# Local Seminar

Our seminar is partially supported by Sonata Bis 3 Narodowe Centrum Nauki Poland Grant No NCN 2014/10/E/ST1/00688.

Here is the list of talks given during our non-regular local seminar at the Polish Academy of Sciences in Cracow. We usually meet together at chosen Wednesday at 10.30 (SHARP). Each talk is about 1h / 1h 30 min.

Seminar IMPAN 2018/2019

Abstracts:

18.10.2018: Niech p będzie liczbą pierwszą. Podczas referatu przedstawię kilka wyników dotyczących wyznaczania wyraźnych wzorów na p-adyczne waluacje pewnych funkcji partycji liczb naturalnych na potęgi liczby p (jednokolorowych i wielokolorowych).

25.10.2018: Funkcje symetryczne to niezwykle proste obiekty, które w naturalny sposób pojawiają się w różnych dziedzinach matematyki. Mają one zastosowanie w teorii reprezentacji, geometrii algebraicznej, oraz fizyce statystycznej. Ze względu na ich kombinatoryczną naturę, kluczem do zrozumienia wspomnianych związków jest zrozumienie kombinatorycznych reguł opisujących naturalne algebraiczne operacje mające teorio-reprezetacyjne lub geometryczne motywacje. Na wykładzie przedstawimy kilka klasycznych przykładów wspomnianych reguł, oraz opiszemy różne problemy badawcze, które obecnie leżą w centrum zainteresowania kombinatoryki algebraicznej funkcji symetrycznych. W szczególności przedstawimy hipotezę o Schur-dodatniości pewnej nowej klasy funkcji, którą można traktować jak wielowymiarowe uogólnienie wielomianów Macdonalda. Hipoteza ta w naturalny sposób uogólnia słynną ex-hipotezę Macdonalda udowodnioną przez Haimana w 2001 poprzez znalezienie głębokiego związku między kombinatoryką, teorią reprezentacji grup permutacji, a schematami Hilberta. Jeśli czas pozwoli, omówimy potencjalną metodę ataku przedstawionej hipotezy.

8.11.2018: Rozmaitości kwaternionowe są przykładem geometrii Cartana typu projektywnego. Dla nich da się dobrze zdefiniować odpowiedniość twistorową uogólniającą na wyższe wymiary (dim=4n) znaną odpowiedniość Penrosa dla czterowymiarowych rozmaitości konforemnych auto-dualnych. W referacie wprowadzę podstawy dotyczące odpowiedniości twistorowej dla rozmaitości kwaternionowych i jej związki z zespoloną geometrią kontaktową a następnie przedstawię wyniki otrzymane we współpracy z D. Calderbankiem (Bath) dotyczące rozmaitości kwaternionowych z działaniem okręgu.

19.11.2018: Projective duality can be used to study singularities. A matrix is singular precisely when its determinant vanishes, or equivalently, when it belongs to the projective dual to rank-one matrices, the Segre variety. A higher order tensor is singular when its hyperdeterminant vanishes, i.e. when it belongs to the dual of a higher order Segre product. Efficient expressions for hyperdeterminants are mostly unknown and they are difficult to compute. We describe a connection to the exceptional Lie algebra $E_8$. This gives an interpretation of certain hyperdeterminants (of formats $2\times 2\times 2\times 2$ and $3\times 3\times 3$) and certain discriminants (of the Grassmannians $Gr(3,9)$ and $Gr(4,8)$) as sparse $E_8$-discriminants. We give expressions of these high degree invariants in terms of lower degree fundamental invariants, which allow evaluation, and may be useful for Quantum Information Theory as measures of entanglement. This is joint work with Frédéric Holweck.

22.11.2018: When an ideal is defined only by combinatorial means, one expects to have a combinatorial description of its algebraic invariants. An attempt to achieve this description often leads to surprisingly deep combinatorial questions. White's conjecture is an example. It asserts that the toric ideal associated to a matroid is generated by quadratic binomials corresponding to symmetric exchanges. Another example is a question of Herzog and Hibi about existence of a quadratic Gröbner basis of the toric ideal of a matroid. Both problems reduce to questions about arrangements of bases in a matroid. We will review recent progress and state some intriguing problems.

29.11.2018: Jednym z fundamentalnych problemów w Geometrii Algebraicznej jest obliczanie wymiarów systemów liniowych na rozmaitościach rzutowych w przestrzeniach dowolnie wymiarowych. Na płaszczyźnie rzutowej jego hipotetycznym rozwiązaniem jest tzw. hipoteza SHGH. Jej pierwsza wersja została sformułowana w 1961 roku, mimo to hipoteza wciąż pozostaje otwarta. W pracy z 2017 roku D. Cook II, B. Harbourne, J. Migliore i U. Nagel zaproponowali nowe spojrzenie na problem obliczania wymiarów systemów liniowych. Wprowadzili definicję krzywej nieoczekiwanej, tj. elementu systemu liniowego [I(Z+m_1P)]_{m_1+1}, który jest wyjątkowy dla ogólnego punktu P. Celem tego wystąpienia będzie zaprezentowanie uzyskanych dotychczas wyników związanych z tym problemem, na podstawie wspólnych prac z M. Di Marca i A. Oneto, a także T. Bauer, T. Szemberg i J. Szpond.

