## Colloquia

### 2021

9 Feb: **Dr Travis Monk **Wald's martingale and the Moran birth-death process

**Abstract: **Sequential analysis studies random walks occurring between two absorbing barriers. These random walks occur frequently in applications, e.g. hypothesis testing, stock prices, evolutionary processes, etc. Generally we want to find the probability that the random walk hits one barrier before the other, and how long it took to do so. During World War II, Abraham Wald presented an elegant solution to this problem for random walks whose steps are independent and identically distributed. His solution was a very early application of martingale theory, formalised by Joseph Doob a decade later.

In this talk I will outline Wald's work and demonstrate its power by applying it to the Moran process. The Moran process is a toy model of evolution that considers the spread of a mutation in an indigenous population. It can be recast as a random walk, where the mutant population size fluctuates randomly in time. The Moran process has two absorbing barriers: one where mutants go extinct, and another where the indigenous population goes extinct. I will show how Wald's martingale can be applied to the Moran process by eliminating time steps where the process does not change. Therefore we can easily calculate the so-called `fixation probability' of the process, and the conditional distributions of how many times the mutant population size changes before absorption. I will highlight the advantages of Wald's approach to solving such problems over alternative approaches such as Markov chains and computer simulation. I will also provide intuition to clarify why martingales are such powerful tools to study stochastic processes in general.

**Talks will be held on Thursdays 8pm Sydney Time: (UTC 10am) Please write to R. Hazrat** **to get the Zoom address.**

**Talks will be held on Thursdays 8pm Sydney Time: (UTC 10am) Please write to R. Hazrat**

**to get the Zoom address.**

### 2020

4 May: **Toke Carlsen **(Faroe Islands) Graph algebras, groupoids, and symbolic dynamics**Abstract: **I will give an overview of some recent results that link diagonal-preserving isomorphism of graph algebras and isomorphism and equivalence of graph groupoids with continuous orbit equivalence, (eventual) conjugacy, and flow equivalence of symbolic dynamical systems constructed from directed graphs. PowerPoint

21 May: **Mark Lawson **(Edinburgh) Non-commutative Stone dualities**Abstract: **The classical Stone dualities for lattices such as frames, distributive lattices and generalized Boolean algebras can be generalized to a non-commutative setting to pseudogroups, distributive inverse semigroups and Boolean inverse semigroups, respectively. The goal of this talk is to sketch out the how and to motivate the why. I shall not assume any background from inverse semigroups or \'etale groupoids. PowerPoint

28 May: **Enrique Pardo Espino** (Cadiz): Self-similar graphs and their algebras**Abstract:** In this talk, we will explain the origins of the notion of self-similar graph. We give a groupoid model of the algebra associated to a self-similar graph and we provide a characterization of simplicity for these algebras. We briefly talk about further developments on this construction. The contents of this talk are part of a joint paper with R. Exel. PowerPoint

4 June: **Mike Whittaker (**Glasgow): Aperiodic tilings: from the Domino problem to aperiodic monotiles**Abstract: **Almost 60 years ago, Hao Wang posed the Domino Problem: is there an algorithm that determines whether a given set of square prototiles, with specified matching rules, can tile the plane. Robert Berger proved the undecidability of the Domino Problem by producing a set of 20,426 prototiles that tile the plane, but any such tiling is nonperiodic (lacks any translational symmetry). This remarkable discovery began the search for other (not necessarily square) aperiodic prototile sets, a finite collection of prototiles that tile the plane but only nonperiodically. In the 1970s, Roger Penrose reduced this number to two. Penrose's discovery led to the planar einstein (one-stone) problem: is there a single aperiodic prototile? In a crowning achievement of tiling theory, the existence of an aperiodic monotile was resolved almost a decade ago by Joshua Socolar and Joan Taylor. My talk will be somewhat expository, and culminate in both a new direction in aperiodic tiling theory and a new aperiodic monotile. PowerPoint

11 June: ** Kevin Brix **(Copenhagen/Wollongong): Fine structure of C*-algebras associated to topological dynamics**Abstract**: I will report on the story of associating C*-algebras to symbolic dynamical systems (e.g. shift spaces or directed graphs) and the recently articulated program of understanding dynamical relations (such as conjugacy or flow equivalence) in terms of structure-preserving *-isomorphisms of the corresponding C*-algebras. A large body of rigidity results have successfully been obtained for graphs and more general systems. There will be an emphasis on open questions and problems yet to be solved! PowerPoint

12 June: **James Hyde** (Cornell) Finitely Presented Groups that Contain Q **Abstract:** I will present joint work with James Belk and Fancesco Matucci. We construct embeddings of the additive group of rational numbers into several finitely presented groups. We rely upon Thompson's groups, which I will describe.

