Seminars and Colloquia by Series

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.

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.

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

 

Evasiveness conjecture and topological methods in graph theory I

Series
Graph Theory Working Seminar
Time
Tuesday, February 8, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
James Anderson – Georgia Institute of Technology –

In the first talk of this seminar series, we follow the manuscript of Carl Miller and introduce the concept of elusive graph properties—those properties for which any edge-querying algorithm requires all possible queries in the worst case. Karp conjectured in 1973 that all nontrivial monotonic graph properties are elusive, and a celebrated theorem by Kahn in 1984 used topological fixed-point methods to show the conjecture is true in the case of graphs with order equal to a prime power. To set the stage for the proof of this result in later talks, we introduce monotone graph properties and their connection to collapsible simplicial complexes.

On sphere packings and the hard sphere model

Series
Job Candidate Talk
Time
Tuesday, February 8, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/552606446/5315
Speaker
Will Perkins – University of Illinois, Chicago –
The classic sphere packing problem is to determine the densest possible packing of non-overlapping congruent spheres in Euclidean space.  The problem is trivial in dimension 1, straightforward in dimension 2, but a major challenge or mystery in higher dimensions, with the only other solved cases being dimensions 3, 8, and 24.  The hard sphere model is a classic model of a gas from statistical physics, with particles interacting via a hard-core pair potential.  It is believed that this model exhibits a crystallization phase transition in dimension 3, giving a purely geometric explanation for freezing phenomena in nature, but this remains an open mathematical problem. The sphere packing problem and the hard sphere model are closely linked through the following rough rephrasing of the phase transition question: do typical sphere packings at densities just below the maximum density align with a maximum packing or are they disordered?  
 
I will present results on high-dimensional sphere packings and spherical codes and new bounds for the absence of phase transition at low densities in the hard sphere model.  The techniques used take the perspective of algorithms and optimization and can be applied to problems in extremal and enumerative combinatorics as well.
 
 

Stable commutator length on big mapping class groups

Series
Geometry Topology Seminar
Time
Monday, February 7, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Elizabeth Field – University of Utah –

The stable commutator length function measures the growth rate of the commutator length of powers of elements in the commutator subgroup of a group. In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will show that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed. Along the way to proving our main results, we will discuss certain topological properties of a class of infinite-type surfaces and their end spaces which may be of independent interest. This talk represents joint work with Priyam Patel and Alexander Rasmussen.

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