Athanase Papadopoulos

S R Srini- Vasa Varadhan

Misha Gromov

John Milnor

Yakov Sinai

John Nash

Louis Nirenberg

DENNIS SULLIVAN: MATHEMATICS AND PHYSICS; MANIFOLD AND SPACE

Keywords: Abel Prize, Dennis Sullivan, topology, manifold, space, hydrodynamics. AMS Classification: 01A70

This article is a personal overview of the work of Dennis Sullivan who was awarded the 2022 Abel prize.

the good he has done for the mathematical community. In this report, after a brief Vita, I have chosen to start by reviewing the three lectures that Sullivan gave at three International Congresses of Mathematicians (Nice 1970, Vancouver 1974and Berkeley 1986). These lectures reflect some of his subjects of interest during the first part of his career. Then, I will present a list of the incredible amount of topics on which he worked, with special emphasis on dynamics and fluid mechanics. From time to time I will quote some emails I have exchanged with him over the years. Going through this correspondence, I see in hindsight that he was always answering my questions, mathematical or not, and his remarks were always compelling. I will mention in particular some thoughts he shared recently with me and with several of his colleagues and friends on his view on what is important in mathematics.

Born in 1941, in Port Huron (Michigan), Sullivan was brought up in Houston (Texas). He entered Rice University, first as a student in chemical engineering. It was the discovery of topology that made him change his mind and shift to mathematics. In an interview with Shubashree Desikan which appeared in The Hindu on March 24, 2022, Sullivan speaking informally about this episode says: "At Rice University, all the science students, electrical engineers and all the others, took math, physics and chemistry. In the second year, when we did complex variables, one day, the professor drew a picture of a kidneyshaped swimming pool, and a round swimming pool. And he said, you could deform this kidney-shaped swimming pool into the round one. At each point, the distortion is by scaling. A little triangle at this point goes to a similar triangle at the other point. We had a formula for the mapping, because we were taking calculus, and we had a notation for discussing it. This was like a geometric picture. This mapping was essentially unique. The nature of this statement was totally different from any math statement I've ever seen before. It was, like, general, deep, and wow! And true! So then, within a few weeks, I changed my major to math. " Sullivan obtained his PhD at Princeton University in 1966, with a dissertation titled Triangulating homotopy equivalences, with William Browder as advisor. In the few preceding years, algebraic topology had been invigorated with the introduction of new techniques and with a series of outstanding results on the classification of manifolds. Let me say a few words on this period.

The notions of fiber space and fiber bundle became central around the year 1950. Soon after, Serre introduced in his thesis (1951) the crucial idea of using spectral sequences to study the homology of fiber spaces. In 1952, Michail M. Postnikov obtained his famous reconstruction of the cohomology of a space from its homotopy invariants. René Thom, in 1954, obtained a classification of manifolds up to cobordism. In 1956, Milnor proved the existence of exotic differentiable structures on the 7-dimensional sphere. The higher-dimensional analogue of the Jordan curve theorem (the so-called Jordan-Schoenflies theorem) was obtained in 1959-1960 (works of Barry Mazur, Marston Morse and Morton Brown). In 1960, Steven Smale proved the Poincaré conjecture (PL and smooth categories) for dimensions ≥ 7 (and then ≥ 5). In 1961, Christopher Zeeman obtained a proof of the same conjecture for dimension 5, and in 1962 Stallings gave a proof for dimension 6. In 1961, Milnor disproved the so-called Hauptvermutung der kombinatorischen Topologie ("main conjecture of combinatorial topology"), a conjecture formulated in 1908 by Ernst Steinitz and Heinrich Franz Friedrich Tietze, asking whether any two triangulations of homeomorphic spaces are isomorphic after subdivision. In 1965 Sergei P. Novikov proved that the Pontryagin classes with rational coefficients of vector bundles are topological invariants.

Sullivan started working on topology amid all this booming activity. In his thesis, he gave an obstruction to deforming a homotopy equivalence of piecewise linear manifolds to a PL homeomorphism, obtaining further cases where the Hauptvermutung is true. This result is called "the characteristic variety theorem." It was used to provide numerical invariants that classified the combinatorial manifolds in a homotopic type. He summarized his results on this topic in his short paper On the Hauptvermutung for manifolds [START_REF] Sullivan | On the Hauptvermutung for Manifolds[END_REF] which appeared in 1967 and for which he was awarded the Oswald Veblen prize of the AMS.

After his PhD, Sullivan worked successively at the University of Warwick (1966)(1967), the University of California at Berkeley (1967Berkeley ( -1969) ) and MIT (1969MIT ( -1973)). In the year 1973-74, he was invited as a visiting professor at the Université de Paris-Sud, Orsay. The next year, he became a permanent member of the Institut des Hautes Études Scientifiques at Bures-sur-Yvette. The institute is a couple of kilometers from the Orsay campus. Remembering that period, I find in an email from Sullivan (2015) In a 2019 email, Sullivan writes: "My first math hero, as a grad student in Princeton, was René Thom. When IH ÉS offered me a professorship, I was honored to accept it and to become Thom's 'colleague'." He adds: "My second math hero starting in December 71 was the Mozartlike figure Bill Thurston. During the next decade I was in France and Bill was in the US but we had a quite fruitful and intense interaction." I shall say more about Sullivan's relation with Thurston below.

In 1981, Sullivan was appointed to the Einstein Chair at the Graduate Center of the City University of New York. He kept his position at IH ÉS on a part-time basis and started spending half the year in France and the other half in the US, until 1996, where he took up a professorship at the State University of New York at Stony Brook, again on a part-time basis, keeping his Einstein Chair at CUNY, where he conducts, since his appointment there, a weekly seminar on geometry in the broadest sense whose sessions are known to last for several hours (sometimes all day). For me, and I assume it is the same for many others who know this place, the Graduate Center of CUNY is the central point where the heart of New York is beating.

