# Department Seminars & Colloquia

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Pressure functions are key ideas in the thermodynamic formalism of dynamical systems. McMullen used the convexity of the pressure function to construct a metric, called a pressure metric, on the Teichmuller space and showed that it is a constant multiple of the Weil-Petersson metric. In the spirit of Sullivan's dictionary, McMullen applied the same idea to define a metric on the space of Blaschke products.
In this talk, we will discuss Bridgeman-Taylor and McMullen's earlier works on the pressure metric, as well as recent developments in more generic settings. Then we will talk about pressure metrics on hyperbolic components in complex dynamics, as well as unsolved problems.

A major trajectory in the development of statistical learning has been the expansion of mathematical spaces underlying observed data, extending from numbers to vectors, functions, and beyond. This expansion has fostered significant theoretical and computational breakthroughs. One notable direction involves analyzing sets where each set becomes an object of interest for inference. This perspective accommodates the intrinsic and non-ignorable heterogeneity inherent in data-generating processes. Among various theoretical frameworks to analyze sets, a principled approach is viewing a set as an empirical measure. In this talk, I revisit the concept of the median - a robust alternative to the mean as a centroid - and introduce a novel extension of this concept within the space of probability measures under the framework of optimal transport. I will present theoretical results and a generic computational pipeline that leverages existing algorithmic developments in the field, with examples. Furthermore, the potential benefits of this novel approach for scalable inference and scientific discovery will be explored.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

Let G be a numerical semigroup. We prove an upper bound for the Betti numbers of the semigroup ring of G which depends only on the width of G, that is, the difference between the largest and the smallest generators of G. In this way, we make progress towards a conjecture of Herzog and Stamate. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.
This is a joint work with A. Moscariello and A. Sammartano.

This is a one-day workshop with young geometric topologists. Follow the link for more details

https://sites.google.com/site/hrbaik85/workshop-and-conferences-at-kaist/yggt-at-kaist?authuser=0

https://sites.google.com/site/hrbaik85/workshop-and-conferences-at-kaist/yggt-at-kaist?authuser=0

It has been well known that any closed, orientable 3-manifold can be obtained by performing Dehn surgery on a link in S^3. One of the most prominent problems in 3-manifold topology is to list all the possible lens spaces that can be obtained by a Dehn surgery along a knot in S^3, which has been solved by Greene. A natural generalization of this problem is to list all the possible lens spaces that can be obtained by a Dehn surgery from other lens spaces. Besides, considering surgeries between lens spaces is also motivated from DNA topology. In this talk, we will discuss distance one surgeries between lens spaces L(n, 1) with n ≥ 5 odd and lens spaces L(s, 1) for nonzero s and the corresponding band surgeries from T(2, n) to T(2, s), by using our Heegaard Floer d-invariant surgery formula, which is deduced from the Heegaard Floer mappping cone formula. We give an almost complete classification of the above surgeries.

The analysis on the limiting behavior of solution is pivotal for equations in geometric analysis, mathematical physics and application in optimization. In 80s, Rene Thom conjectured that if an analytic gradient flow has a limit, then it approaches to the limit along a unique asymptotic direction. This represents a higher-order question following the seminal works by Lojasiewicz and L. Simon. In 2000, this conjecture was affirmatively proved by Kurdyka, Mostowski, and Parusinski for finite dimensional gradient flows. In this talk, we present a generalization of this to the class of PDEs which are gradient flows or equations with gradient structure. Our result reveals both the rate and the direction of convergence to the limit. This is a joint work with Pei-Ken Hung at UIUC.

In the 1970's J. Levine produced a surjection from the knot concordance group to the so called algebraic concordance group. This captured the known features of the knot concordance group to that point and classifies high dimensional concordance. During this survey talk we will explore the construction of the algebraic concordance group and explain some of its consequences.

In the 1980's Casson and Gordon produced the first non slice knots which are trivial in Levine's algebraic concordance group, and in 2003 Cochran-Orr-Teichner produced the first no slice knots undetectable by Casson and Gordon's invariants. They do so by producing a filtration of the concordance group by subgroups a knot in the 1.5th term of this filtration has vanishing Casson-Gordon invariants. Since then this work has been central to the study of knot concordance. We will introduce this filtration and review just enough of the theory of L^2 homology to prove that the successive quotients of this filtration are nontrivial.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball.
Day 1: Jones polynomial and its categorification.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball.
Day 2: Numerical invariants from Khovanov homology and their applications.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball.
Day 3: Recent developments of Khovanov homology and its applications to low-dimensional topology.

When does a topological branched self-covering of the sphere enjoy a holomorphic structure? William Thurston answered this question in the 1980s by using a holomorphic self-map of the Teichmuller space known as Thurston's pullback map. About 30 years later, Dylan Thurston took a different approach to the same question, reducing it to a one-dimensional dynamical problem. We will discuss both characterizations and their applications to various questions in complex dynamics.

Deep learning has shown remarkable success in various fields, and efforts continue to develop investment methodologies using deep learning in the financial sector. Despite numerous successes, the fact is that the revolutionary results seen in areas such as image processing and natural language processing have not been seen in finance. There are two reasons why deep learning has not led to disruptive change in finance. First, the scarcity of financial data leads to overfitting in deep learning models, so excellent backtesting results do not translate into actual outcomes. Second, there is a lack of methodological development for optimizing dynamic control models under general conditions. Therefore, I aim to overcome the first problem by artificially augmenting market data through an integration of Generative Adversarial Networks (GANs) and the Fama-French factor model, and to address the second problem by enabling optimal control even under complex conditions using policy-based reinforcement learning. The methods of this study have been shown to significantly outperform traditional linear financial factor models such as the CAPM and value-based approaches such as the HJB equation.

This talk presents a uniform framework for computational fluid dynamics in porous media based on finite element velocity and pressure spaces with minimal degrees of freedom. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. Since the fluid model in porous media can be seen as a combination of the Stokes and Darcy equations, different conformities of finite element spaces are required depending on viscous parameters, making it challenging to develop a robust numerical solver uniformly performing for all viscous parameters. Therefore, we propose a pressure-robust method by utilizing a velocity reconstruction operator and replacing the velocity functions with a reconstructed velocity. The robust method leads to error estimates independent of a pressure term and shows uniform performance for all viscous parameters, preserving minimal degrees of freedom. We prove well-posedness and error estimates for the robust method while comparing it with a standard method requiring an impractical mesh condition. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for our robust method.