# 세미나 및 콜로퀴엄

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(This is a reading seminar for graduate students.) Let $X$ be a scheme and $U$ be its open subscheme. If $X$ is noetherian, then any coherent sheaf on $U$ always extends to $X$. By contrast, extension problem of algebraic vector bundles is far from being true in this naive sense; there is a counterexample even for $(mathbf A^3,mathbf A^3setminus0)$. Nevertheless, if $X$ is regular, then the Poincaré duality for $K$-theory shows that a coherent sheaf on $X$ extending a given algebraic vector bundle on $U$ is resolved by a bounded complex of algebraic vector bundles. Together with Waldhausen's approximation theorem stating that $K$-theory essentially depends on derived categories, this suggests that the right objects we should consider for this problem are perfect complexes. We will prove that the failure of extension of perfect complexes on $U$ to $X$ in the derived category is captured by the cokernel of $K_0(X)to K_0(U)$, which is proved by Thomason-Trobaugh. As an analogue to Quillen's localization theorem for $G$-theory of noetherian schemes, it then directly gives the proto-localization theorem for $K$-theory of quasi-compact quasi-separated schemes except that the proto-localization theorem doesn't have surjectivity of $K_0(X)to K_0(U)$. If possible, we will measure to what degree this map is surjective by introducing the non-connective Bass $K$-theory spectrum.

The p-curvature conjecture of Grothendieck--Katz gives an arithmetic criterion for certain differential equations on algebraic varieties to have algebraic solutions. We describe a proof of this conjecture for rank two connections on generic algebraiccurves (joint work with Anand Patel and Ananth Shankar). We also consider a different problem of characterizing surface group representations with finite (or bounded) orbits under the mapping class group action, and give a complete solution in the speciallinear rank two case for positive-genus surfaces (joint work with Indranil Biswas, Subhojoy Gupta, and Mahan Mj). An ingredient common to these works is a type of topological "local-to-global principle" for certain surface group representations.

Unlike classical enumerative problems over the complex numbers, there are no fixed number of points of interest over non-algebraically closed fields (in particular, over the real numbers). In differential topology, one instead finds the fixed signed intersection number; a difference between the number of positive points and negative points. By using this idea and tools from mathbb{A}^1-homotopy theory, Kass-Wickelgren and Levine built mathbb{A}^1-enumerative geometry as a toolkit to find such "signed" intersection numbers. First, I will survey background materials and some known results. Then, I will describe the joint work with Ethan Cotterill and Ignacio Darago on counting the number of inflection points of linear systems on hyperelliptic curves.

The positive definite coefficients in Poisson equations are crucial for the stability and error estimates of numerical methods. However, modelling contacts of classical materials and metamaterials proposes second order PDEs with sign changing coefficients. In this talk we consider numerical methods for Poisson-type model problems with sign changing coefficients. To overcome the difficulty from sign changing coefficients we develop novel hybridized discontinuous Galerkin (HDG) methods with a new error analysis which does not rely on the elliptic regularity assumption.

I will give a gentle overview of my work with Petter Brändén on Lorentzian polynomials. Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. No specific background beyond linear algebra and multivariable calculus will be needed to enjoy the talk.

I advertise the talk to people with interest in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, Potts model partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint works with Petter Brändén, Christopher Eur, Jacob Matherne, Karola Mészáros, and Avery St. Dizier.

The geometry of compact moduli spaces of log surfaces is mysterious in general, as opposed to moduli of curves. Thus, describing an example with its geometric properties is already valuable. To do so, we consider an 'almost K3' stable log surface (an extension of Hacking's idea), which is a pair where the log canonical divisor is positive but very close to 0. We study compactified moduli spaces of such log surfaces, constructed using the techniques of Kollár, Shepherd-Barron, Alexeev, and Hacking. I will describe recent joint works with Anand Deopurkar on a compactification of the moduli space of (X, D) where X is a quadric surface and D is a canonical genus 4 curve, obtaining a new birational model of the moduli space M_4 of smooth curves of genus 4. As a generalization, I will survey on the moduli of 'almost K3 stable log quadrics', which are Q-Gorenstein degenerations of a pair of smooth quadric surface with a curve of bidegree (d,d).