To talk about Sullivan's work, I start with the lecture that he gave at the 1970 ICM, which took place in Nice. The invitation to that Congress came just after Sullivan proved (concurrently with and independently of D. Quillen) the Adams conjecture, which concerns the homotopy theory of sphere bundles associated with vector bundles. The conjecture was considered as one of the most important conjectures in topology. Whereas this led Quillen to develop algebraic K-theory, Sullivan's approach was based on the "arithmetization" of geometry and topology, more precisely on the introduction of Galois theory in the geometry of manifolds in the form of concepts like localisation, rationalization, profinite and p-adic homotopy theory. Sullivan's proof of Adam's conjecture, which he obtained in 1967, is based on the construction of a functor from abstract algebraic varieties into profinite homotopy theory. This approach led him to the study of the absolute Galois group itself through its actions on new geometric objects, and at the same time gave rise to strong relations between number theory and homotopy theory. In fact, Sullivan studied the action of the absolute Galois group in homotopy theory via classifying spaces and Postnikov towers. The ICM talk is titled Galois symmetry in manifold theory at the primes [START_REF] Sullivan | Galois symmetry in manifold theory at the primes[END_REF]. A more detailed version of this work appeared in the paper [START_REF] Sullivan | Genetics of homotopy theory and the Adams conjecture[END_REF], in which Sullivan describes his (still open) unrequited Jugendtraum (childhood dream). Let me quote from the introduction to that paper some sentences which are characteristic of Sullivan's personal thinking, based on rich and inspiring analogies: "We are studying the structure of homotopy types to deepen our understanding of more complicated or richer mathematical objects such as manifolds or algebraic varieties. The relationship between these two types of objects is I think rather strikingly analogous to the relationship in biology between the genetic structure of living substances and the visible structure of completed organisms or individuals. The specifications of the genetic structure of an organism and of the homotopy structure of a manifold have similar texture; they are both discrete, combinatorial, rigid, interlocking and sequential."

Sullivan's second ICM talk (Vancouver 1974) is titled : "Inside and outside manifolds" [START_REF] Sullivan | Inside and outside manifolds[END_REF]. The title is characteristic of Sullivan's taste for literary turns of phrase. The paper has two parts, which express his two approaches at the time for the exploration of manifolds: topology and dynamical systems. The first part, on "outside" manifolds, is an exposition of the complete classification theory of simply-connected manifolds of dimension at least 5. The main detailed reference for this part is a set of notes on geometric topology based on lectures that Sullivan gave at MIT in 1970, carrying the subtitle "Localization, periodicity, and Galois symmetry" [START_REF] Sullivan | Geometric topology. Localization, periodicity and Galois symmetry[END_REF]. These notes were widely circulated and they had a major influence on algebraic and geometric topology not only in the West but also in the Soviet Union where they were translated into Russian just after Sullivan's ICM 1974 talk. In the second part of the lecture, "Inside manifolds", Sullivan presents a series of qualitative results on (in particular Smale's Axiom A, structural stability, and the asymptotic properties of leaves of foliations) within individual manifolds. This second part, as Sullivan writes, "focuses attention on the classical goals and problems of analysis situs."

A new version of the 1970 MIT notes was edited in 2005 by Andrew Ranicki [START_REF] Sullivan | Geometric topology. Localization, periodicity and Galois symmetry[END_REF]. It concludes with a 10 pages postscript by Sullivan in which he recounts, in his personal literary style, the genesis and later developments of these notes, as well as episodes from his own mathematical education with the mathematicians who have inspired him. René Thom, who, as we recalled, was Sullivan's first hero, is abundantly quoted. The postscript contains several open problems and conjectures. At several points, the tone is philosophical. In his book Surgery on compact manifolds, C. T. C. Wall writes about these notes: "It is difficult to summarise Sullivan's work so briefly: the full philosophical exposition should be read." The postscript is punctuated with details on Sullivan's family life. After recalling his work in the 1970s, Sullivan writes in these notes: "About this time dynamical systems, hyperbolic geometry, Kleinian groups, and quasiconformal analysis which concerned more the geometry of the manifold than its algebra began to distract me (see ICM report 1974), and some of the work mentioned above was left incomplete and unpublished." This mention of quasiconformal mappings brings us to his 1986 ICM talk titled Quasiconformal homeomorphisms in dynamics, topology, and geometry. The four contexts in which quasiconformal mappings are studied in this paper are the following:

(1) Feigenbaum's numerical discoveries in 1-dimensional dynamics. Here, quasiconformal homeomorphisms are used to define a distance between real analytic dynamical systems, by first complexifying them. This distance is contracting under the Feigenbaum renormalization operator.

(2) The theory of quasiconformal manifolds, including developments of de Rham cohomology, Atiyah-Singer index theory and Yang-Mills connections on these manifolds. This is joint work between Sullivan and Donaldson. Combined with other works of Donaldson and Freedman, this theory provides a complete picture of the structure of quasiconformal manifolds, namely, each topological manifold has an essentially unique quasiconformal structure, except in dimension 4 where, by results of Freedman and Donaldson-Sullivan, the statement is false.

(3) The quasiconformal deformation theory of expanding systems f : U 1 → U where U 1 is a domain of the Riemann sphere and f a d-sheeted (d > 1) onto covering. Sullivan writes: "The analytic classification of expanding systems of a given topological dynamics type on the invariant set is a kind of Teichmüller theory." Here, an infinite-dimensional Teichmüller space of expanding analytical dynamical systems near their fractal invariant sets is embedded in the Hausdorff measure theories possible for the transformation on the fractal, and the Hausdorff measure theories of fractals are embedded in the theory of Gibbs states.

(4) The quasiconformal theory of the geodesic flow of negatively curved manifolds via the action of the fundamental group on the sphere at infinity of the universal cover. Constant curvature is given a characterization among variable negative curvature in terms of a uniform quasiconformality property of the geodesic flow.

