15 MAY 2025
14:00 - 16:00

An imaginary sort of a first-order structure is a possible universe of an interpretable structure. The significance of imaginary sorts was established in the abstract setting of Classification Theory by Shelah, where all notions of stability theory, especially algebraic closure, were interpreted with imaginary sorts taken into account. Imaginaries were studied for specific theories by Poizat who showed that algebraically closed fields and differentially closed fields eliminate imaginaries, i.e. each imaginary sort is in definable bijection with a definable set of n-tuples. In valued fields imaginary sorts differ qualitatively from definable sets; an example is the value group. In this case it is a question of finding a useful basis for the imaginary sorts, meaning that every imaginary sort can be embedded into a product of basis elements. A basis for imaginary sorts for algebraically closed valued fields was given by Haskell-Hrushovski-Macpherson. Rideau-Hrushovski-Martin, and more recently Hils-Rideau-Kikuchi, proved that for the field of $p$-adic numbers the $p$-adic lattices provide a basis for imaginary sorts.
In this talk I will describe joint work with Ehud Hrushovski which gives a description of imaginary sorts and their basis for products and restricted products of structures in terms of a basis for imaginary sorts for the factors. In 1959 Feferman and Vaught proved that the theory of a product is determined by the theory of the factors and the theory of the Boolean algebra of subsets of the index set. This yields decidability results and descriptions of definable sets for products in terms of the factors. Our results show that imaginaries also admit a description relative to the factors. The proofs use tools from geometric stability theory including liaison groups, and descriptive set theory of smooth Borel equivalence relations, including a dichotomy theorem due to Harrington-Kechris-Louveau and Glimm-Efros. As an application, we describe a basis for the imaginary sorts for the ring of adeles of rational numbers in terms of the family of $p$-adic lattices for �almost all� $p$, and prove weak elimination of imaginaries for the adeles in these sorts.

Joint work with: Ehud Hrushovski
Zoom room information:
https://us06web.zoom.us/j/81916335336?pwd=5zQT4lutMiBoY5Xp1pkFGtbqiaGozg.1
Meeting ID: 819 1633 5336
Passcode: 559618
Venue: Niavaran, Lecture Hall 1

14 MAY 2025
14:00 - 15:00

For any k + 1 ≥ 3 positive integers t, n_1, . . . , n_k, there exists a positive integer N such that if every t-element subset of the set {1, . . . , N} is colored using one of the k colors 1, . . . , k, then there exists an index i ∈ {1, . . . , k} and a subset S ⊆ {1, . . . , N} with |S| = n_i, such that every t-element subset of S is colored with color i. This is a generalization of Ramseys Theorem, which is often formulated in the language of graph theory by:
? interpreting {1, 2, . . . , N} as the vertex set of the complete graph K_N,
? considering the case t = 2, and
? coloring each 2-element subset (edge) with one of k colors.
Ramseys Theorem then implies that for any two positive integers s and t, there exists a minimum integer N such that in any red-blue coloring of the edges of K_N , one can find either a red complete subgraph Ks or a blue complete subgraph Kt. The smallest such N is known as the Ramsey number R(s, t), or more generally R(G, H) when considering arbitrary graphs G and H. Several variants of Ramsey numbers have been developed. Notably, the bipartite Ramsey number, the multipartite Ramsey number, the m-bipartite Ramsey number, and the pair bipartite Ramsey number extend the classical notions to bipartite and multipartite settings.
In this talk, we will present recent results concerning some of these extended versions. In particular, we will focus on:
? The multicolor bipartite Ramsey number BR(G_1, G_2, . . . , G_t), defined as the smallest integer n such that any t-edge-coloring of the complete bipartite graph K_{n,n} contains a monochromatic copy of G_i for some 1 ≤ i ≤ t.
? The pair bipartite Ramsey number Bp(G_1, G_2) = (a, b), the smallest ordered pair such that every red-blue edge-coloring of K_{c,d} with c ≥ d guarantees a blue copy of G_1 or a red copy of G_2, provided c ≥ a and d ≥ b.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

14 MAY 2025
17:30 - 19:00

I will start with a review of KSB/A stability, especially their local version and then discuss joint work with Janos Kollar, showing that it is enough to check these conditions, including flatness, up to codimension 2. This implies that we have a very good understanding of this stability condition in general, because local KSB-stability is trivial at codimension 1 points, and quite well understood at codimension 2 points, since we have a complete classification of 2-dimensional slc singularities.