18 June: **Joan Bosa **(Barcelona) The realization problem for von Neumann regular rings**Abstract: **The realization problem for von Neumann (vN) regular rings asks whether all conical refinement monoids arise from monoids induced by the projective modules over a vN regular ring. We will quickly overview this problem, and show the last developments on it. This is joint work with P. Ara, E.Pardo and A.Sims. PowerPoint

June 25th: **Pere Ara** (Barcelona) Graded K-Theory, Filtered K-theory and the classification of graph algebras**Abstract: **We prove that an isomorphism of graded Grothendieck groups of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered K-theory and consequently an isomorphism of filtered K-theory of their associated graph C*-algebras. As an application, we show that, since for a finite graph E with no sinks, the graded Grothendieck group of L(E) coincides with Krieger's dimension group of its adjacency matrix, our result relates the shift equivalence of graphs to the filtered K-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graAbstract: We prove that an isomorphism of graded Grothendieck groups of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered K-theory and consequently an isomorphism of filtered K-theory of their associated graph C*-algebras. As an application, we show that, since for a finite graph E with no sinks, the graded Grothendieck group of L(E) coincides with Krieger's dimension group of its adjacency matrix, our result relates the shift equivalence of graphs to the filtered K-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph C*-algebras. This result was only known for irreducible graphs. This is a joint work with Roozbeh Hazrat and Huanhuan Li. PowerPoint

July 2: **Aidan Sims **(Wollongong) Graded K-theory for Z_2-graded graph C*-algebras**Abstract:** While there is no universally agreed-upon definition of Z_2-graded K-theory for C*-algebras, a very natural way to define it is using Kasparov's celebrated KK-bifunctor: KK is naturally a Z_2-graded theory, and Kasparov proved that if applied to trivially-graded C*-algebras A, the groups KK_*(\mathbb{C}, A) are the K-groups of A. So it is natural to define K^{gr}_*(A) as KK_*(\mathbb{C}, A) for Z_2-graded C*-algebras A in general. I will discuss recent work with Adam Sierakowski and with honours students Quinn Patterson and Jonathan Taylor, building on previous work with Kumjian and Pask, that uses deep ideas of Pimsner to compute the graded K-theory, defined in this way, of relative graph C*-algebras carrying Z_2-gradings determined by binary labellings of the edges of the graph: the formulas that emerge strongly suggest that this notion of graded K-theory captures the right sort of information. PowerPoint

July 9: **Xiao-Wu Chen** (Hefei): Leavitt path algebra via the singularity category of a radical-square-zero algebra**Abstract: **We will recall some previous work primarily by Paul Smith, and show that the Leavitt path algebra is closely related to the singularity category of a finite dimensional radical-square-zero algebra. Recently, we

apply such a link to confirm Keller's conjecture for a radical-square-zero algebra. More precisely, we prove that

for such an algebra, the singular Hochschild cochain complex is B_infinity-isomorphic to the Hochschild cochain complex of the dg singularity category. This is based on a joint work with Huanhuan Li and Zhengfang Wang.PowerPoint

July 16: **Jean Renault** (d'Orleans) : Groupoids Extensions.**Abstract: **I shall present a groupoid version of the Mackey normal subgroup analysis in a C*-algebraic framework. More precisely, the main result is a description of the C*-algebra of a locally compact groupoid with Haar system, possibly endowed with a twist, which is an extension by a group bundle. The natural expression of this result uses Fell bundles over groupoids. When the group bundle is abelian, one obtains a twisted groupoid C*-algebra. I will give some applications. This talk is based on a joint work with M.Ionescu, A.Kumjian, A.Sims and D.Williams. PowerPoint