Moduli spaces of local systems on surfaces are widely studied in geometry. Focusing on the special linear rank two case, after motivating our Diophantine study we use mapping class group dynamics and differential geometric tools to establisha structure theorem for the integral points on the moduli spaces, generalizing work of Markoff (1880). We also give an effective analysis of integral points for nondegenerate algebraic curves on these spaces. Along the way, we present other related resultsconnecting the geometry and arithmetic of the moduli spaces to elementary observations on surfaces.

We present a locally conservative enriched Galerkin finite element method that can be applied for solving parabolic equation as well as Stokes equation in a unified fashion. We present the reason why local conservation is important by establishing continuous and discrete maximum principle of coupled flow and transports. We also present that the resulting system can be solved effectively using a fast solver based on algebraic Multigrid method. Finally, a number of application areas will be presented, which include modeling of Tornado and enhanced Oil recovery.

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Room B232, IBS(기초과학연구원)
Discrete Math
이다빈 (IBS 이산수학그룹)
Integrality of set covering polyhedra and clutter minors

Given a finite set of elements V">V and a family C">C of subsets of V">V, the set covering problem is to find a minimum cardinality subset of V">Vintersecting every subset in the family C">C. The set covering problem, also known as the hitting set problem, admits a simple integer linear programming formulation. The constraint system of the integer linear programming formulation defines a polyhedron, and we call it the set covering polyhedron of C">C. We say that a set covering polyhedron is integral if every extreme point is an integer lattice point. Although the set covering problem is NP-hard in general, conditions under which the problem becomes polynomially solvable have been studied. If the set covering polyhedron is integral, then it is straightforward that the problem can be solved using a polynomial-time algorithm for linear programming.

In this talk, we will focus on the question of when the set covering polyhedron is integral. We say that the family C">C is a clutter if every subset in C">C is inclusion-wise minimal. As taking out non-minimal subsets preserves integrality, we may assume that C">C is a clutter. We call C">C ideal if the set covering polyhedron of it is integral. To understand when a clutter is ideal, the notion of clutter minors is important in that C">C is ideal if and only if so is every minor of it. We will study two fundamental classes of non-ideal clutters, namely, deltas and the blockers of extended odd holes. We will characterize when a clutter contains either a delta or the blocker of an extended odd hole as a minor.

This talk is based on joint works with Ahmad Abdi and Gérard Cornuéjols.

(This is a reading seminar for graduate students.) In previous talks, we investigated perfect complexes on a scheme and Waldhausen categories for which $K$-theory spectrum is defined. We'll define algebraic $K$-theory of quasi-compact quasi-separated schemes with these notions. Basic properties of algebraic $K$-theory will be discussed; properties include functoriality, excision property, relation with inverse limits and Poincaré duality. For this, we need various models of algebraic $K$-theory which give homotopy equivalent $K$-theory spectra most of which are proved by dependency of the derived categories of $K$-theory spectra.

Over the past decade, electrical bioimpedance has been undergoing a rebirth as enhanced methodologies and new theories are greatly extending its use in the field of neuromuscular disease (NMD). Simply put, NMDs change the structure and internal composition of skeletal muscle which, in turn, alter the electrical properties muscle. Thus, the capability of measuring the electrical properties of muscle with accuracy has great potential to provide valuable new insights to inform medical assessment and diagnosis of NMDs. One technique well-suited for measuring the electrical properties of muscle is electrical bioimpedance, where an electrical current is applied to the muscle using two electrodes and the resultant voltage is measured using two additional electrodes. However, the accuracy to detect onset of disease, track disease progression and response to therapy using surface electrodes placed on the skin is limited: data are largely influenced by skin and subcutaneous fat (SF) overlying the muscle. Here, we will present a new source separation (SS) technique that, unlike existing blinded SS techniques principal component analysis (PCA) and independent component analysis (ICA), can distinguish muscle from SF with the accuracy being 99.2%.