These three ICM talks give us an idea of part of Sullivan's mathematical interests until the year 1986. I will mention some other results of him, but before that I would like to list all the fields and the subjects on which he worked.

We already mentioned his work on the topology of manifolds. His results on this topic concern, among others, the classification of manifolds in several categories: smooth, PL, topological, Lipschitz, bi-Lipschitz and quasiconformal. Closely related is his work on algebraic topology: homology and homotopy theories, including rational homotopy theory, intersection homology, differential cohomology, homotopical algebra, characteristic classes, K-theory, flat bundles, minimal models, Galois symmetry and string topology. Then come the more geometrically specialized topics: foliations, minimal surfaces, geometric structures and geometry of manifolds: affine manifolds, complex manifolds, Kähler manifolds, hyperbolic geometry, Kleinian groups, universal Teichmüller theory, laminations, the soleniod and circle packings. Under the heading "dynamics", I mention 1-dimensional dynamics, renormalization, universality, holomorphic dynamics, potential theory, optimal control, measurable dynamics, and the related topics of ergodic theory, probability, Brownian motion, Diophantine approximation, chaos and fractals (in particular, self-similar structures arising in the theory of Kleinian groups, in KAM theory and in iteration theory of holomorphic dynamics). Finally, let me also mention in a nutshell: C * algebras, harmonic analysis, singularity theory, field theories, sigma models, and logic.

I do not know of any mathematician who has such a broad spectrum of interests, except Leonhard Euler.

I would like to say a few more words on Sullivan's French period. As a visitor at Orsay, Sullivan gave a course on the theory of rational homotopy of differential forms that he had newly developed. During the same period, he came with a new interpretation of the Godbillon-Vey invariant, a subject that was keeping busy geometers working on foliation theory which was a hot topic at that time at Orsay. The interpretation was completely new, using the notion of currents. Sullivan included this result in his paper Cycles for the dynamical study of foliated manifolds and complex manifolds [START_REF] Sullivan | Cycles for the dynamical study of foliated manifolds and complex manifolds[END_REF], which appeared in 1976 and which contains a wealth of other results. It was also during the same year that he introduced the topologists at Orsay to Thurston's theory of surface homeomorphisms; this was the major impetus to the seminar on Thurston's work which took place at Orsay in 1976-77 and which gave rise to the famous book "Travaux de Thurston sur les surfaces."

Talking about Orsay and Bures, and on a more personal level, let me recall an episode, in 1983. I was a PhD student at Orsay. François Laudenbach, who was my thesis advisor, had asked me a question related to pseudo-Anosov homeomorphisms. I came to see him one day, with a four pages manuscript, containing a proof of a theorem which answers the question. He told me: "I will show it to Sullivan." After he talked with Sullivan, he decided that this was my PhD dissertation. (For the record, I had some trouble later on, convincing people at the registrar's office that these 4 pages constitute a thesis dissertation.) I already quoted Sullivan saying that he had two heroes, Thom and Thurston, and I would like to say a few more words on his relation with the latter.

Sullivan was always a major promoter of Thurston's ideas, and he was probably the person who best understood, from the first years, their originality and their broadness. The main themes discussed at Sullivan's seminar at IH ÉS and at his New York Einstein Chair seminar included 1-dimensional dynamics and the so-called kneading theory established by Milnor and Thurston. The other topics include Kleinian groups (discrete isometry groups of hyperbolic 3-space) and holomorphic dynamics, two topics which eventually became a single topic, after Sullivan established a complete dictionary between them. Here, again, Thurston's ideas were often at the forefront, and Sullivan spent years trying to understand and to explain them. He was the first to learn from Thurston his result on the characterization of postcritically finite rational maps of the sphere, that is, rational maps whose forward orbits of critical points are eventually periodic. The proof of this theorem, like the proofs of several of Thurston's major theorems, uses a fixed point argument for an action on a Teichmüller space. The rational map in the theorem is obtained through an iterative process as a fixed point of that map on Teichmüller space. This became a major element in Sullivan's analogies between the iteration theory of rational maps and the theory of Kleinian groups. Thurston's theorem, together with Sullivan's dictionary between the theory of discrete subgroups of PSL(2, C) and complex analytic iteration, constitute now the two most fundamental results in the theory of iterations of rational maps. Regarding this theory, in 1985, Sullivan published a paper in which he gave the proof of a longstanding question formulated by Fatou and Julia in the 1920s and which became known as the Sullivan's No-wandering-domain Theorem. This theorem says that every component of the Fatou set of a rational function is eventually periodic. A fundamental tool that was introduced by Sullivan in his proof is that of quasiconformal mappings, the main topic of his 1986 ICM lecture and one of the major concepts used in classical Teichmüller theory.

Let me also mention that in the realm of conformal geometry, Thurston introduced the subject of discrete conformal mappings, and in particular the idea of discrete Riemann mappings. In 1987, Sullivan, together with Burton Rodin, proved an important conjecture of Thurston on approximating the Riemann mapping using circle packings. This result, the authors write, is in the setting of Thurston's "provocative, constructive, geometric approach to the Riemann mapping theorem," see [START_REF] Rodin | The convergence of circle packings to the Riemann mapping[END_REF].