July 23: **Be'eri Greenfeld** (Bar Ilan) How do algebras grow?**Abstract: **The question of `how do algebras grow?', or, which functions can be realized as growth functions of algebras (associative/Lie/etc., or algebras having certain additional algebraic properties) is a major problem in the junction of several mathematical fields, including noncommutative algebra, combinatorics of (infinite) words, symbolic dynamics, self-similarity and more. We provide a novel paradigm for tackling this problem (in fact, family of problems), thereby resolving several open problems posed by experts regarding possible growth types of finitely generated associative algebras and Lie algebras. We also considerthe set of growth functions as a space, and point out odd properties it admits (arbitrarily rapid holes, and convergence to outer points - with respect to some plausible notion of limits). PowerPoint

July 30: **Arun Ram **(Melbourne): Teaching mathematics in the next life**Abstract:** For many years Ive been thinking about how to teach mathematics with honesty and inspiration. This has resulted in ideas like "proof machine", "marking apocalypse", "rotary dial phenomenon", and "just do it". And then a virus came, and the new life began, online, on Zoom. This will be a talk about the adventures of the past life and the preparations for the next. PowerPoint

August 6: **Bob Gray** (East Anglia) Undecidability of the word problem for one-relator inverse monoids

**Abstract: ** It is a classical result of Magnus proved in the 1930s that the word problem is decidable for one-relator groups. This result inspired a series of investigations of the word problem in other one-relator algebraic structures. For example, in the 1960s Shirshov proved the word problem is decidable in one-relator Lie algebras. In contrast, it remains a longstanding open problem whether the word problem is decidable for one-relator monoids. An important class of algebraic structures lying in between monoids and groups is that of inverse monoids. In this talk I will speak about a recent result which shows that there exist one-relator inverse monoids of the form Inv<A|w=1> with undecidable word problem. This answers a problem originally posed by Margolis, Meakin and Stephen in 1987. I will explain how this result relates to the word problem for one-relator monoids, the submonoid membership problem for one-relator groups, and to the question of which right-angled Artin groups arise as subgroups of one-relator groups.PowerPoint

August 13: **Cristobal Gil Canto **(Malaga) Invariant ideals in Leavitt path algebras.

**Abstract: ** As well-known examples of Leavitt path algebras arise the so-called primary colours: they respectively correspond to the ideal generated by the set of line points, the vertices that lie on cycles without exits and the one generated by the set in extreme cycles. It is known that these ideals are invariant under isomorphism. In this talk we will analyze the invariance of another key piece of a Leavitt path algebra. We will see that though the ideal generated by the vertices whose tree contains infinitely many bifurcation vertices or at least one infinite emitter is not invariant, we will find its natural replacement (which is indeed invariant). We will also give some procedures to construct invariant ideals from previous known invariant ideals. In order to do that, on the one hand, we will introduce a topology in the set of vertices of a graph. And on the other hand, via category theory, we will think of the saturated and hereditary set of a graph as a functor. This a joint work together with Dolores Martín Barquero and Cándido Martín González.

August 20: **Jose Burillo **(Barcelona): The irrational-slope Thompson's groups**Abstract:** Irrational-slope Thompson's groups were introduced by Cleary in two papers in 1995 and 2000, where he proved they are FP_\infty. These are groups of PL maps of [0,1] whose breakpoints are in some irrational subring of R and the slopes are also irrational numbers. Interest in these groups grew recently when it was asked whether they can be obtained as subgroups of Thompson's group F. In this paper we will introduce the golden ratio group F_\tau, describe how to work with it in terms of binary trees and also algebraically. We will show a presentation for F_\tau and show that elements admit a unique normal form, in similar fashion as F. We will study its metric properties and undistorted copies of F inside, and finally, if time permits, we will say a few words about the irrational versions of Thompson's groups T and V. This is joint work with Brita Nucinkis and Lawrence Reeves. PowerPoint

