# Seminars & Colloquia

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Let G be a finite group. The minimal ramification problem famously asks about the minimal number µ(G) of ramified primes in any Galois extension of Q with group G. A conjecture due to Boston and Markin predicts the value of µ(G). I will report on recent progress on this problem, as well as several other problems which may be described as
minimal ramification problems in a wider sense, notably: what is the smallest number k = k(G) such that there exists a G-extension of Q with discriminant not divisible by any (k + 1)-th power?, and: what is the smallest number m = m(G) such that there exists
a G-extension of Q with all ramification indices dividing m? Apart from partial results over Q, I will present function field analogs.
(If you would like to join the seminar, contact Bo-Hae Im to get the zoom link.)

For the mass-critical/supercritical pseudo-relativistic nonlinear Schrödinger equation, Bellazzini, Georgiev and Visciglia constructed a class of stationary solutions as local energy minimizers under an additional kinetic energy constraint, and they showed the orbital stability of the energy minimizer manifold. In this talk, by proving its local uniqueness, we show the orbital stability of the solitary wave, not that of the energy minimizer set. The key new aspect is reformulation of the variational problem in the non-relativistic regime, which is, we think, more natural because the proof heavily relies on the sub-critical nature of the limiting model. By this approach, the meaning of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Finally, using the non-relativistic limit, we obtain the local uniqueness and the non-degeneracy of the minimizer. This talk is based on joint work with Sangdon Jin.

We will describe a construction of infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This is an analog of a construction of J. Park in the context of additive Chow groups. The construction allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process.

Zoom ID: 352 730 6970. Password : 9999. All times are in KST.

Zoom ID: 352 730 6970. Password : 9999. All times are in KST.

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Room B232, IBS (기초과학연구원)
Discrete Math
Pascal Gollin (IBS Discrete Mathematics Group)
Enlarging vertex-flames in countable digraphs

Room B232, IBS (기초과학연구원)

Discrete Math

A rooted digraph is a vertex-flame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by Joó and the analogue of Lovász’ result for countable digraphs was shown.
In this talk, I present a strengthening of this result stating that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame.
Joint work with Joshua Erde and Attila Joó.

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in convex domains in ${\bf R}^N$, where $N\ge 2$.
Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the heat flow,
we see that log-concavity is the only power concavity preserved by the Dirichlet heat flow.
This is a joint work with Paolo Salani (Univ. of Florence) and Asuka Takatsu (Tokyo Metropolitan Univ.)

The logarithmic analog of SH(k) is logSH(k).
In logSH(k), topological cyclic homology is representable.
Furthermore, the cyclotomic trace map from K to TC is representable too when k is a perfect field with resolution of singularities.
In the second talk, I will explain the construction of logSH(k) and how we can achieve these results.
This work is joint with Federico Binda and Paul Arne Østvær.

Zoom connection details will be provided a few days before the talk. The date and the time may change if the circumstances of the speaker require it. Please check it often.

Zoom connection details will be provided a few days before the talk. The date and the time may change if the circumstances of the speaker require it. Please check it often.

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

The rapid development of high-throughput sequencing technology in recent years is providing unprecedented opportunities to profile microbial communities from a variety of environments, but analysis of such multivariate taxon count data remains challenging. I present two flexible Bayesian methods to analyze complex count data with application to microbiome study. The first project is to develop a Bayesian sparse multivariate regression method that model the relationship between microbe abundance and environmental factors. We extend conventional nonlocal priors, and construct asymmetric non-local priors for regression coefficients to efficiently identify relevant covariates and their effect directions. The developed Bayesian sparse regression model is applied to analyze an ocean microbiome dataset collected over time to study the association of harmful algal bloom conditions with microbial communities. For the second project, we develop a Bayesian nonparametric regression model for count data with excess zeros. The approach provides straightforward community-level insights into how characteristics of microbial communities such as taxa richness and diversity are related to covariates. The baseline counts of taxa in samples are carefully constructed to obtain improved estimates of differential abundance. We apply the model to a chronic wound microbiome dataset, comparing the microbial communities present in chronic wounds versus in healthy skin.

Co-authors; Kurtis Shuler (Sandia National Lab), Marilou Sison-Mangus (Ocean Sciences, UCSC), Irene A. Chen (Chemistry and Biochemistry, UCLA), Samuel Verbanic (Chemistry and Biochemistry, UCLA)

Co-authors; Kurtis Shuler (Sandia National Lab), Marilou Sison-Mangus (Ocean Sciences, UCSC), Irene A. Chen (Chemistry and Biochemistry, UCLA), Samuel Verbanic (Chemistry and Biochemistry, UCLA)

Diophantine approximation is a branch of number theory where one studies approximation of irrational numbers by rationals and quality of such approximations. In this talk, we will consider intrinsic Diophantine approximation, which deals with approximating irrational points in a closed subset $X$ in $\mathbb{R}^n$ via rational points lying in $X$. First, we consider $X = S^1$, the unit circle in $\mathbb{R}^2$ centered at the origin. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Additionally, we present similar results for approximations of $S^1$ by a few different sets of rational points. This is joint work with Dong Han Kim (Dongguk University, Seoul).

