Seminars and Colloquia by Series

Topological Methods in Convexity

Series
Geometry Topology Student Seminar
Time
Wednesday, February 16, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin Shu – Georgia Tech

Topological methods have had a rich history of use in convex optimization, including for instance the famous Pataki-Barvinok bound on the ranks of solutions to semidefinite programs, which involves the Borsuk-Ulam theorem. We will give two proofs of a similar sort involving the use of some basic homotopy theory. One is a new proof of Brickman's theorem, stating that the image of a sphere into R^2 under a quadratic map is convex, and the other is an original theorem stating that the image of certain matrix groups under linear maps into R^2 is convex. We will also conjecture some higher dimensional analogues.

Control of tissue development and cell diversity by cell cycle dependent transcriptional filtering

Series
Mathematical Biology Seminar
Time
Wednesday, February 16, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Maria Abou Chakra – University of Toronto

Please Note: Meeting Link: https://bluejeans.com/426529046/8775

Cell cycle duration changes dramatically during development, starting out fast to generate cells quickly and slowing down over time as the organism matures. The cell cycle can also act as a transcriptional filter to control the expression of long gene transcripts which are partially transcribed in short cycles. Using mathematical simulations of cell proliferation, we identify an emergent property, that this filter can act as a tuning knob to control gene transcript expression, cell diversity and the number and proportion of different cell types in a tissue. Our predictions are supported by comparison to single-cell RNA-seq data captured over embryonic development. Additionally, evolutionary genome analysis shows that fast developing organisms have a narrow genomic distribution of gene lengths while slower developers have an expanded number of long genes. Our results support the idea that cell cycle dynamics may be important across multicellular animals for controlling gene transcript expression and cell fate.

Recording link: https://bluejeans.com/s/QhCWmELH6AC

Approximating TSP walks in sub cubic graphs

Series
Graph Theory Seminar
Time
Tuesday, February 15, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Youngho Yoo – Georgia Institute of Technology –

We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths $\frac{5n+n_2}{4}-1$ and $\frac{5n}{4} - 2$ respectively. We characterize the extremal subcubic examples meeting this bound. We also give a quadratic-time combinatorial algorithm for finding such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of $\frac{9}{7}$.

On the sum-product problem

Series
Job Candidate Talk
Time
Tuesday, February 15, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
George Shakan – CRM –

Let A be a subset of the integers of size n. In 1983, Erdos and Szemeredi conjectured that either A+A or A*A must have size nearly n^2. We discuss ideas towards this conjecture, such as an older connection to incidence geometry as well as somewhat newer breakthroughs in additive combinatorics. We further highlight applications of the sum-product phenomenon. 

Abelian cycles in the homology of the Torelli group

Series
Geometry Topology Seminar
Time
Monday, February 14, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Erik Lindell – University of Stockholm –

The mapping class group of a compact and orientable surface of genus g has an important subgroup called the Torelli group, which is the kernel of the action on the homology of the surface. In this talk we will discuss the stable rational homology of the Torelli group of a surface with a boundary component, about which very little is known in general. These homology groups are representations of the arithmetic group Sp_{2g}(Z) and we study them using an Sp_{2g}(Z)-equivariant map induced on homology by the so-called Johnson homomorphism. The image of this map is a finite dimensional and algebraic representation of Sp_{2g}(Z). By considering a type of homology classes called abelian cycles, which are easy to write down for Torelli groups and for which we can derive an explicit formula for the map in question, we may use classical representation theory of symplectic groups to describe a large part of the image.