August 27: **Ralf Meyer** (Goettingen) Aperiodicity and related properties for crossed product inclusions**Abstract: **In recent work with Bartosz Kwaniewski, we have vastly generalised the condition that was introduced by Kishimoto in order to prove that reduced crossed products for outer group actions on simple C*-algebras are again simple. We call this condition aperiodicity, and it applies to arbitrary inclusions of C*-algebras, without requiring a crossed product structure. We relate this to topological non-triviality conditions in the special case of actions of inverse semigroups or étale groupoids (which are possibly non-Hausdorff). In that generality, we define an essential crossed product, which is a quotient of the reduced crossed product. If the action satisfies Kishimoto's condition, then the coefficient algebra detects ideals in this essential crossed product. And in the simple case, we also get criteria for the essential crossed product to be simple. We also relate aperiodicity to other properties that have been used to study the ideal structure of crossed products. This includes unique pseudo-expectations and the almost extension property, which assume that the set of pure states on the coefficient algebra that extend uniquely to the crossed product is dense. PowerPoint

September 3: **Lia Vas **(Philadelphia) The Graded Classification Conjecture for graph algebras**Abstract: **The ordinary (pointed) K_0-group is not a complete invariant of algebras typically associated to a directed graph. When these algebras are considered as graded algebras and the definition of the K_0-group is adjusted to reflect the existence of this grading, the situation becomes more interesting. The Graded Classification Conjecture states that this adjusted version of the (pointed) K_0-group is a complete invariant of a Leavitt path algebra over a field (and this statement can be adapted for other graph algebras). We shall discuss the context in which this conjecture has been formulated, the current status of the conjecture, and some ongoing research. PowerPoint

September 10: **Volodymyr Mazorchuk** (Uppsala): Adjunction in the absence of identity**Abstract:** In this talk I plan to present and discuss a rather weak bicategorical setup in which one can talk about genuine adjunctions. I will roughly describe the main motivation coming from representation theory of finitary 2-categories (or bicategories) and make some parallells with the structure theory of finite semigroups. I will try to explain how this approach simplifies some results but also makes some other results much more difficult. This is a joint work with Hankyung Ko and Xiaoting Zhang. PowerPoint

September 17: **Murray Elder** (Sydney) Some new kinds of automatic groups**Abstract:** I will describe some generalisations of the notion of an automatic group, and how far they are away from automatic (in a precise sense). Relevant papers are PowerPoint

https://arxiv.org/abs/2008.02381 and https://arxiv.org/abs/2008.02511

September 24: **Jason Bell **(Waterloo): The growth of algebras**Abstract:** We give an overview of the theory of growth functions for associative algebras and explain their significance when trying to understand algebras from a combinatorial point of view. We then give a classification for which functions can occur as the growth function of a finitely generated associative algebra up to asymptotic equivalence. This is joint work with Efim Zelmanov. PowerPoint

30 September - October 2: **CRMDS-IPM** **Joint Workshop on Operator Algebras**

A joint workshop of CRMDS and Institute for Research in Fundamental Sciences, Tehran (IPM, conveyer, **Prof. Massoud Amini, **mamini@ipm.ir) on Operator Algebras will be held 30Sep 2-Oct via Zoom. Program

**Alcides Buss** (Brazil) : Amenability for actions of groups on C*-algebras**Abstract: **In this lecture I will explain recent developments in the theory of amenability for actions of groups on C*-algebras and Fell bundles, based on joint works with Siegfried Echterhoff, Rufus Willett, Fernando Abadie and Damián Ferraro. Our main results prove that essentially all known notions of amenability are equivalent. We also extend Matsumura’s theorem to actions of exact locally compact groups on commutative C*-algebras and give a counter-example for the weak containment problem for actions on noncommutative C*-algebras. PowerPoint

**Alex Kumjian** (USA): Pushouts of groupoids by abelian group bundles**Abstract:** Given a groupoid extension of a locally compact Hausdorff groupoid by a bundle of abelian groups on which it acts, we construct a pushout twist over the groupoid semidirect product of the groupoid acting on the dual of the bundle regarded as a topological space. We then show that the C*-algebra of the original extension groupoid is isomorphic to the twisted groupoid associated to the pushout. We will also discuss examples. This talk is based on current joint work with Marius Ionescu, Jean Renault, Aidan Sims and Dana Williams. PowerPoint

