

Reader: Dr. Kenneth Brown, Pratt School of Engineering, Duke University

### Arc-Intersection Queries Amid Triangles in Three Dimensions and Related Problems

Series
Combinatorics Seminar
Time
Tuesday, July 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Esther EzraBar Ilan University

Let T be a set of n triangles in 3-space, and let \Gamma be a family of
algebraic arcs of constant complexity in 3-space. We show how to preprocess T
into a data structure that supports various "intersection queries" for
query arcs \gamma \in \Gamma, such as detecting whether \gamma intersects any
triangle of T, reporting all such triangles, counting the number of
intersection points between \gamma and the triangles of T, or returning the
first triangle intersected by a directed arc \gamma, if any (i.e., answering
arc-shooting queries). Our technique is based on polynomial partitioning and
other tools from real algebraic geometry, among which is the cylindrical
algebraic decomposition.

Our approach can be extended to many other intersection-searching problems in
three and higher dimensions. We exemplify this versatility by giving an
efficient data structure for answering segment-intersection queries amid a set
of spherical caps in 3-space, and we lay a roadmap for extending our approach
to other intersection-searching problems.

Joint work with Pankaj Agarwal, Boris Aronov, Matya Katz, and Micha Sharir.

### 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 JungGeorgia 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 MousaviGeorgia 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 SchmidtGeorgia 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
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