3.01.2019:

We introduce and prove a conjecture of Naito-Sagaki which gives a branching rule for the restriction of a complex, finite-dimensional, irreducible representation of the special linear group, to the symplectic group. The rule was first conjectured in 2005.

7.01.2019:

We'll give a survey of various interesting results about the set of points on the modular hyperbola $xy \equiv 1 \pmod p$ for a prime $p$. These results show that this curve is not a typical curve $f(x,y) \equiv 0 \pmod p$ and also have many surprising applications to other, seemingly unrelated, areas. Some of these properties can also be extended to points satisfying the congruence $xy \equiv 1 \pmod n$ for a composite $n$, where they become even more special.

14.02.2019:

Bender and Canfield proved in 1991 that the generating series of maps in higher genus is a rational function of the generating series of planar maps. In this talk, I give the first bijective proof of this result. Our approach starts with the introduction of a canonical orientation that enables us to construct a bijection between 4-valent bicolorable maps and a family of unicellular blossoming maps.

22.02.2019:

In the first part of this talk, I will define a sequence of polynomials resembling the Chebyshev polynomials of the first kind, and present results on their irreducibility and zero distribution. I will also consider $2\times 2$ Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval $(-1,1)$, while those of the other half lie on the unit circle.

If time allows, I will also talk about various other number-theoretic polynomials, in particular, polynomials with gcd powers as coefficients, and once again consider their irreducibility and zero distribution.

27.02.2019:

One form of the famous Cayley formula counting labelled trees says that the number of factorizations of the long cycle (1,2,...,n) in the symmetric group S_n, into (n-1) transpositions, is n^{n-2}. This formula appeals naturally for three generalizations:

1) Replace S_n by an arbitrary finite-dimensional reflection group W, the long cycle by a Coxeter element, and transpositions by reflections.

Such a generalization was conjectured by Loojienga and proved case-by-case by Deligne, Tits, Zagier in the 1970's. The problem is still poorly understood and only very recently have case-free proofs been given, by Jean Michel and by Theodosios Douvropoulos.

2) Stay in S_n but consider "higher genus" factorizations, in which the number of transpositions is now n-1+2g for a «genus» parameter g. This generalization was made by David Jackson using representation theory in the 1980's.

3) Kirchoff's matrix-tree theorem: put weights x_{i,j} on the transposition (i,j) and express the weighted number of factorizations in terms of spectral parameters of the Laplacian matrix.

I will first discuss a common generalization to 1. and 2. obtained with Christian Stump a few years ago. If time allows I will also mention a very recent work with Theodosios Douvropoulos, in which we give a common generalization of 1,,2.,.3, under certain conditions. The problem invites us to consider towers of groups and natural generalizations of the well known Jucys-Murphy elements in the symmetric group, that seem to be interesting on their own.

6.03.2019:

In this talk we will explain how to use the geometry of affine buildings to study representations of algebraic groups. What is the role of symmetry?

20.03.2019:

I will report on the recent work on a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior.

Definition. Let $X\subseteq \P^N$ be a smooth, complete, complex variety. Let $m \geq 1$ be an integer. The $m$-th osculating

space to $X$ at $P$ is the linear subspace $Osc^{(m)}_P(X)$ determined at a point $P \in X$ by the partial derivatives of order $\leq m$ of the coordinate functions, with respect to a system of local parameters for $X$ at $P$, evaluated at $P$.

It is natural to expect that the space $Osc_P^{(m)}(X)$ has at a general point $P\in X$ the (projective) dimension $\binom{n+m}{n}-1$, where $n$ is the dimension of $X$. If this dimension is lower than $\binom{n+m}{n}-1$ at every point, then we say that $X$ is hypo-osculating of order $m$.

In this talk I will identify unexpected plane curves of degree $4$ as sections of a rational surface $B$ of degree $7$ in $\P^5$ with its hypo-osculating spaces of order $2$.

8.06.2019:

D. Guan has constructed examples of non-Kähler, simply-connected holomorphically symplectic manifolds from Kodaira surface. Another construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. I will discuss deformation theory of these manifolds, in particular, that holomorphically symplectic deformations of these manifolds are unobstructed. Using this we obtain local Torelli theorem, which implies existence of symmetric quadratic form q on H^2 (M) analogous to BBF-form of hyperkähler manifolds. This is joint work with M. Verbitsky.

12.06.2019:

Smooth rational curves play a fundamental role for the structure of a K3 surface. In this talk, I will focus on the case of low degree curves and explain some old and some new results for different polarizations, extending work of Miyaoka and Degtyarev. This is joint work with S. Rams.

19.06.2019:

I will report on a joint work in progress with Ch. Lehn and G. Mongardi in which we explore the possibility of extending the Kawamata-Morrison cone conjecture to singular irreducible holomorphic symplectic varieties.The conjecture for smooth IHS varieties was recently established by Amerik-Verbitsky, based upon previous works due to Markman and Amerik-Verbitsky (see also Markman-Yoshioka).