The geometry and cohomology of modular curves play an important role in constructing the Langlands correspondence for GL(2,mathbb{Q}). Similarly, the tower of algebraic varieties called Shimura varieties plays an important role in the Langlands programme for reductive groups over number fields. It is then a natural question to ask if there is a ‘p-adic local analogue’ of modular curves and Shimura varieties, which is now commonly referred to as 'local Shimura varieties' or ‘Rapoport—Zink spaces'. 

In this survey talk, I will try to motivate the concept of local Shimura varieties starting from the review of modular curves and introducing classical examples of local Shimura varieties (that have been known even before the general concept of local Shimura varieties emerged). If time permits, I will introduce my contribution and work in progress on constructing more general classes of local Shimura varieties.

Circle graphs are the intersection graphs of chords on a circle; vertices correspond to chords, and two vertices are adjacent if their chords intersect. We prove that every circle graph with clique number k has chromatic number at most $4k^2$. Joint with James Davies.

Recently, the classification of isoparametric hypersurfaces in spheres has been completed. Therefrom, various new research projects in geometry have been initiated. The study of minimal lagrangian submanifolds via isoparametric hypersurfaces is one of the most active projects a la mode. In this talk, we have an introduction to the study of Isoparametric hypersurfaces and minimal Lagrangian submanfolds, and discuss the relationship between them.

The current focus of our research is to reveal fundamental design principles of the biological clock mechanism and pathogenesis of circadian disorders. We develop unique mouse models to simulate human diseases that can cause circadian disorders directly and indirectly, and unravel how the diseases compromise circadian rhythms including wake-sleep cycles, the most salient feature of circadian rhythms in animals, at the molecular level. In the first half of my talk, I will discuss how cytoplasmic congestion, normally associated with metabolic diseases and aging, can disrupt the clock mechanism and how sleep disorders in patients with these diseases can be treated at the core clock level. In the second half of my talk, I will discuss arguably the most upstream event in rhythm generation. We recently found that phosphorylation of the circadian pacemaker protein, PER is regulated by a strong Hill-type reaction, the basis of robustness in many signaling pathways. Phosphorylation kinetics of PER increases proportionally to concentration of PER: the more PER is generated, the faster PER phosphorylates, similar to O2 binding to hemoglobin. Since PER phosphorylation is considered the time-generating step, we believe this is the most upstream event and simple mutations in Per genes (SNPs) may cause sleep disorders in many humans.  

The circadian clock is an autonomous molecular mechanism that controls biochemical, physiological, and behavioral processes with a periodicity of 24 h in living organisms and can be entrained by environmental cues. The clock is sustained by a coordinated interplay of positive and negative transcriptional-translational feedback loops driven by circadian factors, a core group of proteins that either possess intrinsic transcriptional activity or modulate gene expression. We previously reported that the circadian factor PERIOD 2 (PER2) forms a stable complex with the tumor suppressor and checkpoint protein p53. The PER2:p53 complex undergoes time-of-day–dependent nuclear-cytoplasmic shuttling, thus generating an asymmetric distribution of each protein in different cellular compartments. In unstressed cells, PER2 mediates p53’s stability by binding to its C-terminal domain and preventing p53 from being ubiquitylated at sites targeted by the RING finger–containing E3 ligase and oncoprotein mouse double minute 2 homolog (MDM2). We found that PER2, p53, and MDM2 co-exist as a trimeric and stable complex in the nuclear compartment, although p53 is released from the complex to become transcriptionally active after cells experience genotoxic stimuli. More recently, we found that PER2 could also act as a bona fide substrate for MDM2 in the absence of p53. Indeed, PER2 was efficiently ubiquitylated in vitro and in cells at numerous sites by MDM2 in a process that was independent of PER2 phosphorylation. Accordingly, PER2’s half-life is critically influenced by the abundance and enzymatic activity of MDM2, as shown in cells in which MDM2 expression was either enhanced or silenced and its catalytic activity was pharmacologically inhibited. As a consequence, direct manipulation of MDM2 expression influenced period length by reducing PER2 stability. Our results uncover previously unknown regulatory players that likely impact our view of how other mechanisms crosstalk and modulate the clock itself. Furthermore, it exposes an uncharacterized druggable node that is often found to be deregulated during tumorigenesis. 

(This is a reading seminar for graduate students.)

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)to K_0(X)to K_0(X-Z)to 0$ for a closed immersion $Zto X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the second half of Quillen's algebraic $K$-theory. 