There is one field which I did not stress upon yet; this is physics. Sullivan is also a physicist, in the tradition of Euler and Riemann, who were also physicists. In fact, Riemann considered himself more a physicists than a mathematician. He was thoroughly involved in gravitation, electricity, magnetism and electrodynamics and he adopted a completely physical approach to the theory of functions of a complex variable. In a note contained in his Collected Works, he writes: "My main work consists in a new formulation of the known natural lawsexpressing them in terms of other fundamental ideas-so as to make possible the use of experimental data on the exchanges between heat, light, magnetism, and electricity." During his stay at IH ÉS, Sullivan was discussing with physicists, Oscar Lanford, Henri Epstein, David Ruelle, etc. In the Postface to the 2005 edition of his 1970 MIT notes, he writes: "At a physics lecture at IHES in 1991 I learned the astonishing (to me) fact that the fundamental equations of hydrodynamics in three dimensions were not known to have the appropriate solutions." This brings us to hydrodynamics, one of Sullivan's favorite topics, and in particular, the Navier-Stokes equation. This is the main partial differential equation which describes the flow of an incompressible fluid. It is good to remember here that a special form of this equation was discovered by Euler back in the 18th century (Euler's equation is special in that it does not take viscosity into account). In the interview that appeared in The Hindu which we already quoted, Sullivan, talking about the Navier-Stokes equation, declares that his fascination for this subject goes back to his youth spent in Texas. He says, again with his informal tone: "If you're in Texas, as a student of chemical engineering, there's the petrochemical industry, the oil industry, and organic chemistry and plastics, all around Houston. If you are good in science, and you work on that and become an engineer you can get a good job and have a nice work at a research center. So it's a good thing to do. In fact, during the summers, I had jobs at various such places. Once I had to study the computer methods that they were using to do what's called secondary recovery. You know, when they find oil, because of the pressure, if they drill a hole, the pressure makes it shoot up, right. But after they drill for 20 years, the pressure goes down. What they do then is go to another part of the field, and they drill and they pump in water to create pressure that will push the oil back to their wells, and for this they have to solve the linearised version of the Navier-Stokes equation. I didn't know that name, then but it's a linearised version of the Navier-Stokes equation. While at the summer job where I was studying the possible computer programs I had a certain question there. That was around 1960. [. . . ] So in a sense, I was aware that there's this huge industry related to fluid flow through porous media. It was astonishing to me to find out as I found out in the 1990s, that that equations in three dimensions, the beautiful equations, are not solved."

In a recent email (2021), Sullivan writes on the same subject: "Back to 1991-92: After a talk at IHES by computational physicists using Fourier modes to numerically analyze the 3D incompressible Navier-Stokes Equation one learned about the lack of a mathematical answer to the question of long time existence of solutions in certain classes. That this beautiful and widely used equation that designed airplanes helped to more efficiently transport oil and to understand how to stabilize aneurisms in or near the brain or heart etc. held such mysteries was astonishing. For example, why was this 3D problem so hard? The same Navier-Stokes problem in 2D could be treated by exactly the same Calderon-Zygmund analysis tools that treated the Beltrami equation, so important in the Ahlfors-Bers treatment of Teichmüller theory. In addition half of the non-trivial theory of quasiconformal mappings in 2D extended to higher dimensions as in Mostow rigidity for 3D."

Let me also recall that the Navier-Stokes equations are among the Clay Mathematics Institute Millennium Prize problems. The statement of the Clay problem is the following: "Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations."

Talking about physics, let me also mention that Sullivan's work on 1-dimensional dynamics, period doubling and chaos was motivated by physical experimentation. In the Postface to his MIT notes, he writes: "For example, one new project began in the 1980s in fractal geometrythe Feigenbaum universal constant associated to period doublingpresents a new kind of epistemological problem. Numerical calculations showed that a certain mathematical statement of geometric rigidity in dynamics was almost certainly true, but the available mathematical technique did not seem adequate. Thus assuming the result was true there had to be new ideas in mathematics to prove it. The project consumed the years up to the birth of my second son Thomas in July '88."

Recently (2021), Sullivan, emailed emailed me together with a few friends and colleagues some basic and fundamental questions and thoughts on "What is a manifold", describing this as his mathematical lifelong quest. I read in one of these emails: "3D fluids and turbulence (as in the real world) is an area that has been puzzling me since the 90-91 because it seems its natural setting or comfort zone is not really identified yet. I am thinking, focusing really on What is a manifold ". In another mail, I read: "Riemann introduced the notion of manifold, Gelfand, then Grothendieck and then Connes recast the geometrical aspect in terms of the algebra and the quality of functions. The two viewpoints are in duality."

A few months ago, he sent me to read a 1992 paper by Hikosaburo Komatsu on hyperfunctions and microfunctions, with the comments: "Here is a marvelous example of a transfer of information explaining the history of ideas. [. . . ] I liked so much how the paper first explained the ease of the 1-dimensinal case (in terms of what we know from before 1900), and then the modern story with the competitions between wave front set perspective of one school and the sheaf/algebra perspective of another. [. . . ] I think this sheaf of hyperfunctions is a key point for me in a three decade hiatus to define smooth structures in real dimension one that will accommodate the vastly successful theory of dynamics in one dimension which provides the tool and the language to be able to 'explain' some of the missing parts of the numerical universalities discovered numerically by physicists in the 70's . The non-missing part achieved by the end of the 80's only worked for integer critical points using Teichmüller theory, even though the numerical theorems were known for all critical exponents." In another email, again, on the same topic: "The thread with which I am preoccupied is the discussion of space where physical reality seems to take place which, by the way, can be reformulated with charts in terms of the notion of function or with algebra in terms of ideals in an algebra of functions".

We are back to the most fundamental questions: What is a manifold? What is a function? What is space?

  Sullivan's third ICM talk (Berkeley 1986) [8] is a survey of results and conjectures, mainly due to him, on quasiconformal homeomorphisms and their use in four different contexts. Here, a homeomorphism ϕ : X → Y between metric spaces (X, | |) and (Y, | |) is said to be (K-)quasiconformal if for some K > 0 lim sup r→0 sup |ϕ(x)ϕ(y)| where |x -y| = r and x is fixed inf |ϕ(x)ϕ(y)| where |x -y| = r and x is fixed < K.