**Ralf Meyer **(Germany) Groupoid models and C*-algebras of diagrams of groupoid correspondences**Abstract: **A groupoid correspondence is a generalised morphism between étale groupoids. Topological graphs, self-similarities of groups, or self-similar graphs are examples of this. Groupoid correspondences induce C*-correspondences between groupoid C*-algebras, which then give Cuntz-Pimsner algebras. The Cuntz-Pimsner algebra of a groupoid correspondence is isomorphic to a groupoid C*-algebra of an étale groupoid built from the groupoid correspondence. This gives a uniform construction of groupoid models for many interesting C*-algebras, such as graph C*-algebras of regular graphs, Nekrashevych's C*-algebras of self-similar groups and their generalisation by Exel and Pardo for self-similar graphs. If possible, I would also like to mention work in progress to extend this theorem to relative Cuntz-Pimsner algebras, which would then cover all topological graph C*-algebras. Groupoid correspondences form a bicategory. This structure is already used to form the groupoid model of a groupoid correspondence. It also allows us to define actions of monoids or, more generally, of categories on groupoids by groupoid correspondences. Passing to C*-algebras, this gives a product system where the unit fibre is a groupoid C*-algebra. If the monoid is an Ore monoid, then the Cuntz-Pimsner algebra of this product system is again a groupoid C*-algebra of an étale groupoid, which is defined directly from the action by groupoid correspondences. For more general monoids, the two constructions become different, however. We show this in a special case that is related to separated graph C*-algebras and their tame versions. PowerPoint 1 PowerPoint 2 PowerPoint 3

**Chris Phillips** (USA): Crossed Products by Automorphisms of C(X;D)**Abstract**: We consider crossed products of the form C*(Z, C(X;D), α) in which D is simple, X is compact metrizable, α induces a minimal homeomorphism h: X→ X, and a mild technical assumption holds. In a number of examples inaccessible via methods based on finite Rokhlin dimension, either because D is not Z-stable or because X is infinite dimensional, we prove structural properties of the crossed product, such as (tracial) Z-stability, stable rank one, real rank zero, and pure infiniteness. The method is to find a centrally large subalgebra of the crossed product which is a direct limit of "recursive subhomogeneous algebras over D". With a better understanding of such direct limits, many more examples would become accessible. This is joint work with Dawn Archey and Julian Buck. PowerPoint

**Jean Renault**: KMS states and groupoid C*-algebras**Abstract:** I will illustrate the use of groupoids in the study of KMS states and weights on C*-algebras. The KMS condition, which was introduced in quantum statistical mechanics to characterize equilibrium states, plays a crucial role in the theory of von Neumann algebras. The study of KMS states and their phase transitions on specific C*-algebras, in particular graph algebras, is an active field of research where the groupoid techniques are well suited. PowerPoint

**Aidan Sims **(Australia): Reconstruction of groupoids, and classification of Fell algebras**Abstract:** I will the history of reconstruction of groupoids from pairs of operator algebras, from Feldman and Moore’s results on von Neumann algebras through Kumjian’s and then Renault’s results about C*-algebras of twists, and including some recent results about groupoids that are not topologically principal. I will finish by outlining how Kumjian’s theory leads to a Dixmier-Douady classification theorem for Fell algebras. PowerPoint 1 PowerPoint 2

**Gabor Szabo **(Belgium): Dynamical criteria towards classifiable transformation group C*-algebras**Abstract:** In this talk I will report on joint work with David Kerr regarding the structure and classification of certain transformation group C*-algebras. It is a general important question when free minimal actions of amenable groups on compact spaces give rise to crossed product C*-algebras that fall within the scope of Elliott's program. After some years of research where this had been partially settled for special classes of groups with methods related to noncommutative dimension theory, Kerr's notion of almost finiteness opens the door to systematically study this problem for all amenable groups. I will give an overview of these techniques and the current state-of-the-art, culminating in our result that asserts the classifiability of such crossed products if the underlying space is finite-dimensional and the group has subexponential growth. PowerPoint

**Dana Williams **(USA): Morita equivalence, the equivariant Brauer group, and beyond**Abstract:** I will give a brief survey of work on the equivariant Brauer group together with the necessary preliminaries as well as generalizations involving groupoid C*-algebras.PowerPoint1 PowerPoint 2

October 15** Itamar Stein **(Ashdod) Representation theory of the monoid of all partial functions on a set and other Ehresmann semigroups