In X-ray computed tomography (CT), deep learning techniques have shown great potential for reduction of various artifacts (e.g., noise arising from low-dose, streaking artifacts due to sparse view). Most of these approaches learn the relationship between artifact images and artifact-free (ground-truth) images. However, paired training data are not generally available in clinical practice. In this talk, I will introduce simulation-based and unpaired learning methods, which can be used to circumvent such issue.

A Boolean function is a function from the set Q of binary vectors of length n (i.e., the binary n-dimensional hypercube) to $F_2=\{0,1\}$. It has several applications to complexity theory, digital circuits, coding theory, and cryptography.
In this talk we give a connection between Boolean functions and Artificial Neural Network. We describe how to represent Boolean functions by Artificial Neural Network including linear and polynomial threshold units and sigmoid units. For example, even though a linear threshold function cannot realize XOR, a polynomial threshold function can do it. We also give currently open problems related to the number of (Boolean) linear threshold functions and polynomial threshold functions.

Many modern applications such as machine learning require solving large-dimensional optimization problems. First-order methods are widely used to solve such problems, since their computational cost per iteration mildly depends on the problem dimension. However, they suffer from relatively slow convergence rates, and this talk will discuss recent progress on the acceleration of first-order methods, particularly using the close relationship between convex optimization methods and maximally monotone operators.

 Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $mathcal{L}$ on $X$, then the group $textrm{Aut}_k(X,mathcal{L})$ of automorphisms of the polarized abelian variety $(X,mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over $k.$ 

In this talk, we give a classification of such finite groups for the case when $k$ is a finite field and $g$ is a prime number. If $g=2,$ then we need a notion of maximality in a certain sense, and for $g geq 3,$ we achieve a rather complete list without conveying maximality.

Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both  the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and  non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems.

In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin approach on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes.

Goldman parametrizes the $mathrm{PSL}_3(mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g- 10$ coordinates are canonical. We prove that the $mathrm{PSL}_3(mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we establish a version of the action-angle principle and show that the Hitchin component can be decomposed into a product of smaller Hitchin components. 

(This is a reading seminar for graduate students.)

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)to K_0(X)to K_0(X-Z)to 0$ for a closed immersion $Zto X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the first half of Quillen's algebraic $K$-theory. 

Millions of Korean workers follow night or nonstandard shifts. A similar number of Koreans travel overseas each year. This leads to misalignment of the daily (circadian) clock in individuals: a clock that controls sleep, performance, and nearly every physiological process in our body. A mathematical model of this clock can be used to optimize schedules to maximize productivity and minimize jetlag. We simulate this model in a smartphone app, ENTRAIN (www.entrain.org), which has been installed in phones over 200,000 times in over 100 countries. I will discuss techniques we have been developing and clinically testing to determine sleep stage and circadian time from wearable data, for example, as collected by our app or used in many commercially available sleep trackers. This project has led us to develop new techniques to: 1) estimate phase from noisy data with gaps, 2) rapidly simulate of population densities from high dimensional models and 3) determine how mathematical models of sleep and circadian physiology can be used with machine learning techniques to improve predictions.

(This is a reading seminar for graduate students.)

Algebraic K-theory originated with the Grothendieck-Riemann-Roch theorem, a generalization of Riemann-Roch theorem to higher dimensional varieties. For this, we shall discuss the definitions of $K_0$-theory of a variety, its connection with intersection theory, $lambda$-operation, $gamma$-filtration, Chern classes and Adams operations.

I report on work with M. Gubinelli and T. Oh on the
renormalized nonlinear wave equation
in 2d with monomial nonlinearity and in 3d with quadratic nonlinearity.
Martin Hairer has developed an efficient machinery to handle elliptic
and parabolic problems with additive white noise, and many local
existence questions are by now well understood. In contrast not much is
known for hyperbolic equations. We study the simplest nontrivial
examples and prove local existence and weak universality, i.e. the
nonlinear wave equations with additive white noise occur as scaling
limits of wave equations with more regular noise.

It remains  to bound the terms on the expansion of the logarithm of the
transmission coefficient.

I report on joint work with Mihaela Ifrim, Xian Liao and Daniel Tataru.

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph $G$ does not contain $K_{2,2}$ as an induced subgraph yet has at least $c(\frac{n}{2})$edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced $K_{2,2}$, which allows us to extend their result to $k$-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.

The Gross-Pitaevskii equation is essentially the defocusing nonlinear
Schrödinger equation with a nonzero boundary condition at infinity. This
makes the phase space very nonlinear and we will need a good metric and
smooth structure on it.