17 JUN 2026
14:00 - 15:00

We first show the capabilities of AI in mathematics by considering different philosophies of mathematics and their relationship with AI. Then we use attribution techniques and exploratory data analysis to give sufficient conditions for a given statement and make conjectures. As a demonstration, we apply this process to Stanley's problem of e-positivity of graphs. Guided by AI, we rediscover that one sufficient condition for a graph to be e-positive is that it is co-triangle-free, and that the number of claws is the most important factor for e-positivity. Based on the most important factors identified in the saliency map analysis of neural networks, we suggest that the classification of e-positive graphs is more likely related to continuous graph invariants rather than discrete ones, and we make some conjectures connecting continuous graph invariants with e-positivity.

This week's talk will be on our Zoom room:
https://us06web.zoom.us/j/86247043799?pwd=KH1lKfYt1rGLzSgKk31MwprllTiOef.1
Meeting ID: 862 4704 3799
Passcode: 362880
Venue: Niavaran, Khosrovshahi Lecture Hall

25 FEB 2026
14:00 - 15:00

The arboreal gas with fugacity $\lambda$ is the probability measure on forests in a graph that assigns weight proportional to $\lambda^{\# edges}$ to a forest. A longstanding conjecture due to Kahn says that edge events in the arboreal gas are negatively correlated, similar to the case of spanning tree measures (the case when $\lambda = \infty$).
This problem was one of the key motivations for the introduction of the class of Lorentzian polynomials (by Gurvits, Anari-Liu-Oveis Gharan-Vinzant and Branden-Huh). While Lorentzian polynomials have been used to settle a number of other problems in discrete probability and combinatorics, they do not address this particular problem.
I will show in this talk that the conjecture is true for most graphs when $\lambda$ is sufficiently large. This is arguably the first general result on this specific problem.
More generally, Random Cluster measures $\mu^{RC}_{q, \lambda}$ for $q < 1$ were conjectured by Grimmett and Winkler to exhibit edge negative correlation. I will explain why this is true in the mean field case, on $K_n$ with edge weights all $1$. I will also relate this to a curious new correlation conjecture involving bridges in graphs and discuss applications.
This is joint work with Recep Altar Ciceksiz (Umea University)

This week's talk will be on our Zoom room:
https://us06web.zoom.us/j/86247043799?pwd=KH1lKfYt1rGLzSgKk31MwprllTiOef.1
Meeting ID: 862 4704 3799
Passcode: 362880
Venue: Niavaran, Khosrovshahi Lecture Hall

25 FEB 2026
13:30 - 15:00

In this talk, we will explore the area of function field arithmetic, with a focus on Drinfeld's elliptic sheaves and their generalizations, as well as analogies to the number field setting. Drinfeld modules, introduced in 1974 as analogues of elliptic curves in the function field setting, play a central role in this context. To establish a Langlands correspondence, Drinfeld studied moduli spaces of elliptic sheaves, or equivalently, shtukas. After a brief introduction to the function field framework, we will examine some well-known generalizations of elliptic sheaves, concentrating on generalized D-elliptic sheaves and presenting results on their moduli spaces. In the final part of the talk, we will explore the connections between (generalized) shtukas and (generalized) elliptic sheaves.

Zoom details:
Zoom link: https://kocun.zoom.us/j/99715471656
Meeting ID: 997 1547 1656
Passcode: 848084
ICS file:
https://researchseminars.org/seminar/FGC-IPM/ics
Venue: , (Online Lecture)

18 FEB 2026
15:30 - 17:00

Deep ideas silently reappear across diverse fields of mathematics. They shape nature. Determine which questions are natural to be asked, which objects are fundamental, what is a meaningful equivalence, etc. Many phenomena are deeply rooted in their essence. In this talk, we try to recall some of these metaphorical ideas. We will discuss some examples of geometrization (resp. categorification/decategorification) process, including Hecke stacks and Hecke operators.
Venue: Niavaran, Khosrovshahi Lecture Hall