Recent progress on Hadwiger's conjecture

Series
Job Candidate Talk
Time
Monday, February 14, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Luke Postle – University of Waterloo –

Link: https://bluejeans.com/398474745/0225

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t \ge 1$. Hadwiger's Conjecture is a vast generalization of the Four Color Theorem and one of the most important open problems in graph theory. Only the cases when $t$ is at most 6 are known. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t (\log t)^{0.5})$ and hence is $O(t (\log t)^{0.5})$-colorable.  In a recent breakthrough, Norin, Song, and I proved that every graph with no $K_t$ minor is $O(t (\log t)^c)$-colorable for every $c > 0.25$,  Subsequently I showed that every graph with no $K_t$ minor is $O(t (\log \log t)^6)$-colorable.  Delcourt and I improved upon this further by showing that every graph with no $K_t$ minor is $O(t \log \log t)$-colorable. Our main technical result yields this as well as a number of other interesting corollaries.  A natural weakening of Hadwiger's Conjecture is the so-called Linear Hadwiger's Conjecture that every graph with no $K_t$ minor is $O(t)$-colorable.  We prove that Linear Hadwiger's Conjecture reduces to small graphs. In 2005, Kühn and Osthus proved that Hadwiger's Conjecture for the class of $K_{s,s}$-free graphs for any fixed positive integer $s \ge 2$. Along this line, we show that Linear Hadwiger's Conjecture holds for the class of $K_r$-free graphs for every fixed $r$.

The k-Cap Process on Geometric Random Graphs

Series
ACO Student Seminar
Time
Friday, February 11, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Mirabel Reid – Georgia Tech CS –

The k-cap (or k-winners-take-all) process on a graph works as follows: in each
iteration, exactly k vertices of the graph are in the cap (i.e., winners); the next round
winners are the vertices that have the highest total degree to the current winners,
with ties broken randomly. This natural process is a simple model of firing activity
in the brain. We study its convergence on geometric random graphs revealing rather
surprising behavior

Stochastic and Convex Geometry for the Analysis of Complex Data

Series
Job Candidate Talk
Time
Thursday, February 10, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/532559688
Speaker
Eliza O’Reilly – California Institute of Technology –

Many modern problems in data science aim to efficiently and accurately extract important features and make predictions from high dimensional and large data sets. While there are many empirically successful methods to achieve these goals, large gaps between theory and practice remain.  A geometric viewpoint is often useful to address these challenges as it provides a unifying perspective of structure in data, complexity of statistical models, and tractability of computational methods.  As a consequence, an understanding of problem geometry leads both to new insights on existing methods as well as new models and algorithms that address drawbacks in existing methodology.

 In this talk, I will present recent progress on two problems where the relevant model can be viewed as the projection of a lifted formulation with a simple stochastic or convex geometric description. In particular, I will first describe how the theory of stationary random tessellations in stochastic geometry can address computational and theoretical challenges of random decision forests with non-axis-aligned splits. Second, I will present a new approach to convex regression that returns non-polyhedral convex estimators compatible with semidefinite programming. These works open a number of future research directions at the intersection of stochastic and convex geometry, statistical learning theory, and optimization.

Measure theoretic Rogers-Shephard and Zhang type inequalities

Series
Analysis Seminar
Time
Wednesday, February 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Michael Roysdon – Tel Aviv University –

This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures, the first of which is a reverse form of the Brunn-Minkowsk inequality, and the second of which can be seen to be a reverse affine isoperimetric inequality; the feature of both inequalities is that they each provide a classification of the n-dimensional simplex in the volume case. The covariogram of a measure plays an essential role in the proofs of each of these inequalities. In particular, we will discuss a variational formula concerning the covariogram resulting in a measure theoretic version of the projection body, an object which has recently gained a lot of attention--these objects were previously studied by Livshyts in her analysis of the Shephard problem for general measure.

 

The talk will be on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

 

The slice-ribbon conjecture and 3-stranded pretzel knots

Series
Geometry Topology Student Seminar
Time
Wednesday, February 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hugo Zhou – Georgia Tech

This is an expository talk about the slice-ribbon conjecture. A knot is slice if it bounds a disk in the four ball. We call a slice knot ribbon if it bounds a slice disk with no local maxima. The slice-ribbon conjecture asserts all slice knots arise in this way. We also give a very brief introduction to Greene, Jabuka and Lecuona's works on the slice-ribbon conjecture for 3-stranded pretzel knots.

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