A knot is a smooth embedding of an oriented circle into the three-sphere, and two knots are concordant if they cobound a smoothly embedded annulus in the three-sphere times the interval. Concordance gives an equivalence relation, and the set of equivalence classes forms a group called the concordance group. This group was introduced by Fox and Milnor in the 60's and has played an important role in the development of low-dimensional topology. In this talk, I will present some known results on the structure of the group. Also, I will talk about a knot that has infinite order in the concordance group, though it bounds a smoothly embedded disk in a rational homology ball. This is joint work with Jennifer Hom, Sungkyung Kang, and Matthew Stoffregen.

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https://kaist.zoom.us/j/89760039979?pwd=M2k4Vk9pVz
Applied mathematics
Won-Kwang Park (Kookmin University)
Application of MUSIC algorithm in real-world microwave imaging

https://kaist.zoom.us/j/89760039979?pwd=M2k4Vk9pVz

Applied mathematics

MUltiple SIgnal Classification (MUSIC) is a well-known, non-iterative imaging technique in inverse scattering problem. Throughout various researches, it has been confirmed that MUSIC is very fast, effective, and stable. Due to this reason MUSIC has been applied to various inverse scattering problems however, it has not yet been designed and used to identify unknown anomalies from measured scattering parameters (S-parameters) in microwave imaging. In this presentation, we apply MUSIC in microwave imaging for a fast identification of arbitrary shaped anomalies from real-data and establish a mathematical theory for illustrating the feasibilities and limitations of MUSIC. Simulations results with real-data are shown for supporting established theoretical results.

I will discuss various near-rationality concepts for smooth projective varieties. I will introduce the standard norm variety associated with a symbol in mod-l Milnor K-theory. The standard norm varieties played an important role in Vovedsky's proof of the Bloch-Kato conjecture. I will then describe known near-rationality results for standard norm varieties and outline an argument showing that a standard norm variety is universally R-trivial over an algebraically closed field of characteristic 0. The talk is based on joint work with Chetan Balwe and Amit Hogadi.

Zoom connection details will become available some days ahead of the talk.

Zoom connection details will become available some days ahead of the talk.

We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with Richard P. Thomas.

Zoom connection details will be provided later. The above times are in Korean Standard Time. This is the same time as 9AM of June 18, 2021 (Friday) in the UK (GMT+1)

Zoom connection details will be provided later. The above times are in Korean Standard Time. This is the same time as 9AM of June 18, 2021 (Friday) in the UK (GMT+1)

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Room B232, IBS (기초과학연구원)
Discrete Math
Mark Siggers (Kyungpook National University)
The list switch homomorphism problem for signed graphs

Room B232, IBS (기초과학연구원)

Discrete Math

A signed graph is a graph in which each edge has a positive or negative sign. Calling two graphs switch equivalent if one can get from one to the other by the iteration of the local action of switching all signs on edges incident to a given vertex, we say that there is a switch homomorphism from a signed graph $G$ to a signed graph $H$ if there is a sign preserving homomorphism from $G’$ to $H$ for some graph $G’$ that is switch equivalent to $G$. By reductions to CSP this problem, and its list version, are known to be either polynomial time solvable or NP-complete, depending on $H$. Recently those signed graphs $H$ for which the switch homomorphism problem is in $P$ were characterised. Such a characterisation is yet unknown for the list version of the problem.
We talk about recent work towards such a characterisation and about how these problems fit in with bigger questions that still remain around the recent CSP dichotomy theorem.

The standard machine learning paradigm optimizing average-case performance performs poorly under distributional shift. For example, image classifiers have low accuracy on underrepresented demographic groups, and their performance degrades significantly on domains that are different from what the model was trained on. We develop and analyze a distributionally robust stochastic optimization (DRO) framework over shifts in the data-generating distribution. Our procedure efficiently optimizes the worst-case performance, and guarantees a uniform level of performance over subpopulations. We characterize the trade-off between distributional robustness and sample complexity, and prove that our procedure achieves this optimal trade-off. Empirically, our procedure improves tail performance, and maintains good performance on subpopulations even over time.

This is part VI of the lectures on infinity-categories.
I'll keep talking about the simplicial nerve construction in contrast to the ordinary nerve functor. To finish off this whole series, some overview of the theory of infinity-categories will be given, including how similar and different it is to the ordinary category theory.

Zoom ID: 352 730 6970, Password: 9999

Zoom ID: 352 730 6970, Password: 9999

Morel and Voevodsky constructed the A^1-motivic homotopy category SH(k).
One purpose of motivic homotopy theory is to incorporate various cohomology theories into a single framework so that one can discuss relations between cohomology theories more efficiently.
However, there are still lots of non A^1-invariant cohomology theories, e.g. Hodge cohomology and topological cyclic homology.
There is no way to represent these in SH(k).
In the first talk, I will explain the construction of logDM^{eff}(k) for a perfect field k, which is strictly larger than Voevodsky's triangulated categories of motives DM^{eff}(k) because Hodge cohomology is representable in logDM^{eff}(k).
This work is joint with Federico Binda and Paul Arne Østvær.

Zoom ID: 352 730 6970, Password: 9999

Zoom ID: 352 730 6970, Password: 9999

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties.
We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank
for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural
parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese,
and consider various techniques to provide equations on the secants.
In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

A notion of sublinear expander has played a central role in the resolutions of a couple of long-standing conjectures in embedding problems in graph theory, including e.g. the odd cycle problem of Erdos and Hajnal that the harmonic sum of odd cycle length in a graph diverges with its chromatic number. I will survey some of these developments.