Farzam Kharajinezhadian

Saeid Rashidi
email: rashidi.saeid@gmail.com

Mojtaba Banifakhr
email: banifakhr.mojtaba@stu.yazd.ac.ir

Mohammadali Kavousi

Automatic Breast cancer

Keywords: Breast cancer, lesions classification, Convolutional Neural Networks, fusion of classifiers

Introduction

Breast cancer is the most common cancer in women and the second most common cancer overall. Evidence suggests that breast cancer incidence rates increase every year [START_REF] Gholizadeh | A Study of Deep Convolutional Neural Network for Diagnosing Breast Cancer in Thermographic Images[END_REF]. In fact, one in 8 women in the United States and two in 5 are diagnosed with cancer [2]. The number of cancer cases is projected to increase to 6.2 million people by 2030 [START_REF] Suganthi | Anisotropic diffusion filter based edge enhancement for segmentation of breast thermogram using level sets[END_REF]. The average annual cost of health care for a new cancer patient in the United States was $21,222 during 2001-2008 [START_REF] Kandlikar | Infrared imaging technology for breast cancer detection-Current status, protocols and new directions[END_REF]. Therefore, early diagnosis is essential for guiding appropriate and timely treatments and a higher chance of survival [START_REF] Ng | A review of thermography as promising non-invasive detection modality for breast tumor[END_REF].

Various imaging techniques have been developed to improve the sensitivity and specificity of breast cancer diagnosis [START_REF] Kandlikar | Infrared imaging technology for breast cancer detection-Current status, protocols and new directions[END_REF]. Conventional screening techniques include complicated imaging methods. Meanwhile, there are some limitations to the use of these techniques, such as the pain experienced during mammography, damage caused by long-term exposure to radiation, and high costs [START_REF] Suganthi | Anisotropic diffusion filter based edge enhancement for segmentation of breast thermogram using level sets[END_REF]. Generally, the sensitivity of mammography depends on the physical characteristics of the tumor, and mammographic sensitivity is rendered a less sensitive tool in younger women with dense breast tissues. Ultrasound is another breast cancer screening tool, which is mostly used in young women to reduce unnecessary radiation exposure. Nonetheless, certain issues must be taken into account in this method, including noise conditions and operators' imaging expertise. In fact, they could cause failure in the discovery of deep breast microtissue [START_REF] Kuhl | Mammography, breast ultrasound, and magnetic resonance imaging for surveillance of women at high familial risk for breast cancer[END_REF]. The final technique is thermography. Infrared imaging is a non-contact method, used widely by the medical community in recent years [START_REF] Lahiri | Medical applications of infrared thermography: a review[END_REF]. Given the fact that infrared thermal imaging does not use ionizing radiation or additional invasive techniques, patients undergoing examination by this method feel no discomfort. Owing to its high sensitivity and specificity, thermography has been extensively used as an auxiliary screening tool [START_REF] Ghafarpour | A review of the dedicated studies to breast cancer diagnosis by thermal imaging in the fields of medical and artificial intelligence sciences[END_REF]. Thermography results can be correct 8-10 years before detection by mammography [START_REF] Gholizadeh | A Study of Deep Convolutional Neural Network for Diagnosing Breast Cancer in Thermographic Images[END_REF]. In addition, infrared thermography shows skin surface temperature as a qualitative and quantitative tool. It is also able to minimize small temperature changes in the anomalies of arteries and reflects angiogenic areas due to breast cancer [START_REF] Keyserlingk | Functional infrared imaging of the breast[END_REF]. Given the higher chemical changes and activities of blood reservoirs in cancer tissues, relative to adjacent areas, cancer breast tissues are detected in thermography due to high temperature [START_REF] Ng | A review of thermography as promising non-invasive detection modality for breast tumor[END_REF].

Various image processing and artificial intelligence techniques have been proposed to detect breast cancer by thermography images in the last few decades, which can be more efficient and reduce costs. The clinical interpretation of a breast thermogramis primarily based on the asymmetry analysis of these heat patterns subjectively. The heat patterns are symmetrically distributed in both breasts. Therefore, bidirectional asymmetric analysis has been widely evaluated in previous studies. In fact, it is the classic breast screening method in thermography [START_REF] Gogoi | Singular value based characterization and analysis of thermal patches for early breast abnormality detection[END_REF].

There is an asymmetric temperature distribution of heat in two breasts with hot areas due to abnormal growth of blood vessels [START_REF] Gholizadeh | A Study of Deep Convolutional Neural Network for Diagnosing Breast Cancer in Thermographic Images[END_REF]. While the quantitative analysis of these hot areas provides useful information about breast pathology, its success requires the extraction of some features from the left and right breast to perform asymmetric analysis [START_REF] Borchartt | Breast thermography from an image processing viewpoint: A survey[END_REF]. Normally, the extracted features are the first moments, such as mean, standard deviation, and the difference between the maximum and minimum temperature of two sides of the breast [START_REF] Borchartt | Thermal feature analysis to aid on breast disease diagnosis[END_REF]. Some studies use morphology information, such as the number of veins with higher temperatures, tissue features, the number of hot areas, the geometry of the hot center and the histogram of thermal images [START_REF] Acharya | Thermography based breast cancer detection using texture features and support vector machine[END_REF]. Ultimately, the extracted features are classified by artificial intelligence algorithms. In this regard, some of the algorithms applied in previous studies include fuzzy classification, Bayesian network, support vector machine (SVM), neural networks and wavelet transform [START_REF] Acharya | Thermography based breast cancer detection using texture features and support vector machine[END_REF][START_REF] Nicandro | Evaluation of the diagnostic power of thermography in breast cancer using bayesian network classifiers[END_REF][START_REF] Pramanik | Wavelet based thermogram analysis for breast cancer detection[END_REF].

A literature review revealed some limitations to the use of breast thermography, including lack of researchers, especially image processing engineers' access to free databases, the small number of images on the database, and, most important of all, different breast morphologies (various geometrical shapes) in people.

This significance is also related to the asymmetry analysis, which is the main thermography technique. Lack of proper selection and separation of breast area will lead to failure to implement analyses such as the first and second moments by using the asymmetry technique. Another problem is reduced sensitivity and accuracy of result classification in terms of being sick or healthy .