**Abstract: **Given a finite semigroup S, we can study its linear representations (for this talk - over the field of complex numbers). Semigroups with natural combinatorial structure are clearly of major interest. An important example of such semigroup is the monoid of all partial functions on an n element set, denoted PT_n. A description of its simple modules by induced left Schützenberger modules was obtained in the fifties by Munn and Ponizovskii as part of a more general work on the representation theory of finite semigroups. Unlike group algebras, semigroup algebras are seldom semisimple and therefore have (none-semisimple) projective modules. We give a description of the indecomposable projective modules of PT_n which is similar in spirit to the Munn-Ponizovskii construction of the simple modules. Moreover, we generalize both results and describe the simple and the indecomposable projective modules of a certain class of Ehresmann semigroups, with the case of PT_n being a natural example. This is a joint work with Stuart Margolis. PowerPoint

October 22 **Nora Szakacs **(York) Simplicity of Nekrashevych algebras of contracting self-similar groups

**Abstract: **A self-similar group is a group G acting on the infinite |X|-regular rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups like Grigorchuk's 2-group of intermediate growth are of this form. Nekrashevych associated C*-algebras and algebras with coefficients in a field to self-similar groups. In the case G is trivial, the algebra is the classical Leavitt algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contracting groups are finitely presented. In this talk we discuss the simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm which, given an automaton generating the group, outputs the characteristics over which the algebra is non-simple. We apply our results to several families of contracting groups like Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields. This work is joint with Benjamin Steinberg (City College of New York). PowerPoint

October 29:** Julius Jonusas** (Vienna) Canonical topologies for monoids**Abstract: **The problem of determining which topologies are compatible with the multiplication and inversion in a group has an extensive history that can be traced back to Markov. For example, it has been shown by Gaughan in the 1960s that the symmetric group on a countable set has a unique Polish topology which makes composition and inversion continuous. In the same way we will explore to what extent the algebraic structure of a monoid structure determines the topologies which make the multiplication of the monoid continuous, such topologies are known as semigroup topologies. In particular, we will investigate which monoids have a unique Polish semigroup topology and which have automatic continuity. If M is a monoid equipped with a semigroup topology, then automatic continuity, in this context, means that every homomorphsim from M to a second countable topological monoid is necessarily continuous. PowerPoint

October 29: **Tony Bak** (Bielefeld) Solution to the sandwich classification problem in arbitrary groups and applications to classical-like groups over arbitrary rings**Abstract: ** Let G be an arbitrary group and F an arbitrary subgroup. For each mixed commutator subgroup K = [F, H] of G, we define the notion of an F-cocommutator subgroup over K. The set of F-cocommutator subgroups over K forms a sandwich of subgroups of G, which is denoted by Sand(K). It has a largest member C(K) called the full cocommutator subgroup over K and if F is perfect then K is its smallest member. C(K) is the replacement in the setting of arbitrary groups for the notion of full congruence subgroup in the setting of classical-like groups over rings when F is the elementary subgroup and the K's are replacements for the relative elementary subgroups of a classical-like group. The MAIN THEOREM is: A subgroup H of G is F-normal if and only if it belongs to a sandwich Sand(K) for some K. Moreover K is unique. We show that the known classification of E-normal subgroups of a classical-like group G(R) over a quasi-finite ring R, where E is the elementary subgroup of G(R), is a consequence of the Main Theorem and we use the Main Theorem to extend this result to classical-like groups G(R) over an arbitrary ring R. PowerPoint

November 5: Nikolai Vavilov (St. Petersburg) 50 SHADES OF PROOF **Abstract: **Qui dit Mathématiques, dit démonstration. The only problem is that there is no obvious standard of proof, common for different areas of mathematics at different times. For vast majority of mathematicians proofs are not mere texts, and are intimately related to individual and collective understanding. From this viewpoint FORMAL PROOFS are not higher forms of traditional proofs, they ARE NOT mathematical PROOFS at all. Rather, they play a role of testimonies, or experimental evidence, urging us to find a real proof that might give such an understanding. I plan to discuss and illustrate by a medley of historical examples of various levels, the difference between proofs, verifications, and their intermediate forms, as far as their reliability, transparency, and durability. PowerPoint