# Seminars & Colloquia

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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.

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.

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자연과학동(E6) Room 1401
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 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.

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 $(\mathbf A^3,\mathbf A^3\setminus0)$. 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.

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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">VV and a family C">CC of subsets of V">VV, the set covering problem is to find a minimum cardinality subset of V">VVintersecting every subset in the family C">CC. 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">CC. 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">CC is a clutter if every subset in C">CC is inclusion-wise minimal. As taking out non-minimal subsets preserves integrality, we may assume that C">CC is a clutter. We call C">CC 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">CC 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.