According to what was mentioned and with regard to our knowledge or authors of previous works [START_REF] Ghafarpour | A review of the dedicated studies to breast cancer diagnosis by thermal imaging in the fields of medical and artificial intelligence sciences[END_REF][START_REF] Zadeh | A model for diagnosing breast cancerous tissue from thermal images using active contour and Lyapunov exponent[END_REF][START_REF] Zadeh | Clustering and screening for breast cancer on thermal images using a combination of SOM and MLP[END_REF]

Materials and Proposed Methods

In this study, we used the breast thermography images derived from the BUS database [START_REF] Proeng | Image Processing and Image Analyses Applied to Mastology[END_REF], Based on [START_REF] Proeng | Image Processing and Image Analyses Applied to Mastology[END_REF] suitable result. There are other images (e.g., 1-e), where the breast area cannot be accurately detected. In some images (e.g., 1-F), the breast might have been biopsied or operated on shortly after imaging, which required dressing of the chest area. This leads to lack of responsiveness of asymmetry.

Convolutional Neural Network (CNN) is one of the most important deep learning techniques, through which several layers can be trained in a robust manner [START_REF] Chambolle | Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage[END_REF]. CNN is extremely efficient and has been a dominant method in computer vision tasks. Figure 2 convolutional operations are applied to each layer. In the next step, the network's output is calculated, and the output results of the networks are used to tune the parameters related to network training by estimating the network error rate. To this end, the output of the network is compared to the accurate response (favorable solution) using an error function and the error rate is calculated. In the next step, the backpropagation stage initiates based on the estimated error rate. At this stage, the gradient of each parameter is calculated based on the chain rule, and all parameters are changed based on their effect on the error created in the network. The feed-forward stage starts following tuning all parameters. Ultimately, network training is terminated after repeating an appropriate number of steps. Now, the following steps must be taken to perform the process based on the mentioned pre-training tasks. In the first stage, the input image is applied to the network. However, preprocessing must be performed on the images before entering the images into CNNs as input in order to prepare them for processing. In this regard, preparation could involve cutting, changing the size of the image, normalizing the image and zeroing. Therefore, the size of the input image must be changed, if required, to match the dimensions of the input layer of the network.

Therefore, the input images must be the standard size of these pretrained networks in the first stage. Afterwards, the size of the collected color image is changed to 224×224×3. The third dimension is related to the red, green and blue colors. Each entry of the three-dimensional matrix is valued at 0-255, and the minimum and maximum values show the lowest and highest amount of brightness, respectively.

Following the preprocessing of images and adjusting them to the standard size, the first convolutional layer receives the image as input. In the second stage, the images must be convolved by the convolution kernels of the first layer. Each convolution layer comprises three sublayers. In the first sublayer, the pre-processed image is convolved by prespecified convolution kernels. However, these convolution kernels have different amounts and dimensions depending on the type of convolutional network used. In addition, the output dimensions can be the same size as the input image or smaller and larger depending on the image zeroing number. Moreover, several convolution kernels can be used at this stage. Hereon, the convolution operator for data means a local operator that does not change with displacement. The input-tooutput mapping for the convolution operator is shown in the equation below:

: × × → ′ × ′ × ′
As observed, the convolution operator presented here is mapping from 3D data to another three-dimensional data and is in the set of real numbers. The following equation shows the operation of the convolution operator mathematically:

Where w is a convolutional filter bank and a four-dimensional data. In fact, the fourth dimension is the filter number in the filter bank, and the filter itself is a three-dimensional weight mass. In other words, in any location that the convolutional 3D filter number 1 is placed after sliding over the 3D data, the dot product is found by multiplying the corresponding values in each vector and adding them together. Since only the value of one pixel is attained in each placement position, a 2D feature map is obtained by sliding the filter number 1 over the entire data. Similarly, the next filters are applied, and a 2D feature map is obtained per each filter. The ultimate 3D feature map will be developed by putting the feature maps in a row in the third dimension.

After applying the first convolutional sublayer, the output of the first sublayer is given to the non-linear sublayer. At this stage, a non-linear activation function is applied to the obtained values to achieve higherlevel features. In addition, a Rectified Linear Unit (RLU) activation function is applied in all three models as the non-linear function. In general, there is a higher tendency for using RLUs in deep convolutional networks, compared to other non-linear functions since the mentioned function can be easily solved and does not engage several computational resources. Based on experience, the mentioned function has acceptable accuracy. The following equation defines the RLU:

Ultimately, the obtained values are given to the third sublayer to perform the combination operation. In the combination sublayer, a statistical summary of neighboring pixels of the central pixel replaces the value of the central pixel in the merge window. The combination increases the stability of features and reduces sensitivity to unwanted changes. The output data dimensions of this sublayer can be equal to the dimensions of the input data or can have different dimensions depending on the values of zeroing or step parameters. Therefore, another use of this sublayer is maintaining more valuable features and eliminating less significant features in case of a decrease of dimensions in the combination operations. There are different types of combinations, the most popular of which is max pooling, which is described below:

'' max{ : ' , ' } ijk i j k y y i i i p j j j p       
At this point, an entire convolutional layer is applied to the preprocessed image. This step is repeated a certain number depending on the type of convolutional network used. After finishing the convolutional layers, one or several fullyconnected layers can be used for the final mapping of the obtained features. While a fullyconnected layer is exactly similar to a convolutional layer, their difference is that no sporadic interactions occur in the former, and just like traditional neural networks, a complete connection is established between this layer and the previous layer. The output of the last layer is a one-dimensional vector, and the number of members of this vector is equal to the number of classification classes.

In fact, this layer carries out the classification process. The last fullyconnected layer of all convolutional architecturesis connected to a softmax layer. In fact, a softmax layer performs the classification process in convolutional networks. The layer encompasses a number of neurons that is equal to the number of classes in the classification problem and is used for the final mapping of features and applying the classification [START_REF] Spanhol | Breast cancer histopathological image classification using convolutional neural networks[END_REF].