18 FEB 2026
14:00 - 15:00

A graph property is called increasing if it is closed under addition of edges. For an increasing property, the probability of its occurrence in the binomial random graph G(n,p) transitions rapidly (in asymptotics) from 0 to 1 as p crosses the so-called probability threshold. Since the original paper of Erdos and Renyi the task of determining the asymptotic behaviour of threshold probabilities for increasing properties has been a central topic in probabilistic combinatorics. While the asymptotic order of the probability threshold has been determined for many natural increasing graph properties, a general solution remains unknown, and determining the exact asymptotics is even more challenging. In the talk I will provide an answer to the latter question for a class of increasing properties generated by d-regular graphs. This family of d-regular graphs contains asymptotically almost all d-regular graphs and, in particular, it contains the square of a cycle. This resolves a conjecture of Kahn, Narayanan, and Park, regarding the asymptotics of the threshold for the appearance of the square of a Hamilton cycle.
Zoom room information:
https://us06web.zoom.us/j/86247043799?pwd=KH1lKfYt1rGLzSgKk31MwprllTiOef.1
Meeting ID: 862 4704 3799
Passcode: 362880

Venue: Niavaran, Khosrovshahi Lecture Hall

5 JAN 2026
10:00 - 11:30

This talk presents a unified framework for finiteness results concerning arithmetic points on algebraic curves, exploring the analogy between number fields and function fields. The number field setting, joint work with F. Janbazi, generalizes and extends classical results of Birch-Merriman, Siegel, and Faltings. We prove that the set of Galois-conjugate points on a smooth projective curve with good reduction outside a fixed finite set of places is finite, when considered up to the action of the automorphism group of a proper integral model. Motivated by this, we consider the function field analogue, involving a smooth and proper family of curves over an affine curve defined over a finite field. In this setting, we show that for a fixed degree, there are only finitely many etale relative divisors over the base, up to the action of the family's automorphism group (and including the Frobenius in the isotrivial case). Together, these results illustrate both the parallels and distinctions between the two arithmetic settings, contributing to a broader unifying perspective on finiteness
Venue: Niavaran, Khosrovshahi Lecture Hall

31 DEC 2025
14:00 - 15:00

The study of signed graphs with two or three distinct eigenvalues has attracted increasing attention. Huang's results on the Sensitivity Conjecture, in which a signing of a d-dimensional cube graph with two distinct eigenvalues is applied, has increased the importance of the problem. In this talk, we present several properties of such signed graphs. In particular, we discuss a generalization of the following to signed graphs: The order of a graph with an eigenspace of prescribed codimension is bounded by {t+1 choose 2}. It is shown that for regular graphs the bound can be reduced by 1, and that this reduced bound is attained by a regular graph G if and only if G is an extremal strongly regular.

Zoom room information:
https://us06web.zoom.us/j/86247043799?pwd=KH1lKfYt1rGLzSgKk31MwprllTiOef.1
Meeting ID: 862 4704 3799
Passcode: 362880
Venue: Niavaran, Khosrovshahi Lecture Hall

24 DEC 2025
14:00 - 15:00

A fundamental problem in computational algebra is computing a basis for a finite Galois extension field of a given field. A particularly effective representation can be obtained through a normal basis, which is known to exist for every such extension. An interesting and fundamental problem is the complexity of computing a normal basis while taking advantage of the structure of the Galois group of the extension. Assume K/F is a finite Galois extension with G=Gal(K/F). It is well known that there exists an element a in K such that the orbit of a under the action of G forms a basis of K as an F-vector space. The element a is said to be a normal element of K/F. In this talk, we present efficient randomized algorithms that test whether a given element of K is normal in the case where G is either a finite abelian or a metacyclic group. Moreover, we show how to perform conversions between the power basis of K/F and a normal basis with the same asymptotic cost as testing normality. This is a joint work with Mark Giesbrecht and Eric Schost.

Zoom room information:
https://us06web.zoom.us/j/86247043799?pwd=KH1lKfYt1rGLzSgKk31MwprllTiOef.1
Meeting ID: 862 4704 3799
Passcode: 362880
Venue: Niavaran, Khosrovshahi Lecture Hall

17 DEC 2025
14:00 - 15:00

The celebrated Erdos-Pósa Theorem asserts a qualitative duality between feedback vertex sets and collections of vertex-disjoint cycles. In this talk, I discuss a variation of this result that considers cycles intersecting a prescribed set of vertices or edges. As a consequence, the directed tree-width of a directed graph is shown to be linearly bounded in terms of its cycle-width, improving the previously known quadratic upper bound. The corresponding problem in bidirected graphs will also be addressed. Joint work with N. Bowler, F. Gut, R. W. Jacobs, and F. Reich.

Zoom room information:
https://us06web.zoom.us/j/86247043799?pwd=KH1lKfYt1rGLzSgKk31MwprllTiOef.1
Meeting ID: 862 4704 3799
Passcode: 362880
Venue: Niavaran, Khosrovshahi Lecture Hall