# Department Seminars & Colloquia

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Room B232, IBS (기초과학연구원)
Discrete Math
Mihyun Kang (TU Graz)
The genus of a random graph and the fragile genus property

In this talk we shall discuss how quickly the genus of the Erdős-Rényi random graph grows as the number of edges increases and how dramatically a small number of random edges can increase the genus of a randomly perturbed graph. (Joint work with Chris Dowden and Michael Krivelevich)

(This is a reading seminar for graduate students.) Algebraic $K$-theory of schemes has a sequence of $K$-groups for a closed immersion which is exact except only one place, namely $K_0$ of the open immersion of the complement, as the proto-localization theorem shows. Following the idea of Hyman Bass, we define a non-connective spectrum of a scheme whose non-negative part coincides with the usual algebraic $K$-theory defined by perfect complexes. This non-connective $K$-theory spectrum in particular gives a long exact sequence for a closed immersion. Our aim is to construct it, which is done by descending induction and requires the projective bundle formula that is also important regardless of our situation.

We introduce the mechanical model designed by W. Malkus and L. Howard for the Lorenz system. Some mechanical equation explaining this will be derived. Based on this mechanical equations, the Fourier modes and velocity of the system will be presented as a complete system following Lorenz's model. Some key factors, including Rayleigh number, will be introduced to compare Malkus's model and the fluid convection, key interest of Lorenz that gave birth to his model.

Abstract: Linkage is a classical topic in algebraic geometry and commutative algebra. Fix an affine space A. We say two subschemes X, Y of A are linked if their union is a complete intersection in A and X and Y do not have a common component. Two linked subschemes share several properties in common. Linkage has been studied by various people, Artin-Nagata, Peskine-Szprio, Huneke-Ulrich, to name a few.

In 2014, Niu showed that if Y is a generic link of a variety X, then LCT (A, X) <= LCT (A, Y), where LCT stands for log canonical threshold. In this talk, we show that if Y is a generic link of a determinantal variety X, then X and Y have the same log canonical threshold. This is joint work with Lance E. Miller and Wenbo Niu.

The models of directed polymers are based on Gibbs measures on paths with the reference measure usually describing a process with independent increments where the energy of the interaction between the path and the environment is given by a space-time random potential accumulated along the path. One of the intriguing phenomena that these models exhibit is the localization/delocalization transition between high/low temperature regimes. In order to study directed polymers in Euclidean space, we will first review a new metrization of the Mukherjee-Varadhan topology, introduced as a translation-invariant compactification of the space of probability measures on Euclidean spaces. This new metrization allows us to prove that the asymptotic clusterization (a natural continuous analogue of the asymptotic pure atomicity property) holds in the low temperature regime and that the endpoint distribution is geometrically localized with positive density if and only if the system is in the low temperature regime.

In the first part of the talk, I will present my recent working paper on continuous time game between a suspect and a defender. Economically, this is a sequential game model with asymmetric information, imperfect observation, and optimal stopping. Mathematically, search of a Markov equilibrium boils down to finding a solution of the system of equations (a HJB equation from the suspect’s optimal control, variational inequality from the defender’s optimal stopping, and a SDE from the filtering equation).

In the second part of the talk, I will roughly overview subjects in mathematical finance, and place where my past, current, and future research topics lie.

(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.

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자연과학동(E6) Room 1501
Algebraic Geometry
June Huh (IAS/Princeton Univ.)
Lorentzian polynomials: theory

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
Dabeen Lee (IBS Discrete Mathematics Group)
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.