At this point, the output result is used to calculate the network error rate in order to tune its parameters (i.e., network training). To this end, the network's output is compared to a correct solution using an error function, which leads to the calculation of the error rate. The experimental error estimation method is shown below: 2 shows the characteristics of each layer in the CNN applied in the present study.

Weeused Matlab, which was applied on a high-performance computing system, which encompasses several computational clusters and integration among them leads to the centralized management of the tasks. Hardware characteristics were: two nodes, 2x NVIDIA® Tesla K80 GPUs, 8x16GB memory, and2x Intel® Xeon® E5-2695 v3 @ 2.30GHzprocessor. It is worth noting that only one specific model was used on an HPC system in all results obtained from various techniques.

Table 3 exhibits characteristics related to the neural network adjustment parameters.

3.Results

In this study, three evaluation criteria of accuracy, sensitivity and specificity are used to facilitate the assessment of function of our proposed breast anomaly detection system:

The variables of Sen, ACC and SPE are related to sensitivity, accuracy and specificity of the model, whereas the variables of Tp, Tn, Fp and Fn show the number of true positives, true negatives, as well as false positives and false negatives, respectively. Therefore, accuracy will be high if both sensitivity and specificity are high. Out of the 1960 available images, 70% of the data are considered as training data and 30% of the data (588 samples) are considered as a test. Table 2 presents the results related to the sensitivity, accuracy and specificity obtained from the test data.

The level of accuracy and area under the ROC Curve will increase when there is an increase in the amount of sensitivity and specificity. Figure 3 includes a diagram related to the amount of accuracy of the validation data used for the three neural network models.

In addition, Figure 4 shows the diagram related to the amount of error of the validation data applied for the three neural network models.. We propose votingmethod for combining classifiers.

Discussion

In the previous section, result of simulations presented. In this section, brief discussion presented.The present study was performed to implement and evaluate a decision support system (DSS) to assist the detection of breast cancer with the use of deep neural networks. The use of deep neural networks in the detection of breast cancer by thermographic images can be discussed in two areas, including the application of deep neural networks in comparison to previous works and analysis of the deep neural network itself .

Most previous works relied on combining stochastic tissue features with a classifier such as SVM, including an article by Acharya and Francis [START_REF] Acharya | Thermography based breast cancer detection using texture features and support vector machine[END_REF][START_REF] Wakankar | Automatic diagnosis of breast cancer using thermographic color analysis and SVM classifier[END_REF]. While the accuracy values of the mentioned research and present study were almost equal, the former used a much lower number of images, compared to the latter [1960 images]. In addition, the accuracy of the proposed model was higher than other articles [START_REF] Schaefer | Thermography based breast cancer analysis using statistical features and fuzzy classification[END_REF][START_REF] Ng | Integrative computer-aided diagnostic with breast thermogram[END_REF][START_REF] Haddadnia | Diagnosis of breast cancer using a combination of genetic algorithm and artificial neural network in medical infrared thermal imaging[END_REF] but similar to studies [START_REF] Borchartt | Thermal feature analysis to aid on breast disease diagnosis[END_REF][START_REF] Araújo | Interval symbolic feature extraction for thermography breast cancer detection[END_REF][START_REF] Lashkari | Full intelligent cancer classification of thermal breast images to assist physician in clinical diagnostic applications[END_REF] with a close error rate.

The traditional breast anomaly prediction approach is the two-way analysis of asymmetry, the accuracy of which depends on the proper separation of the left and right breasts from a thermogram. Similar to some studies, Schaefer and Ng proposed the manual extraction of features, which involved a difficult task by the system's developer [START_REF] Schaefer | Thermography based breast cancer analysis using statistical features and fuzzy classification[END_REF][START_REF] Ng | Integrative computer-aided diagnostic with breast thermogram[END_REF]. Nonetheless, the proposed method detects breast cancer without the need for the feature extraction process in thermal patterns .

The architecture of the proposed method can also be assessed. Some studies have analyzed certain aspects of computational costs (e.g., memory use and inference time), which was similar to a study by Bianco et al., who evaluated the effect of computational costs on detection accuracy.At this point, we compare and analyze the architectures used in the proposed method. VGG16 has five convolutional blocks and one block that is fully connected to four layers. This method is improved by replacing large filters with core size (11 and 5 in the first and second layers, respectively) using size three filters one after the other. However, there are some barriers to the use of the VGG algorithm, including large computational necessities, both in terms of time and memory. On the other hand, the ResNet algorithm is deeper than the VGG algorithm. The mentioned algorithm also includes feedback, which is due to the error backpropagation education algorithm. The deeper the algorithm becomes and the more increase occurs in the number of layers, the impact of weights and layers might become zero due to the presence of gradient function in the education function by entering deep space. This led to the proposal and use of ResNet algorithms that include feedback [START_REF] Szegedy | Inception-v4, inception-resnet and the impact of residual connections on learning[END_REF]. Similar to other neural networks, data placement order during convolutional network training affects the method of network training.

Weight tuning is based on calculating the error between the predicted value and the expected value. Evidently, the more network training data there is, the more the model adapts to the training event. If the data consecutively given to the network do not all belong to one class, the network adapts itself during the training in such a way that it can better distinguish the data of these classes. Therefore, the order of data placement is disturbed before entering the convolutional network and data are not entered into a class in a row.

Compared to previous studies implemented on this database [START_REF] Yousefi | Low-rank Convex/Sparse Thermal Matrix Approximation for Infrared-based Diagnostic System[END_REF], the proposed method achieves better results. For example in [START_REF] Yousefi | Comparative analysis on thermal non-destructive testing imagery applying candid covariance-free incremental principal component thermography (ccipct)[END_REF] final accuracy is 94.8% and less than our accuracy also in [START_REF] Ibarra-Castanedo | Nondestructive testing with thermography[END_REF] accuracy is 87.3% about 10% less than our method. More than this result we can refer [START_REF] Yousefi | Infrared nondestructive testing via semi-nonnegative matrix factorization[END_REF] and [START_REF] Yousefi | Application of sparse non-negative matrix factorization in infrared non-destructive testing[END_REF] that achives 86.4% and 90.6% respectively. Despite all the benefits of deep neural network models, they have some limitations as well. One of the major drawbacks of these methods is the accurate tuning of parameters based on the duration of each program run to achieve favorable results. This forced the authors to run the program several times. Access to a computer hardware system with optimal processing capability is another feature required for such tasks in the field of deep neural networks. Another limitation is the lack of access to large datasets. there is currently only one rich database available to the public. Notably, a deep neural network requires a rich database to achieve good accuracy in such tasks.