Venue: online
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Meeting ID: 9086116889
Passcode: 362880
Website: http://math.ipm.ac.ir/agnt/
Venue: , (Online)

14 MAY 2025
15:30 - 17:00

The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two Dehn surgeries with different surgery coefficients r and r' results in distinct oriented three-manifolds. Earlier works on the conjecture imply that one needs to consider only the case r'=-r and r=2 or 1/n for some integer n. In this talk, I'll explain how one can remove the case of r=1/n, reducing the conjecture to the case r=2. Furthermore, in the case r=2, the Alexander polynomial of the knot is trivial. The key tools in the proof are a persistence module associated to any integer homology sphere using instanton gauge theory and an exact triangle relating instanton Floer homology groups of 1/n-surgeries on a knot. This talk is based on a joint work with Tye Lidman and Mike Miller Eismeier.

Online via Zoom:
Meeting ID: 908 611 6889
Passcode: 362880
Venue: , (Zoom Lecture)

7 MAY 2025
14:00 - 15:00

A signed graph (G, σ) is a graph together with an assignment σ which assigns a sign (positive or negative) to each edge. Sign of a cycle in (G, σ) is the product of signs of its edges. A subset of vertices is said to be balanced if it induces no negative cycle.
The balanced chromatic number of a signed graph is the minimum number of balanced sets that covers all vertices of (G, σ). The parameter is studied under various equivalent terms.
In this talk, after presenting some basic properties of balanced coloring and balanced chromatic number, we will have a look at how structural conditions, such as forbidden minors, impacts the balanced chromatic number.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

1 MAY 2025
14:00 - 16:00

Intuitively, generic sampling is a method for sampling types over monster models of a first-order theory with respect to a Keisler measure. In practice, this involves studying countably iterated Morley products of relatively tame Keisler measures. Given such a measure, one can straightforwardly derive a corresponding Borel measure on the space of L-structures over the natural numbers. In particular, if the original monster model is a graph (respectively, a hypergraph), then generic sampling yields measures on the space of all graphs (respectively, hypergraphs) on the natural numbers.
A graphon is a symmetric measurable map from $[0,1]^{2}$ to $[0,1]$. These graph-like objects also give rise to a notion of sampling. It turns out that all Sym(N)-invariant measures on the space of graphs (over the natural numbers) are mixtures of sampling procedures arising from graphons. We remark that graphons admit a higher-arity generalization to hypergraphs, called hypergraphons, which exhibit a similar property.
We prove a representation theorem: (hyper)graphon sampling can be encoded within the framework of generic sampling. We remark that generic sampling in practice can give rise to non-Sym(N)-invariant measures and so generic sampling is strictly more general than what arises from (hyper)graphons. This is joint work with Nathanael Ackerman, Cameron Freer, James Hanson, and Rehana Patel.

Zoom room information:
https://us06web.zoom.us/j/81916335336?pwd=5zQT4lutMiBoY5Xp1pkFGtbqiaGozg.1
Meeting ID: 819 1633 5336
Passcode: 559618
Venue: Niavaran, Lecture Hall 1

30 APR 2025
14:00 - 15:00

Imagine a game played between two players, Maker and Breaker, on an infinite complete graph G. The players alternately claim edges of G, beginning with Maker, and play continues forever. Maker's aim is that by the end of the game she should have claimed all edges of an infinite complete subgraph of G, and Breaker's aim is to prevent this. I will discuss a number of simple variants of this game and examine who has a winning strategy in each case.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

30 APR 2025
17:30 - 19:00

Let $X$ be a smooth connected projective $K3$ surface over the complex numbers and let $Spl(r; c_1, c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. In 1984, Mukai proved that $Spl(r; c_1, c_2)$ is a smooth algebraic space of dimension $2rc_2-(r-1)c_1^2-2r^2+2$ with a natural symplectic sstricture, i.e., it has a non-degenerate closed holomorphic 2-form. In my talk, I will present a useful method to construct isotropic and Lagrangian subspaces of $Spl(r; c_1, c_2)$. This is joint work with Barbara Fantechi.

https://zoom.us/join
Meeting ID: 9086116889
Passcode: 362880
Website: http://math.ipm.ac.ir/agnt/
Venue: , (Online)