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## WS 2019/2020

### 15.10.2019 um 16:15 Uhr in 69/125:

#### Mandira Mondal (Chennai Mathematical Institute, India)

##### Density function for the second coefficient of the Hilbert-Kunz function on projective toric varieties

### 05.11.2019 um 16:15 Uhr in 69/125:

#### Ilya Smirnov (Stockholm University)

##### Improving Lech's inequality

Lech's inequality is a fundamental inequality in a local ring that uniformly relates the multiplicity and the colength of a finite colength ideal. For monomial ideals there is a nice combinatorial proof and the general proof reduces to this case. Lech's inequality is never sharp in dimension at least two and in my talk I will present a way to fix it.

### 12.11.2019 um 16:15 Uhr in 69/125:

#### Yairon Cid Ruiz (MPI Leipzig)

##### Noetherian operators, primary submodules and symbolic powers

We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an equivalent description of primary submodules in terms of differential operators. As a consequence, we introduce a new notion of differential powers which coincides with symbolic powers in many interesting non-smooth settings, and so it could serve as a generalization of the Zariski-Nagata Theorem.

### 19.11.2019 um 16:15 Uhr in 69/125:

#### Andrew Newman (TU Berlin)

##### Enumerating simplicial complexes up to homotopy equivalence

The Dedekind numbers enumerate labeled simplicial complexes on n vertices and grow at a rate which is doubly exponential in n. Here we show that such a rate of growth also holds for simplicial complexes on n vertices even enumerating only up to homotopy equivalence. This is accomplished by exhibiting many possible homology groups which are realizable by simplicial complexes on n vertices. This is direction of research is motivated by surprising properties observed in homology of random complexes and the proof relies on the probabilistic method.

### 19.11.2019 um 17:15 Uhr in 69/125:

**Mitra Koley** (Chennai Mathematical Institute, India)

##### Hilbert-Kunz functions and Hilbert-Kunz multiplicities of Rees algebras

First we define Hilbert-Kunz functions and Hilbert-Kunz multiplicity of a local/graded ring. Then we discuss Hilbert-Kunz functions and Hilbert-Kunz multiplicities of Rees algebras. This is joint work with J.K. Verma and K. Goel.

### 26.11.2019 um 16:15 Uhr in 69/125:

#### Joseph Samuel Doolittle (FU Berlin)

##### Relative Complexes as a Construction Tool

Many properties defined for simplicial complexes allow an equivalent definition for relative complexes. Cohen-MaCaulay, shellability, partitonability, and homology have fundamentally equivalent definitions for both simplicial and relative complexes. We take advantage of this to find relative complexes with interesting properties that would be intensely challenging to find directly in simplicial complexes. Then through repeated gluing, we can construct simplicial complexes with our desired interesting properties. This technique which was first used to great effect by Duval, Goeckner, Klivans, and Martin in 2016. We will illustrate the technique with two recent results in joint work with Goeckner and Lazar.

### 26.11.2019 um 17:15 Uhr in 69/125:

**Ulrich von der Ohe** (Università degli Studi di Genova, Italy)

##### Algebraic decomposition of functions from evaluations

The task of decomposing certain functions in terms of given vector space bases is historically motivated by physics and recently finds applications in signal and image processing. Prony's work from 1795 remains central to this subject. In this talk, we discuss instances and variations of this problem, their Prony structures and relations. The talk is based on recent joint work with Stefan Kunis and Tim Römer.

### 28.11.2019 um 16:15 Uhr in 69/125:

**Bogdan Ichim** (University of Bucharest, Romania)

##### On a class of Gorenstein polytopes

We present a new class of Gorenstein polytopes, which was found by experimental computations with Normaliz.

### 03.12.2019 um 16:15 Uhr in 69/125:

#### Marvin Hahn (Goethe Universität Frankfurt)

##### Mustafin models of plane curves and syzygy bundles

Mustafin varieties are degenerations of projective spaces, which are induced by point configurations in a Bruhat Tits building. In this talk, we use these degenerations to construct certain models of plane curves. Motivated by recent advances towards a p-adic Simpson correspondence, we then use these models to construct families of syzygy bundles which admit strongly semistable reduction. This talk is based on a joint work with Annette Werner.

### 14.01.2020 um 16:15 Uhr in 69/125:

#### Marta Panizzut (FU Berlin)

##### Tropical cubic surface

### 21.01.2020 um 16:15 Uhr in 69/125:

#### Graham Keiper (McMaster University, Canada)

##### Decompositions of Toric Ideals of Finite Simple Graphs

I will discuss recent joint work which allows us to construct new finite simple graphs from two known ones in a specified way such that the corresponding toric ideals split. This construction more generally behaves well with respect to generators of the toric ideals of the graphs used in the construction. In some cases the technique allows us to recover the graded betti numbers of the resulting graph given that this information is known for the graphs used to construct it. I will also discuss more general results about the independence of generators of toric ideals.

### 28.01.2020 um 16:15 Uhr in 69/125:

#### Jorge Alberto Olarte (RWTH Aachen)

##### The moduli space of Harnack curves

Harnack curves are a special family of plane real algebraic curves which are characterized by having amoebas that behave in the simplest possible way. In this talk, we will show how to compute the moduli space of Harnack curves with a given Newton polygon. Moreover, we will show how this moduli space can be compactified by considering collections of Harnack curves that can be glued using Viro's patchworking method to obtain another Harnack curve. This involves using abstract tropical curves. We show that this compactifcation has a cell structure whose poset is isomorphic to the face poset of the secondary polytope of the Newton polygon. We end by discussing several questions for future work.