However, the standard procedure of patient care for diagnosing NMDs consists of inserting needles electrodes into the muscle to measure the electrical activity at rest and during muscle contraction. To take advantage of this, we have designed an enhanced needle device also integrating impedance recording capabilities. Our new needle improves the accuracy measuring the electrical properties by recording these properties and their direction dependence directly in the muscle, the latter also known as anisotropy. Ongoing work in this area promises exciting and valuable new applications in the years to come.

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Room B232, IBS(기초과학연구원)
Discrete Math
Patrice Ossona de Mendez (CNRS)
A model theoretical approach to sparsity

I will review the current status of 2D Coulomb gas with $beta = 1$ (hence "normal matrices" in the title).

When the potential is nice enough (algebraic Hele-Shaw) we can apply the method that has been developed to study complex orthogonal polynomials.

I will explain such method (called Riemann-Hilbert method) and its usefulness in studying the partition function of the Coulomb system.

Consider the complex Monge-Amp`ere equation (i∂∂u¯ )n = dµ in a bounded domain in Cn, where u is a plurisubharmonic function and dµ is a positive Radon measure. We give a sharp condition on the right hand side to guarantee the existence of a H¨older continuous solution. Namely, there exists a H¨older continuous plurisubharmonic function ϕ such that dµ ≤ (i∂∂ϕ¯ )n in the sense of measures. In particular, the answer to a question of Ahmed Zeriahi is always positive. Then, we will discuss its counterparts on compact complex manifolds and some applications.

We survey recent applications of weak plurisubharmonic solutions to complex MongeAmp`ere equations, using pluripotential theory, for problems arising from complex geometry and algebraic geometry. On one hand it is used to construct singular K¨ahler-Einstein metrics, which may occur as the limits of the Kahler-Ricci flow or the limits of families of Calabi-Yau metrics when the K¨ahler class hits the boundary of the K¨ahler cone. On the other hand it played a crucial role in studying structure of the K¨ahler cone. In particular, we will discuss a weak form of a conjecture due to Demailly and Paun in 2004.

(This is a reading seminar for graduate students.) Quillen's $K$-theory is defined for categories in which a suitable notion of exactness can be spoken. In much more generality, Waldhausen introduced the $S$-construction for categories equipped with suitable notions of cofibrations and weak equivalences. We will discuss the definition of S-construction, its dependence on the derived category, the cofinality theorem, and its comparison to Quillen's $K$-theory.

We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani.

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via three different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.

Since the early 2000s, it has turned out that well developed mathematical tools can play crucial roles in quantum information theory. One monumental work was made by Hastings in 2009. He disproved a long standing conjecture in quantum information theory, which is called the additivity conjecture of Holevo capacities. A natural way to prove this result will be covered in this talk based on the theory of i.i.d. random unitary matrices. Another outstanding application of random matrix theory in quantun information theory is to provide a systematic way to produce PPT entangled states in high dimensional tensor product spaces. This construction comes from i.i.d. random Gaussian matrices and I will try to explain why this application is important in view of quantun information theory.

Directed polymer models are well known Gibbs measures on random walk paths. Canonically they are defined so as to tilt the path distribution towards regions of space-time where an independent random field happens to be large, and as a result the paths tend to exhibit superdiffusive Kardar-Parisi-Zhang type fluctuation exponents, somehow betraying their random walk upbringing. Constructing these models on in the discrete space-time setting with a finite time horizon is straightforward, but extending them to infinite time horizons is difficult even in the fully discrete setting. I will review some relatively recent progress in the discrete and semi-discrete setting by myself and several other authors, some previous work of myself, Khanin, and Quastel on constructing continuous space-time models in the finite time horizon setting, and some attempts in progress to connect the two.

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Room B232, IBS (기초과학연구원)
Discrete Math
박진영 (Rutgers University)
The number of maximal independent sets in the Hamming cube

Let Qn be the n-dimensional Hamming cube (hypercube) and N=2n. We prove that the number of maximal independent sets in Qn is asymptotically 2n2N/4, as was conjectured by Ilinca and Kahn in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof of the upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in Qn. This is joint work with Jeff Kahn.