4-Conclusion

Breast cancer has been the cause of mortality of many women worldwide. A review of articles revealed that making advancements in the breast cancer detection field could be a great help to this area from the perspective of computer sciences. While thermography can be an aid in the early detection of breast cancer, it certainly has advantages and disadvantages. Different architectures of CNN and fusion with voatingmethod were used in the present study to classify breast cancer patients using thermography images. The present study aimed to make significant improvements in the accuracy of breast anomaly detection by using deep neural networks and raw thermal images without extracting features. Three pre-trained architectures of Google Net, ResNet18 and VGG16 and fusion of these classifiers with voating method, were used to classify breast cancer in thermal images. According to the results, the GoogleNet architecture was the most table one among the three architectures and yielded excellent results regarding the classification of thermography images. Meanwhile, VGG had a better performance, compared to ResNet. Therefore, CNNs were recognized to be extremely efficient in breast cancer screening. The most important novelty of the present study, compared to previous works in the field of breast thermography, was the lack of effective involvement of different breast morphologies in the results and the lack of manual extraction of features.

In addition, the accuracy of the results was due to the proper tuning of the parameters and choosing an appropriate loss function.It is recommended that the classification accuracy be increased by using larger datasets for training and validation. In fact, CNNs will yield better results if there is an increase in the number of training samples. In addition, the deep learning approach will definitely perform well with the use of color datasets, compared to grayscale datasets for static protocols.

  accumulated dataset, contain 2673 images werecollected in three batches in HunanXiangya Hospital over the years, of which 1425 were Benign images and 1248 Malignantmammograms from405 patients. For this paper we use randomly selected.163 images.As observed, the images are analyzed individually in two pattern categories. Some images are in the form of Figure1-a, where the chest area is symmetrical and elliptical, which are isolated in previous studies by parabolic separation techniques or methods such as a decision tree. Some other images are in the form of figure1-b, where the chest area is spherical and can be isolated by methods such as circular Hough transform or contour detection. However, some images (e.g., figures 1-c and 1-d) are not morphologically symmetrical, and the simultaneous use of a single technique on both left and right breast might not yield a

Fig. 1

 1 Fig.1 Sample of dataset images

  function that determines the penalty for incorrect prediction of Ẑ instead of Z. In the next stage, backpropagation starts based on the estimated error rate. At this point, the gradient of each parameter is calculated based on the chain rule, and all parameters change based on their impact on the error made in the network. next step (i.e., feed-forward) starts following updating the parameters, and the network training stage terminates after repeating an appropriate number of these stages. Nevertheless, one of the most important challenges of deep learning and CNNs is their training. Training these networks is very difficult due to the high volume of the layers and their weights and the need for powerful high-speed processors. Therefore, using deep learning networks does not require the training of the entire network, and pretrained models and algorithms can be applied in this regard. Accordingly, a pre-trained network can be considered instead of designing a deep convolutional neural network from the base, which can be used as an input feature to learn a classification task. During the process, a number of layers that express the generalities of images are frozen based on the existing database so that their weight does not change during training. If there is a low amount of data in the database, only the last layer of the fullyconnected block is trained. Otherwise, a higher number of layers are trained. The latest advancements in the use of CNNs in the field of computer vision have led to the emergence of famous CNN models, including VGG, GoogleNet and ResNet, which can be uploaded. Table

Figure 3 .Figure 4 .

 34 Figure 3. Diagram of accuracy measured at each iteration

Figure 5 .

 5 Figure 5. Diagram of accuracy and error measured at each iteration for proposed method

Table 1 . A summary of the analysis of bilateral asymmetry in mammograms based on screening diagnosis methods of thermography Author Method Type of Classifier Accuracy Sensitivity Specificity # Database images

 1 in the area of breast cancer thermography, it is suggested that breast separation and feature extraction be avoided and diagnosis be made based on the raw thermal image. In this regard, deep learning is a novel technique that can significantly contribute to this area. Various architectures have been proposed for image classification problems in deep networks. In this study, attempts were made to diagnose breast cancer based on thermal images by assessing several deep learning methods.

	Schaefer et al.	Stochastic	tissue	A	classifier	80%	-	-	146
	[16]2009		features		based on hybrid
						fuzzy law
	Ng	et	al.	Patient temperature	Radial	basis	80.95%	81.20%	88.20%	82
	[17]2007		and clinical data	function neural
						networks
	Mookiah et al.	Tissue properties	Decision tree	93.30%	86.70%	100%	50
	[18]2012		and	DWT
				coefficients
	Acharya et al.	Tissue features	SVM		88.10%	85.71%	100%	50
	[13]2012
	Francis et al.	Stochastic	tissue	SVM		90.91%	81.82%	100%	22
	[19]2014		features
	Araujo et al.	Symbolic features	Distance-based	-	85.70%	86.50%	50
	[20]2014				classifier
				Temperature time
				interval
	Zadeh et al.	The first moments	Self-organizing	70%	50%	75%	200
	[21]2012				neural networks
	Sathish et al.	Stochastic	tissue	SVM		90%	87.5%	92.5%	80
	[22]2017		features
	Borchartt et al.	The first moments	SVM		85.71%	95.83%	25%	28
	[12]2011		stochastic

Table 2 . Evaluation of extracted features using the classifications

 2

	Model	TP	TN	FP	FN	ACC	SPE	Sen